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Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term

Faramarz Tahamtani* and Mohammad Shahrouzi

Author Affiliations

Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71454, Iran

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Boundary Value Problems 2012, 2012:50  doi:10.1186/1687-2770-2012-50


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/50


Received:3 December 2011
Accepted:26 April 2012
Published:26 April 2012

© 2012 Tahamtani and Shahrouzi; 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

We consider the semilinear Petrovsky equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M1">View MathML</a>

in a bounded domain and prove the existence of weak solutions. Furthermore, we show that there are solutions under some conditions on initial data which blow up in finite time with non-positive initial energy as well as positive initial energy. Estimates of the lifespan of solutions are also given.

Mathematics Subject Classification (2000): 35L35; 35L75; 37B25.

Keywords:
viscoelasticity; existence; blow-up; life-span; negative initial energy; positive initial energy

1 Introduction

In this article, we concerned with the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M2">View MathML</a>

(1.1)

where Ω Rn is a bounded domain with smooth boundary ∂Ω in order that the divergence theorem can be applied. ν is the unit normal vector pointing toward the exterior of Ω and p > 0. Here, g represents the kernel of the memory term satisfying some conditions to be specified later.

In the absence of the viscoelastic term, i.e., (g = 0), we motivate our article by presenting some results related to initial-boundary value Petrovsky problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M3">View MathML</a>

(1.2)

Research of global existence, blow-up and energy decay of solutions for the initial boundary value problem (1.2) has attracted a lot of articles (see [1-4] and references there in).

Amroun and Benaissa [1] investigated (1.2) with f(u, ut) = b|u|p-2u-h(ut) and proved the global existence of solutions by means of the stable set method in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M4">View MathML</a> combined with the Faedo-Galerkin procedure. In [3], Messaoudi studied problem (1.2) with f(u, ut) = b|u|p-2u-a|ut|m-2ut. He proved the existence of a local weak solution and showed that this solution blows up in finite time with negative initial energy if p > m.

In the presence of the viscoelastic terms, Rivera et al. [5] considered the plate model:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M5">View MathML</a>

in a bounded domain Ω RN and showed that the energy of solution decay exponentially provided the kernel function also decay exponentially. For more related results about the existence, finite time blow-up and asymptotic properties, we refer the reader to [5-16].

In the present article, we devote our study to problem (1.1). We will prove the existence of weak solutions under some appropriate assumptions on the function g and blow-up behavior of solutions. In order to obtain the existence of solutions, we use the Faedo-Galerkin method and to get the blow-up properties of solutions with non-positive and positive initial energy, we modify the method in [17]. Estimates for the blow-up time T* are also given.

2 Preliminaries

We define the energy function related with problem (1.1) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M6">View MathML</a>

(2.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M7">View MathML</a>

We denote by ∥.∥k, the Lk-norm over Ω. In particular, the L2-norm is denoted ∥.∥2. We use the familiar function spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M8">View MathML</a> and throughout this article we assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M10">View MathML</a>.

In the sequel, we state some hypotheses and three well-known lemmas that will be needed later.

(A1) p satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M11">View MathML</a>

(A2) g is a positive bounded C1 function satisfying g(0) > 0, and for all t > 0

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M12">View MathML</a>

also there exists positive constants L0, L1 such that

(A3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M13">View MathML</a>

Lemma 1 (Sobolev-Poincare's inequality). Let p be a number that satisfies (A1), then there is a constant C* = C(Ω, p) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M14">View MathML</a>

(2.2)

Lemma 2 [4]. Let δ > 0 and B(t) ∈ C2(0, ∞) be a nonnegative function satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M15">View MathML</a>

(2.3)

If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M16">View MathML</a>

(2.4)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M17">View MathML</a>, then B'(t) > K0 for t > 0, where K0 is a constant.

Lemma 3 [4]. If Y(t) is a non-increasing function on [t0, ∞) and satisfies the differential inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M18">View MathML</a>

(2.5)

where a > 0, δ > 0 and b R, then there exists a finite time T* such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M19">View MathML</a>

Upper bounds for T* is estimated as follows:

(i) If b < 0, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M20">View MathML</a>

(ii) If b = 0, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M21">View MathML</a>

(iii) If b > 0, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M22">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M23">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M24">View MathML</a>.

3 Existence of solutions

In this section, we are going to obtain the existence of weak solutions to the problem (1.1) using Faedo-Galerkin's approximation.

Theorem 1 Let the assumptions (A1)-(A3) hold. Then there exists at least a solution u of (1.1) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M25">View MathML</a>

(3.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M26">View MathML</a>

as t → 0.

Proof We choose a basis {ωk} (k = 1, 2, ...) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M27">View MathML</a> which is orthonormal in L2(Ω) and ωk being the eigenfunctions of biharmonic operator subject to the homogeneous Dirichlet boundary condition.

