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Uniform blow-up rate for a porous medium equation with a weighted localized source

Weili Zeng1, Xiaobo Lu2* and Qilin Liu3

Author affiliations

1 School of Transportation, Southeast University, Nanjing 210096, China

2 School of Automation, Southeast University, Nanjing 210096, China

3 Department of Mathematics, Southeast University, Nanjing 210096, China

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Citation and License

Boundary Value Problems 2011, 2011:57  doi:10.1186/1687-2770-2011-57

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/57


Received:21 June 2011
Accepted:28 December 2011
Published:28 December 2011

© 2011 Zeng et al; 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

In this article, we investigate the Dirichlet problem for a porous medium equation with a more complicated source term. In some cases, we prove that the solutions have global blow-up and the rate of blow-up is uniform in all compact subsets of the domain. Moreover, in each case, the blow-up rate of |u(t)|is precisely determined.

Keywords:
porous medium equation; localized source; blow-up, uniform blow-up rate

1 Introduction

Let Ω be a bounded domain in ℝN (N ≥ 1) with smooth boundary ∂Ω. We consider the following parabolic equation with a localized reaction term

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M1">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M2">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M3">View MathML</a>

(1.3)

where m ≥ 1, q1 ≥ 0, s1 > 0 and x0 ∈ Ω is a fixed point. Throughout this article, we assume the functions a(x) and v0(x) satisfy the following conditions:

(A1) a(x) and v0(x) ∈ C2(Ω); a(x), v0(x) > 0 in Ω and a(x) = v0(x) = 0 on ∂Ω.

When Ω = B = {x ∈ ℝN; |x| < R}, we sometimes assume

(A2) a(x) and v0(x) are radially symmetric; a(r) and v0(r) are non-increasing for r ∈ [0, R].

Problems (1.1)-(1.3) arise in the study of the flow of a fluid through a porous medium with an internal localized source and in the study of population dynamics (see [1-3]). Porous medium equations (m > 1) with or without local sources have been studied by many authors [4-6].

Concerning (1.1)-(1.3), to the best of authors knowledge, a number of articles have studied it from the point of the view of blow-up and global existence [7-10]. Many studies have been devoted to the case m = 1 [10-13]. The case m = 1, a(x) = 1, q1 = 0, s1 ≥ 1 and m = 1, a(x) = 1, q1, s1 > 1 were studied by Souple [10,11]. Souple [10] demonstrated that the positive solution blows up in finite time if the initial value v0 is large enough. In the case a(x) = 1, q1 = 0, and s1 > 1, Souple [11] showed that the solution v(x, τ) blows up globally and the blow-up rate is precisely determined. The case q1 = 0 and s1 > 0 was studied by Cannon and Yin [12] and Chandam et al. [13]. Cannon and Yin [12] studied its local solvability and Chandam et al. [13] investigated its blow-up properties.

The study of this article is motivated by some recent results of related problems (see [14][15][16]. In the case of a(x)(= constant), the global existence and blow-up behavior have been considered by Chen and Xie [15]. It turns out that if q1 + s1 < m or q1 + s1 = m and a(x)(= constant) is sufficiently small, there exists a global solution of problem (1.1)-(1.3); if q1 + s1 > m, the solution of problem (1.1)-(1.3) blows up for large initial datum while it admits a global solution for small initial datum. Furthermore, Du and Xiang [16] obtained the blow-up rate estimates under some appropriate hypotheses on initial datum. For some related localized models arising in physical phenomena, we refer the readers to [17-19] and the references therein.

For the localized semi-linear parabolic equation of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M4">View MathML</a>

(1.4)

with the Dirichlet boundary condition (1.2) and the initial condition (1.3). In [20], Li and Wang proved that the blow-up set to system (1.2)-(1.4): (a) the system possesses total blow-up when q1 ≤ 1; (b) the system presents single point blow-up patterns when q1 > 1.

