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Global behavior for a diffusive predator-prey system with Holling type II functional response

Yanzhong Zhao

Author Affiliations

Department of Basic Research, Qinghai University, Xining, 810016, China

Boundary Value Problems 2012, 2012:111  doi:10.1186/1687-2770-2012-111


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/111


Received:6 April 2012
Accepted:25 September 2012
Published:9 October 2012

© 2012 Zhao; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A strongly coupled self- and cross-diffusion predator-prey system with Holling type II functional response is considered. Using the energy estimate, Sobolev embedding theorem and bootstrap arguments, the global existence of non-negative classical solutions to this system in which the space dimension is not more than five is obtained.

MSC: 35K50, 35K55, 35K57, 92D40.

Keywords:
predator-prey system; cross-diffusion; global solution

1 Introduction

In this paper, we consider the global existence of non-negative classical solutions to the following diffusion predator-prey system with Holling type II functional response:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M1">View MathML</a>

(1.1)

where Ω is a bounded region in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M3">View MathML</a>) with a smooth boundary Ω; η is the outward normal on Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M4">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M6">View MathML</a> are non-negative smooth functions and are not identically zero; u and v denote the population densities of predator and prey, respectively; α, β, r, a, K, m and c are positive constants, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M7">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M9">View MathML</a> are the diffusion rates of the two species; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M10">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M11">View MathML</a>) are given non-negative constants, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M13">View MathML</a> are self-diffusion rates; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M14">View MathML</a> is the cross-diffusion rate. It means that the diffusion is from one species of high-density areas to the other species of low-density areas. See [1,2] for more details on the ecological backgrounds of this system.

Obviously, the non-negative equilibrium solutions of system (1.1) are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M16">View MathML</a>. For the reaction-diffusion problem of system (1.1), i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M17">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M18">View MathML</a>), the global attraction, persistence and stability of non-negative equilibrium solutions are studied in [3]. The main result can be summarized as follows:

(1) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M19">View MathML</a>, a semi-trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M15">View MathML</a> is globally asymptotically stable;

(2) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M22">View MathML</a>, a unique positive constant solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M23">View MathML</a> is globally asymptotically stable;

(3) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M22">View MathML</a>, a positive constant solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M23">View MathML</a> is locally asymptotically stable.

In view of the study of dynamic behavior of a predator-prey reaction-diffusion system with Holling type II functional response, a natural problem is what the global behavior for a predator-prey cross-diffusion system (1.1) is. To the best of our knowledge, the existing results are very few. In this paper, we consider the space dimension to be less than six, and initial function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M6">View MathML</a> under some smooth conditions, using the energy estimate, Sobolev embedding theorem and bootstrap arguments, we consider the global existence of non-negative classical solutions for system (1.1).

We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M29">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M30">View MathML</a> means that u, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M32">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M33">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M34">View MathML</a> are in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M35">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M36">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M37">View MathML</a>.

2 Auxiliary results

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M38">View MathML</a>be the solution of (1.1). There exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M39">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M40">View MathML</a>

(2.1)

Proof Firstly, the existence of local solutions for system (1.1) is given in [4-6]. Roughly speaking, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M42">View MathML</a>, there exists the maximum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M43">View MathML</a> such that system (1.1) admits a unique non-negative solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M44">View MathML</a>

(2.2)

If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M45">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M46">View MathML</a>.

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M47">View MathML</a>. By use of the maximum principle, the non-negative solution of system (1.1) can be derived from the maximum principle, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M48">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M49">View MathML</a>. This completes the proof of Lemma 2.1. □

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M51">View MathML</a>for the solution to the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M52">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M12">View MathML</a>are positive constants and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M55">View MathML</a>. Then there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M56">View MathML</a>, depending on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M57">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M58">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M59">View MathML</a>

(2.3)

Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M60">View MathML</a>

(2.4)

Proof From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M50">View MathML</a>, it is easy to find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M62">View MathML</a>

(2.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M64">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M65">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M66">View MathML</a> are bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M67">View MathML</a> from (2.1). Multiplying (2.5) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M68">View MathML</a> and integrating by parts over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M69">View MathML</a> yields

(2.6)

Using the Hölder inequality and Young inequality to estimate the right-hand side of (2.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M71">View MathML</a>

(2.7)

with some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M72">View MathML</a>. Substituting (2.7) into (2.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M73">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M74">View MathML</a> depends on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M57">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M58">View MathML</a>. So, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M77">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M78">View MathML</a>, it follows from the elliptic regularity estimate [[7], Lemma 2.3] that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M79">View MathML</a>

From (2.5), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M80">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M81">View MathML</a>. Moreover, (2.4) can be obtained by use of the Sobolev embedding theorem. □

Lemma 2.3Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M82">View MathML</a>is a bounded function satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M83">View MathML</a>

with the boundary condition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M84">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M85">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M86">View MathML</a>. ThenWis in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M87">View MathML</a>.

