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

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

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

v τ - Δ v m = a ( x ) v q 1 ( x , τ ) v s 1 ( x 0 , τ ) , x Ω , τ > 0 , (1.1)

v ( x , τ ) = 0 , x Ω , τ > 0 , (1.2)

v ( x , 0 ) = v 0 ( x ) , x Ω , (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

v τ - Δ v = v q 1 ( x , τ ) v s 1 ( x 0 , τ ) , x Ω , τ > 0 , (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

v τ - Δ v m = a ( x ) v q 1 ( x , τ ) v s 1 ( 0 , τ ) , x B , τ > 0 , (1.5)

v ( x , τ ) = 0 , x B , τ > 0 , (1.6)

v ( x , 0 ) = v 0 ( x ) , x B , (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

C 1 ( a ( x ) ) 1 ( 1 - q 1 ) v ( x , τ ) ( T - τ ) 1 ( q 1 + s 1 - 1 ) C 2 , a s τ T ,

where C 1 = ( ( a ( 0 ) ) s 1 ( 1 - q 1 ) ( q 1 + s 1 - 1 ) ) 1 ( 1 - q 1 - s 1 ) , C 2 = ( a ( 0 ) ( p + q - 1 ) ) 1 ( 1 - q 1 - s 1 ) .

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

a ( x ) a ( 0 ) ln ( T - τ ) - 1 s 1 ln v ( x , τ ) ln ( T - τ ) - 1 s 1 , a s τ T .

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.

### 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

lim τ T ( T - τ ) 1 ( q 1 + s 1 - 1 ) v ( x , τ ) = C ( a ( x ) ) 1 ( 1 - q 1 ) , (2.1)

where C = ( ( q 1 + s 1 - 1 ) ( a ( 0 ) ) s 1 ( 1 - q 1 ) ) 1 ( 1 - q 1 - s 1 ) .

(ii) If q1 = 1, then

lim τ T ln v ( x , τ ) = a ( x ) a ( 0 ) ln ( T - τ ) - 1 s 1 . (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

u t = u p ( Δ u + a ( x ) u q ( x , t ) u s ( 0 , t ) ) , x B , t > 0 , u ( x , t ) = 0 , x B , t > 0 , u ( x , 0 ) = u 0 ( x ) = v 0 m ( x ) , x B , (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

- Δ φ = λ φ , in B , φ ( x ) = 0 , on B . (2.4)

Denote λ be the first eigenvalue and by φ the corresponding eigenfunction with φ(x) > 0 in B, normalized by B a ( x ) φ ( x ) d x = 1 .

### 3 Proof of Theorem 2.2

For convenience, we denote

g ( t ) = u s ( 0 , t ) a n d G ( t ) = 0 t g ( s ) d s .

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

w t = w α ( Δ w + a ( x ) g ( t ) ) , x B , t > 0 , w ( x , t ) = 0 , x B , t > 0 , w ( x , 0 ) = w 0 ( x ) = u 0 1 - q ( x ) , x B , (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

lim t T * w 1 - α ( x , t ) G ( t ) = ( 1 - α ) a ( x ) .

(ii) If α = 1, then

lim t T * ln w ( x , t ) G ( t ) = a ( x ) .

Proof. (i) Assumption (B2) implies wr ≤ 0 (r = |x|), it then follows that w ( 0 , t ) = max x B ̄ w ( x , t ) and Δw(0, t) ≤ 0 for t > 0. From (3.1), we then get

d w 1 - α ( 0 , t ) d t ( 1 - α ) a ( 0 ) g ( t ) , 0 < t < T * .
.

Consequently,

lim t T * sup w 1 - α ( 0 , t ) G ( t ) ( 1 - α ) a ( 0 ) , (3.2)

which implies

lim t T * G ( t ) = a n d lim t T * g ( t ) = .

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

w 1 ( x , t ) = b 1 ( 1 - α ) ( x ) w ( x , t ) , x B 1 , 0 < t < T * .

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

b ( x ) Δ w ( x , t ) m 1 Δ w 1 ( x , t ) - m 2 w ( x , t ) x B 1 , 0 < t < T * .

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

1 1 - α ( w 1 1 - α ) t = b ( x ) Δ w + g ( t ) m 1 Δ w 1 + ( 1 - ε ( t ) ) g ( t ) + ε ( t ) g ( t ) - m 2 w ( 0 , t ) = m 1 Δ w 1 + ( 1 - ε ( t ) ) g ( t ) , x B 1 , t 1 < t < T * .

Set g1(t) = (1 - ε(t))g(t), G 1 ( t ) = t 1 t g ( s ) d s , we then obtain

lim t T * G 1 ( t ) = and lim t T * G 1 ( t ) G ( t ) = 1 .

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

{ ( w * ) t = ( w * ) α ( ( m 1 w * + g 1 ( t ) ) , x B 1 , t 1 < t < T * , w * ( x , t ) = 0 , x B 1 , t > t 1 , w * ( x , t 1 ) = b 1 / 1 α ( x ) w ( x , t 1 ) , x B 1 .