We now restrict ourselves to the problem of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M5">View MathML</a>

(1.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M6">View MathML</a>

(1.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M7">View MathML</a>

(1.7)

where q1 ≥ 0, s1 > 0, and q1 + s1 > m > 0. When m = 1, it was proved in [14] that

(1) If 0 ≤ q1 ≤ 1 and q1 + s1 > 1, then the solution of (1.5)-(1.7) blows up in a finite time T.

(2) If q1 > 1, then x = 0 is the only blow-up point for (1.5)-(1.7).

In the meantime, they obtained the blow-up rate estimate but less precise. Namely,

(i) If 0 ≤ q1 < 1, then for any x B

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M8">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M10">View MathML</a>.

(ii) If q1 = 1, then for any x B

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M11">View MathML</a>

It seems that the results of [14] can be extended to m ≥ 1 and the blow-up rate can be precisely determined. Motivated by this, in this article, we will extend and improve the results of [14].

The purpose of this article is to determine the blow-up rate of solutions for a nonlinear parabolic equation with a weighted localized source, that is, we investigate how the localized source and the local term affect the blow-up properties of the problem (1.5)-(1.7). Indeed, we find that when q1 ≤ 1, the solution of (1.5)-(1.7) blows up at the whole domain with a uniformly blow-up profile.

The rest of this article is organized as follows. The results are stated in Section 2. We then prove these results in Section 3.

2 Preliminaries and Main Results

The following two theorems are our main results.

Theorem 2.1 Assume q1 + s1 > m, (A1) and (A2) hold. Let v(x, t) be the solution of problem (1.5)-(1.7), then v(x,t) blows up provided that the initial value v0(x) is sufficiently large.

The method used in the proof Theorem 2.1 is originally due to [8,18], and bears much resemblance to that of Theorem 3.2 in [15] and Theorem 1.3 in [16]. Therefore, we omitted them here.

For the case q1 > 1, we do not know how to deal with the uniform blow-up rate of problem (1.5)-(1.7). In the following, we focus only on the case of 0 ≤ q1 ≤ 1.

Theorem 2.2 Assume (A1) and (A2) hold. Let v(x, t) be the blow-up solution of (1.5)-(1.7), which blows up in finite time T and v(x, t) is non-decreasing in time, then the following limits hold uniformly in all compact subsets of B.

(i) If 0 ≤ q1 < 1, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M12">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M13">View MathML</a>.

(ii) If q1 = 1, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M14">View MathML</a>

(2.2)

Remark 2.1 The domain we considered here is a ball, it seems that the results of Theorem 2.2 remain valid for the general domain. (It is an open problem in this case.)

To get the blow-up profiles for problem (1.5)-(1.7), we need some transformations. Let u(x, t) = vm(x, τ), t = , then (1.5)-(1.7) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M15">View MathML</a>

(2.3)

where 0 ≤ p = (m - 1)/m < 1, q = q1/m, and s = s1/m.

Under above transformation, assumptions (A1) and (A2) become

(B1) a(x) and u0(x) ∈ C2(B); a(x), u0(x) > 0 in B and a(x) = u0(x) = 0 on ∂B.

(B2) a(x) and u0(x) are radially symmetric; a(r) and u0(r) are non-increasing for r ∈ [0, R].

In our consideration, a crucial role is played by the Dirichlet eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M16">View MathML</a>

(2.4)

Denote λ be the first eigenvalue and by φ the corresponding eigenfunction with φ(x) > 0 in B, normalized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M17">View MathML</a>.

3 Proof of Theorem 2.2

For convenience, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M18">View MathML</a>

Before proving our result, we would like to give a property of the following problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M19">View MathML</a>

(3.1)

where 0 ≤ α ≤ 1 and w = u1-q(x, t).

Lemma 3.1 Assume (B1) and (B2) hold. Let w(x, t) be the solution of Equation (3.1), which blows up in a finite time T* and non-decreasing in time t, then the following limits hold uniformly in all compact subsets of B.