The proof of the above lemma can be found in [[8], Proposition 2.1].

The following result can be derived from Lemma 2.3 and Lemma 2.4 of [9].

Lemma 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M89">View MathML</a>. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M90">View MathML</a>

and there exist positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M91">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M92">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M93">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M94">View MathML</a>), there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M95">View MathML</a>, independent ofwbut possibly depending onn, Ω, p, βand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M92">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M97">View MathML</a>

Finally, one proposes some standard embedding results which are important to obtain the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M99">View MathML</a> normal estimates of the solution for (1.1).

Lemma 2.5There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M100">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M101">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M103">View MathML</a>. By Lemma 2.1, u is bounded. Therefore, X is also bounded. By Lemma 2.2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M104">View MathML</a>. Moreover, X satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M105">View MathML</a>

By Lemma 2.3 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M108">View MathML</a>, we obtain the desired result. □

Lemma 2.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M109">View MathML</a>be a fixed bounded domain and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M110">View MathML</a>. Then for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M111">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M112">View MathML</a>, one has

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M115">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M118">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M119">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M121">View MathML</a>,

whereCis a positive constant dependent onq, n, Ω, T.

3 The existence of classical solutions

The main result about the global existence of non-negative classical solutions for the cross-diffusion system (1.1) is given as follows.

Theorem 3.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M122">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M123">View MathML</a>satisfy homogeneous Neumann boundary conditions and belong to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M124">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M125">View MathML</a>. Then system (1.1) has a unique non-negative solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M126">View MathML</a>if the space dimension is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M127">View MathML</a>.

Proof When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M128">View MathML</a>, the proof is similar to the methods of [10-12]. So, we just give the proof of Theorem 3.1 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M129">View MathML</a>. The proof is divided into three parts.

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M130">View MathML</a>-, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M131">View MathML</a>-estimate and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M132">View MathML</a>-estimate for v.

Firstly, integrating the first equation of (1.1) over Ω, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M133">View MathML</a>

Thus, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M49">View MathML</a>, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M135">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M136">View MathML</a>.

Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M137">View MathML</a>

(3.1)

Secondly, linear combination of the second and first equations of (1.1) and integrating over Ω yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M138">View MathML</a>

So, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M139">View MathML</a>

(3.2)

Further,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M140">View MathML</a>

(3.3)

Then multiplying both sides of the second equation of system (1.1) by v and integrating over Ω, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M141">View MathML</a>

Integrating the above expression in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M142">View MathML</a> yields

(3.4)

Estimating the first term on the right-hand side of (3.4),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M144">View MathML</a>

(3.5)

Substituting (3.5) into (3.4) yields

(3.6)

Select <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M146">View MathML</a> and denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M147">View MathML</a>. Notice the positive equilibrium point of (1.1) exists under condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M22">View MathML</a>, then

(3.7)

By Lemma 2.5, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M101">View MathML</a>. Integrating the above inequality and using the Gronwall inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M151">View MathML</a>

Hence, there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M152">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M153">View MathML</a>. Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M154">View MathML</a>

(3.8)

Secondly, multiplying both sides of the second equation of system (1.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M155">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M156">View MathML</a>) and integrating over Ω, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M157">View MathML</a>

Integrating the above equation over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M158">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M159">View MathML</a>), it is clear that

(3.9)

By Lemma 2.2, it can be found that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M161">View MathML</a>. According to the Hölder inequality and Young inequality, we get

(3.10)

Choose an appropriate number ε satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M163">View MathML</a>. Substituting (3.10) into (3.9) and taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M164">View MathML</a>, we have

(3.11)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M166">View MathML</a>

From (3.11), we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M167">View MathML</a>

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M168">View MathML</a>, it is easy to find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M169">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M170">View MathML</a>. So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M171">View MathML</a>