By the maximum principle, w1(x, t) ≥ w*(x, t) and w r * 0 . Similar to the proof of (4.15) in [15] that

lim t T * ( w * ) 1 - α ( x , t ) G ( t ) = ( 1 - α ) ,

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

lim t T * inf w ( 1 - α ) ( x , t ) G ( t ) ( 1 - α ) a ( x ) . (3.3)

In particular,

lim t T * inf w ( 1 - α ) ( 0 , t ) G ( t ) ( 1 - α ) a ( 0 ) . (3.4)

From (3.2) and (3.4), we deduce

lim t T * w ( 1 - α ) ( 0 , t ) G ( t ) = ( 1 - α ) a ( 0 ) . (3.5)

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

1 1 - α B w 1 - α φ d x - B w 0 1 - α φ d x = - λ 0 t B w φ d x d s + G ( t ) .

Since 0 t B w φ d x d s B φ d x 0 t w ( 0 , s ) d s , so we have

lim t T * 0 t B w φ d x d s G ( t ) = 0 .

It then follows that

lim t T * B w 1 - α φ d x G ( t ) = ( 1 - α ) . (3.6)

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

lim t T * sup w 1 - α ( x , t ) G ( t ) ( 1 - α ) a ( x ) , x B . (3.7)

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

lim t T * sup w 1 - α ( x 1 , t ) G ( t ) = c > ( 1 - α ) a ( x 1 ) .

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

lim t n T * sup w 1 - α ( x 1 , t n ) G ( t n ) = c > ( 1 - α ) a ( x 1 ) .

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

lim t n T * B w 1 - α ( x , t n ) φ ( x ) d x G ( t n ) = lim t n T * B \ B 1 w 1 - α ( x , t n ) φ ( x ) d x + B 1 w 1 - α ( x , t n ) φ ( x ) d x G ( t n ) B \ B 1 ( 1 - α ) a ( x ) φ ( x ) d x + lim t n T * c B φ ( x ) d x > ( 1 - α ) ,

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

lim t T * u 1 - q - p ( x , t ) G ( t ) = ( 1 - q - p ) a ( x ) .

(ii) If p + q = 1, then

lim t T * ln u ( x , t ) G ( t ) = a ( x ) .

Proof. (i) Since ur ≤ 0 and ut ≥ 0, it then follows that u ( 0 , t ) = max x B ̄ u ( x , t ) and Δu(0, t) ≤ 0 for t > 0, which imply limtT* u(0, t) = ∞. Obviously,

u t ( 0 , t ) a ( 0 ) u p + q ( 0 , t ) g ( t ) , 0 < t < T * ,

which implies

lim t T * sup u 1 - p - q ( 0 , t ) G ( t ) ( 1 - p - q ) a ( 0 ) . (3.8)

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

A simple calculation yields

1 1 - r Δ u 1 - r = - r u - ( 1 + r ) | u | 2 + u - r Δ u ( if 0 < r < 1 ) .

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

1 - q 1 - p - q d u 1 - p - q d t = Δ u 1 - q + q ( 1 - q ) u - q - 1 | u | 2 + ( 1 - q ) a ( x ) g ( t ) . (3.9)

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

1 1 - p - q B u 1 - p - q φ d x - B u 0 1 - p - q φ d x = - λ 1 - q 0 t B u 1 - q φ d x d s + G ( t ) + 0 t B q u - q - 1 | u | 2 φ d x d s , (3.10)

for 0 < t < T*. Clearly,

0 t B u 1 - q φ d x d s 0 t u 1 - q ( 0 , t ) d s B φ ( x ) d x , (3.11)

which yields

lim t T * 0 t B u 1 - q φ d x d s G ( t ) = 0 . (3.12)

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

lim t T * ( u 1 ( r , t ) ) r ( g ( t ) ) 1 2 = 0 , a . e . r ( 0 , R ) .

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

lim t T * 0 R ( u 1 ( r , t ) ) r d r ( g ( t ) ) 1 2 = lim t T * u 1 ( R , t ) - u 1 ( 0 , t ) ( g ( t ) ) 1 2 = 0 .

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

lim t T * B q u - q - 1 | u | 2 φ ( x ) d x g ( t ) = q ω n lim t T * 0 R u - q - 1 ( r , t ) u r 2 φ ( r ) r n - 1 d r g ( t ) q ω n R n - 1 lim t T * 0 R u - q - 1 ( r , t ) u r 2 φ ( r ) d r g ( t ) = 4 q ( 1 - q ) 2 R n - 1 ω n lim t T * 0 R ( ( u ( 1 - q ) 2 ) r ) 2 φ ( r ) d r g ( t ) C 0 R lim t T * ( ( u ( 1 - q ) 2 ) r ( g ( t ) ) 1 2 ) 2 d r = 0 , (3.13)

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

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

lim t T * B u 1 - p - q φ d x G ( t ) = ( 1 - p - q ) . (3.14)

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

d u 1 - q d t u p ( Δ u 1 - q + ( 1 - q ) a ( x ) g ( t ) ) , x B , 0 < t < T * ,

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

d v d t = v γ ( Δ v + ( 1 - q ) a ( x ) g ( t ) ) , x B , 0 < t < T * , v ( x , t ) = 0 , x B , t > 0 , v ( x , 0 ) = u 0 1 - q ( x ) , x B .