(i) If 0 ≤ α < 1, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M20">View MathML</a>

(ii) If α = 1, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M21">View MathML</a>

Proof. (i) Assumption (B2) implies wr ≤ 0 (r = |x|), it then follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M22">View MathML</a> and Δw(0, t) ≤ 0 for t > 0. From (3.1), we then get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M23">View MathML</a>

.

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M24">View MathML</a>

(3.2)

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M25">View MathML</a>

Moreover, it is apparent that limt-T* w(0, t)/g(t) = 0, since s > 1 - q.

Set R1 ∈ (0, R), B1 = {x ∈ ℝN, | x |< R1} and b(x) = 1/a(x), x B1. Since a'(r) ≤ 0, we obtain that b'(r) ≥ 0, for 0 ≤ r R1.

We now introduce the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M26">View MathML</a>

By a simple calculation, and note that ∇w(x, t)∇b(x) = ur(r, t)b'(r) ≤ 0, then there exist m1, m2 > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M27">View MathML</a>

Setting ε(t) = m2w(0, t)/g(t). From limtT* w(0, t)/g(t) = 0, we infer that there exists t1 ∈ (0, T*) such that 0 < ε(t) ≤ 1/2 for t1 t < T*.

Hence, in view of (3.1), we observe

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M28">View MathML</a>

Set g1(t) = (1 - ε(t))g(t), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M29">View MathML</a>, we then obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M30">View MathML</a>

Obviously, w1(x, t) is a sup-solution of the following equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M31">View MathML</a>

By the maximum principle, w1(x, t) ≥ w*(x, t) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M32">View MathML</a>. Similar to the proof of (4.15) in [15] that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M33">View MathML</a>

uniformly in all compact subsets of B1,

Therefore, by the arbitrariness of B1, we obtain that the following inequatlity holds uniformly in all compact subsets of B

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M34">View MathML</a>

(3.3)

In particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M35">View MathML</a>

(3.4)

From (3.2) and (3.4), we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M36">View MathML</a>

(3.5)

Multiplying both sides of (3.1) by φ and integrating over B × (0, t), 0 < t < T*

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M37">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M38">View MathML</a>, so we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M39">View MathML</a>

It then follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M40">View MathML</a>

(3.6)

Note that wr ≤ 0, (3.3) and (3.6), it is sufficient to prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M41">View MathML</a>

(3.7)

Assume on the contrary that there exists a point x1 B, x1 ≠ 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M42">View MathML</a>

Then there exists a sequence {tn} such that tn T*

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M43">View MathML</a>

By the continuity of a(x), we deduce that there exists x2 B such that (1 - α)a(x) < c for B1 = {x ∈ ℝn : |x2| ≤ |x| ≤ |x1|}. Using wr ≤ 0, (3.3) and (3.6), it is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M44">View MathML</a>

which is a contradiction to (3.6). Combining (3.3) and (3.7), Lemma 3.1 (i) is proved. Case (ii) can be treated similarly.

The key step in establishing the result of Theorem 2.2 is the following lemma.

Lemma 3.2 Under the assumption of Lemma 3.1, let u(x, t) be the blow-up solution of (2.3), which blows up in a finite time T* and non-decreasing in time t, then the following statements hold uniformly in all compact subsets of B:

(i) If p + q < 1, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M45">View MathML</a>

(ii) If p + q = 1, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M46">View MathML</a>

Proof. (i) Since ur ≤ 0 and ut ≥ 0, it then follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M47">View MathML</a> and Δu(0, t) ≤ 0 for t > 0, which imply limtT* u(0, t) = ∞. Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M48">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M49">View MathML</a>

(3.8)

Notice that p + q < 1 and (3.8), hence limtT* G(t) = ∞ and limtT* g(t) = ∞.