Let Vm be the subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M27">View MathML</a> generated by the first m vectors. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M28">View MathML</a>

(3.2)

where um(t) is the solution of the following Cauchy problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M29">View MathML</a>

(3.3)

with the initial conditions (when m → ∞)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M30">View MathML</a>

(3.4)

The approximate systems (3.3) and (3.4) are the normal one of differential equations which has a solution in [0, Tm) for some Tm > 0. The solution can be extended to the [0, T] for any given T > 0 by the first estimate below.

First estimation. Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M31">View MathML</a> instead of ωk in (3.3), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M32">View MathML</a>

(3.5)

Simple calculation similar to [11] yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M33">View MathML</a>

(3.6)

Combining (3.5) and (3.6), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M34">View MathML</a>

(3.7)

integrating (3.7) over (0, t) and using assumption (A3) we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M35">View MathML</a>

(3.8)

where C1 is a positive constant depending only on ∥u0∥, ∥u1∥, p, and l. It follows from (3.8) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M36">View MathML</a>

(3.9)

Second estimation. Differentiating (3.3) with respect to t, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M37">View MathML</a>

(3.10)

If we substitute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M38">View MathML</a> instead of ωk in (3.10), it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M39">View MathML</a>

(3.11)

Since H2(Ω) ↪ L2p+2(Ω), using Lemma 2, Hölder and Young's inequalities and (3.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M40">View MathML</a>

(3.12)

Combining the relations (3.11), (3.12) and integrating over (0, t) for all t ∈ [0, T] with arbitrary fixed T, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M41">View MathML</a>

(3.13)

From (3.4) and (3.8), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M42">View MathML</a>

(3.14)

where L2 is a positive constant independent of m. In the following, we find the upper bound for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M43">View MathML</a>. Again we substitute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M38">View MathML</a> instead of ωk in (3.3), and choosing t = 0, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M44">View MathML</a>

which combined with the Green's formula imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M45">View MathML</a>

(3.15)

By using (A1), (3.4) and Young's inequality, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M46">View MathML</a>

(3.16)

where L3 > 0 is a constant independent of m.

Owing to (3.8), (3.5) and Young's inequality with (A3), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M47">View MathML</a>

(3.17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M48">View MathML</a>

(3.18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M49">View MathML</a>

(3.19)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M50">View MathML</a>

(3.20)

Now we choose γ > 0 small enough and combining (A3), (3.8), (3.13), (3.14), and (3.16)-(3.20), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M51">View MathML</a>

(3.21)

By using Gronwall's lemma we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M52">View MathML</a>

(3.22)

for all t ∈ [0, T], and L10 is a positive constant independent of m. Estimate (3.22) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M53">View MathML</a>

(3.23)

By attention to (3.9) and (3.23), there exists a subsequence {ui} of {um} and a function u such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M54">View MathML</a>

(3.24)

By Aubin-Lions compactness lemma [18], it follows from (3.24) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M55">View MathML</a>

(3.25)

In the sequel we will deal with the nonlinear term. By (3.9) and Sobolev embedding theorem, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M56">View MathML</a>

(3.26)

and therefore we can extract a subsequence {ui} of {um} such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M57">View MathML</a>

(3.27)

Applying (3.24), (3.27) and letting i → ∞ in (3.3), we see that u satisfies the equation. For the initial conditions by using (3.4), (3.25) and the simple inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M58">View MathML</a>

we get the first initial condition immediately. In the similar way, we can show the second initial condition and the proof is complete.

4 Blow-up of solutions

In this section, we study blow-up property of solutions with non-positive initial energy as well as positive initial energy, and estimate the lifespan of solutions. For this purpose, we assume that g is positive and C1 function satisfying

(A4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M59">View MathML</a>

and we make the following extra assumption on g

(A5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M60">View MathML</a>

From (2.1), (A4) and Lemma 1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M61">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M62">View MathML</a>. It is easy to verify that G(λ) has a maximum at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M63">View MathML</a> and the maximum value is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M64">View MathML</a>.

Lemma 4 Let (A4) hold andu be a local solution of (1.1). Then E(t) is a non-increasing function on [0, T] and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M65">View MathML</a>

(4.2)

for almost every t ∈ [0, T].

Proof Multiplying (1.1) by ut, integrating over Ω, and finally integrating by parts, we obtain (4.2) for any regular solution. Then by density arguments, we have the result.

Lemma 5 Let (A4) hold and u be a local solution of (1.1) with initial data satisfying E(0) < E1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M66">View MathML</a>. Then there exists λ2 > λ1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M67">View MathML</a>

(4.3)

Proof See Li and Tsai [11].