(3.12)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M172">View MathML</a>. It follows from the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M173">View MathML</a>-estimate for v that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M174">View MathML</a>

By Lemma 2.4 and (3.12), we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M175">View MathML</a>

(3.13)

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M176">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M177">View MathML</a>. It is easy to know that E is bounded by use of reduction to absurdity. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M179">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M180">View MathML</a> is bounded, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M181">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M182">View MathML</a> still as q. So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M183">View MathML</a>

(3.14)

Finally, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M185">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M186">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M187">View MathML</a>, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M186">View MathML</a> in (3.9), it follows from (3.8) that there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M189">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M190">View MathML</a>

(3.15)

By embedding theorem, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M191">View MathML</a>

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M98">View MathML</a>-estimate for v.

The second equation of system (1.1) can be written as the following divergence form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M193">View MathML</a>

(3.16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M196">View MathML</a> is the Kronecker sign.

In order to apply the maximum principle [13] to (3.16), we need to prove the following conditions:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M197">View MathML</a> is bounded;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M198">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M199">View MathML</a>,

where ν, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M200">View MathML</a> are positive constants and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M201">View MathML</a>

(3.17)

Next, we will show that the above conditions (1) to (3) are satisfied for (3.16). When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M187">View MathML</a>, it is easy to find that the condition (1) is satisfied by use of (3.15). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M203">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M204">View MathML</a>, the condition (2) is verified. In view of the condition (3), we take appropriate q and r. Rewrite the first equation of system (1.1) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M205">View MathML</a>

(3.18)

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M207">View MathML</a>, it is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M208">View MathML</a> has an upper bound over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M209">View MathML</a> by Lemma 2.1. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M210">View MathML</a>

From (3.14), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M211">View MathML</a>. Therefore, all conditions of the Hölder continuity theorem [[5], Theorem 10.1] hold for (3.18). Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M212">View MathML</a>

(3.19)

We will discuss (2.5) which is the corresponding form of (3.18). It follows from (2.1) and (3.14) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M214">View MathML</a>. From (3.19), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M215">View MathML</a>. Thus, according to the parabolic regularity result of [[5], pp.341-342, Theorem 9.1], we can conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M216">View MathML</a>

(3.20)

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M217">View MathML</a> by Lemma 2.6.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M218">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M219">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M220">View MathML</a>. It means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M221">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M222">View MathML</a>. From (2.1) and (3.14), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M223">View MathML</a>.

Then the condition (3) and (3.17) are satisfied by choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M224">View MathML</a>. According to the maximum principle [[13], p.181, Theorem 7.1], we can conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M225">View MathML</a>. From (2.1), there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M226">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M227">View MathML</a>

(3.21)

Therefore, the global solution to the problem (1.1) exists.

(iii) The existence of classical solutions.

Under the conditions of Theorem 3.1, we consider above global solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M38">View MathML</a> to be classical. By (3.20) and Lemma 2.6, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M229">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M230">View MathML</a>. It follows from Lemma 3.3 in [13] that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M231">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M50">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M233">View MathML</a>.

So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M234">View MathML</a>

(3.22)

Rewrite the second equation of system (1.1) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M235">View MathML</a>

Therefore, we can conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M236">View MathML</a>, u, v, ∇u and ∇v are all bounded. By the Schauder estimate [13], there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M237">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M238">View MathML</a>

(3.23)

Furthermore, by the Schauder estimate, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M239">View MathML</a>

(3.24)

Next, the regularity of v will be discussed. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M240">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M241">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M242">View MathML</a>

(3.25)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M243">View MathML</a>. According to (3.22) to (3.24), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M245">View MathML</a>. Applying the Schauder estimate to (3.25), we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M246">View MathML</a>

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M247">View MathML</a>, we can see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M248">View MathML</a>

(3.26)

Combining (3.24) and (3.26), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M249">View MathML</a>

Therefore, the result of Theorem 3.1 can be obtained for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M250">View MathML</a>, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M251">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M252">View MathML</a>, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M253">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/111/mathml/M254">View MathML</a>. (3.24) and (3.26) are obtained by repeating the above bootstrap argument and the Schauder estimate. This completes the proof of Theorem 3.1. □

Competing interests

The author declares that he has no competing interests.

Authors’ contributions

The work was realized by the author.

Acknowledgements

The author thanks the anonymous referee for a careful review and constructive comments. This work was supported by the Qinghai Provincial Natural Science Foundation (2011-Z-915).

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