A simple calculation yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M50">View MathML</a>

In view of (2.3), we have, for x ∈ Ω, 0 < t < T*

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M51">View MathML</a>

(3.9)

Multiplying both sides of Equation (3.9) by φ and integrating over B × (0, t), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M52">View MathML</a>

(3.10)

for 0 < t < T*. Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M53">View MathML</a>

(3.11)

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M54">View MathML</a>

(3.12)

Setting u1(r, t) = u(1-q)/2(r, t)(r = |x|). We may claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M55">View MathML</a>

Indeed, due to limtT* g(t) = limtT* us(0, t) = ∞, ur ≤ 0, and s > 1 - q, we then have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M56">View MathML</a>

Therefore, by Lebesgue's dominated convergence theorem, we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M57">View MathML</a>

(3.13)

where wn is the surface area of unit ball in ℝN.

Now according to (3.10)-(3.12), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M58">View MathML</a>

(3.14)

On the other hand, By (3.9), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M59">View MathML</a>

where γ = p/(1 - q). Consequently, u1-q is a sup-solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M60">View MathML</a>

By the maximum principle, u1-q v in B × (0, T*). Note that 0 ≤ γ < 1, we know from Lemma 3.1 (i) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M61">View MathML</a>

uniformly in all compact subsets of B.

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M62">View MathML</a>

(3.15)

uniformly in all compact subsets of B.

Next, we prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M63">View MathML</a>

(3.16)

uniformly in all compact subsets of B.

We can verify (3.15) by similar means of (3.7). Therefore, we conclude the proof of case (i).

(ii) Proceeding as (3.8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M64">View MathML</a>

For any compact subset B1 B, there exists t1 ∈ (0, T*) such that u(x, t1) ≥ 1 for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M65">View MathML</a>, and thus ln u(x, t) ≥ 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M66">View MathML</a>.

Direct calculation shows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M67">View MathML</a>

(3.17)

Let λ1 be the first eigenvalue of -Δ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M68">View MathML</a> and by φ1 > 0 the corresponding eigenfunction, normalized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M69">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M70">View MathML</a>. Clearly, limtT* G(t)/G1(t) = 1.

Multiplying both sides of Equation (3.16) by φ1 and integrating over B1 × (t1, t), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M71">View MathML</a>

(3.18)

The result of case (ii) follows by analogy with the argument used in the proof of case (i).

Proof of Theorem 2.2

(i) By Lemma 3.2 (i), we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M72">View MathML</a>

hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M73">View MathML</a>

(3.19)

Integrating equivalence (3.18) between t and T*, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M74">View MathML</a>

(3.20)

Using Lemma 3.2 (i) and substituting p = (m - 1)/m, q = q1/m, s = s1/m, t* = , and u(x, t) = vm(x, τ) into (3.19), we complete the proof of Theorem 2.2 (i).

(ii) To obtain the blow-up rate of the exponent type, we need to be more careful in this case, since exponentiation of equivalents is not permitted. Similar to the proof of Theorem 3 in [14] and Lemma 2.3 in [16], we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M75">View MathML</a>

(3.21)

Thanks to Lemma 3.2(ii) and (3.20), we then get the desired result.

4 Discussion

This article deals with the porous medium equation with local and localized source terms, represented by two factors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/57/mathml/M77">View MathML</a>, respectively. As we all know that, in the absence of weight function, the solutions of model (1.5)-(1.7) have a global blow-up and the rate of blow-up is uniform in all compact subsets of the domain. A natural question is what happens in the model (1.5)-(1.7), where the source term is the product of localized source, local source, and weight function. It is shown by Theorem 2.2 that if 0 ≤ q1 ≤ 1, this equation possesses uniform blow-up profiles. In other words, the localized term plays a leading role in the blow-up profile for this case. Moreover, the blow-up rate estimates in time and space is obtained.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All the authors typed, read, and approved the final manuscript.

Acknowledgements

The authors thank the anonymous referee for their constructive and valuable comments, which helped in improving the presentation of this study. This study was supported by the National Natural Science Foundation of China (60972001), the National Key Technologies R & D Program of China (2009BAG13A06), the Scientific Innovation Research of College Graduate in Jiangsu Province (CXZZ_0163), and the Scientific Research Foundation of Graduate School of Southeast University (YBJJ1140).

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