The choice of the functional is standard (see [19])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M68">View MathML</a>

(4.4)

It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M69">View MathML</a>

(4.5)

and from (1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M70">View MathML</a>

(4.6)

Lemma 6 Let u be a solution of (1.1) and (A4), (A5) hold, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M71">View MathML</a>

(4.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M72">View MathML</a>.

Proof Using the Hölder and Young's inequalities, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M73">View MathML</a>

therefore (4.6) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M74">View MathML</a>

Then, using (4.2), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M75">View MathML</a>

and so by (2.5) and (A5), we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M76">View MathML</a>

(4.8)

if we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M77">View MathML</a> then inequality (4.8) yields the desired result.

Consequently, we have the following result.

Lemma 7 Assume that (A4) and (A5) hold. u be a local solution of (1.1) and that either one of the following four conditions is satisfied:

(i) E(0) < 0

(ii) E(0) = 0 and ψ'(0) > 0

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M78">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M79">View MathML</a>

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M80">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M81">View MathML</a>.

Then ψ' (t) > 0 for t > t*, where

in case (i)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M82">View MathML</a>

(4.9)

in cases (ii), (iv)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M83">View MathML</a>

(4.10)

and in case (iii)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M84">View MathML</a>

(4.11)

Proof Suppose that condition (i) is satisfied. Then from (4.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M85">View MathML</a>

Thus ψ'(t) > 0 for t > t*, and it is easy to see that t* satisfies (4.9).

If E(0) = 0, then by using (4.3) we have ψ" (t) ≥ 0, and since ψ'(0) > 0 we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M86">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M79">View MathML</a> then by Lemma 4, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M87">View MathML</a>

Thus from (4.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M88">View MathML</a>

(4.12)

and integrating (4.12) from 0 to t gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M89">View MathML</a>

where t* satisfies (4.11).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M80">View MathML</a>, this assumption causes that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M90">View MathML</a>

and by using Hölder and Young's inequalities, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M91">View MathML</a>

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M92">View MathML</a>

(4.13)

We see that the hypotheses of Lemma 2 are fulfilled with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M93">View MathML</a>

and the conclusion of Lemma 2.2 gives us

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M94">View MathML</a>

Therefore the proof is complete.

To estimate the life-span of ψ(t), we define the following functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M95">View MathML</a>

(4.14)

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M96">View MathML</a>

(4.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M97">View MathML</a>

(4.16)

Using (4.4)-(4.6) and exploiting Holder's inequality on ψ'(t), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M98">View MathML</a>

Utilizing the last inequality into (4.16) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M99">View MathML</a>

(4.17)

Now we should assume different values for initial energy E(0).

(1) At first if E(0) ≤ 0 then from (4.17) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M100">View MathML</a>

(4.18)

on the other hand by Lemma 7, Y'(t) < 0 for t > t*. Multiplying (4.18) by Y'(t) and integrating from t* to t, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M101">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M102">View MathML</a>

(4.19)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M103">View MathML</a>

(4.20)

Then the hypotheses of Lemma 3 are fulfilled with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M104">View MathML</a> and using the conclusion of Lemma 3, there exists a finite time T* such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M119">View MathML</a>, i.e., in this case some solutions blow up in finite time T*.

(2) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M78">View MathML</a>, then from (4.17) and (4.12) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M105">View MathML</a>

Then using the same arguments as in (1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M106">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M107">View MathML</a>

(4.21)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M108">View MathML</a>

(4.22)

Thus by Lemma 3, there exists a finite time T* such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M109">View MathML</a>

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M80">View MathML</a>. In this case, it is easy to see that by using (4.19) and (4.20) into discussion in part (1), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M110">View MathML</a>

Hence, Lemma 3 yields the blow-up property in this case.

Therefore, we proved the following theorem.

Theorem 2 Assume that (A4) and (A5) hold. u be a local solution of (1.1) and that either one of the following four conditions is satisfied:

(i) E(0) < 0

(ii) E(0) = 0 and ψ'(0) > 0

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M78">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M79">View MathML</a>

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M80">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M81">View MathML</a>holds.

Then the solution u blows up at finite time T*. Moreover, the upper bounds for T* can be estimated according to the sign of E(0):

in case (i)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M111">View MathML</a>

Furthermore, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M112">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M113">View MathML</a>

in cases (ii)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M114">View MathML</a>

in case (iii)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M115">View MathML</a>

Furthermore, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M112">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M116">View MathML</a>

and in case (iv)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M117">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/50/mathml/M118">View MathML</a>. Here α, β, α1, and β1 are given in (4.19)-(4.22), respectively. Note that each t* in the above cases satisfy the same case in Lemma 7.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees for the careful reading of this article and for the valuable suggestions to improve the presentation and style of the article. This study was supported by Shiraz University.

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