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

lim t T * v ( 1 - p - q ) ( 1 - q ) ( x , t ) G ( t ) = ( 1 - p - q ) a ( x ) ,

uniformly in all compact subsets of B.

Thus,

lim t T * inf u ( 1 - p - q ) ( x , t ) G ( t ) ( 1 - p - q ) a ( x ) , (3.15)

uniformly in all compact subsets of B.

Next, we prove that

lim t T * sup u ( 1 - p - q ) ( x , t ) G ( t ) ( 1 - p - q ) a ( x ) , (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

lim t T * sup ln u ( 0 , t ) G ( t ) a ( 0 ) .

For any compact subset B1 B, there exists t1 ∈ (0, T*) such that u(x, t1) ≥ 1 for all x B ̄ 1 , and thus ln u(x, t) ≥ 0 in B ̄ 1 × ( t 1 , T * ) .

Direct calculation shows

d ln u d t = 1 1 - q Δ u 1 - q + q u - q - 1 | u | 2 + a ( x ) g ( t ) . (3.17)

Let λ1 be the first eigenvalue of -Δ in H 0 1 ( B 1 ) and by φ1 > 0 the corresponding eigenfunction, normalized by B 1 a ( x ) φ 1 ( x ) d x = 1 . Set G 1 ( t ) = t 1 t g ( s ) d s . Clearly, limtT* G(t)/G1(t) = 1.

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

B 1 ( ln u ) φ 1 d x - B 1 ( ln u ( x , t 1 ) ) φ 1 d x = - λ t 1 t B 1 u 1 - q φ d x d s + G ( t ) + t 1 t B 1 q u - q - 1 | u | 2 φ d x d s , t 1 < t < T * . (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

u ( 0 , t ) ~ ( ( 1 - q - p ) a ( 0 ) ) 1 ( 1 - q - p ) G ( t ) , as  t T * ,

hence

G ( t ) = g ( t ) = u q ( 0 , t ) ~ ( ( 1 - q - p ) a ( 0 ) ) q ( 1 - q - p ) G ( t ) q ( 1 - q - p ) . (3.19)

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

G ( t ) ~ ( a ( 0 ) ( 1 - q - p ) ) - 1 ( a ( 0 ) ( p + q + s - 1 ) ( T * - t ) ) ( 1 - q - p ) ( 1 - p - q - s ) . (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

lim t T * G ( t ) = a - 1 ( 0 ) ln ( T * - t ) ( - 1 s ) . (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 v q 1 ( x , τ ) and v s 1 ( 0 , τ ) , 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).

### References

1. Diaz, J, Kerser, R: On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium. J Diff Equ. 69, 368–403 (1987). Publisher Full Text

2. Furter, J, Grinfeld, M: Local vs. nonlocal interactions in population dynamics. J Math Biol. 27, 65–80 (1989). Publisher Full Text

3. Okada, A, Fukuda, I: Total versus single point blow-up of solution of a semilinear parabolic equation with localized reaction. J Math Anal Appl. 281, 485–500 (2003). Publisher Full Text

4. Cantrell, R, Cosner, C: Diffusive logistic equation with indefinite weights: population models in disrupted environments. II SIAM J Math Anal. 22, 1043–1064 (1991). Publisher Full Text

5. Levine, H: The role of critical exponents in blow-up theorem. SIMA Rev. 32, 268–288 (1990)

6. Anderson, J: Local existence and uniqueness of solutions of degenerate parabolic equations. Commun Partial Diff Equ. 16, 105–143 (1991). Publisher Full Text

7. Chen, Y, Liu, Q, Gao, H: Bounedeness of global solutions of a porous medium equation with a localized source. Nonlinear Anal. 64, 2168–2182 (2006). Publisher Full Text

8. Fukuda, I, Suzuki, R: Blow-up bebavior for a nolinear beat equation with a localized source in a ball. J Diff Equ. 218, 273–291 (2005). Publisher Full Text

9. Chen, Y, Liu, Q, Gao, H: Bounedeness of global positive solutions of a porous medium equation with a moving localized source. J Math Anal Appl. 333, 1008–1023 (2007). Publisher Full Text

10. Souple, P: Blow-up in non-local reaction-diffusion equations. SIAM J Math Anal. 29(6), 1301–1334 (1998). Publisher Full Text

11. Souple, P: Uniform blow-up profiles and boundary for diffusion equations with nonlocal nonlinear source. J Diff Equ. 153, 374–406 (1999). Publisher Full Text

12. Cannon, R, Yin, M: A class of non-linear non-classical parabolic equations. J Diff Equ. 79, 226–288 (1989)

13. Chandam, J, Peirce, A, Yin, H: The blow-up property of solutions to some diffusion equations with localized nonlinear reactions. J Math Anal Appl. 169, 313–328 (1992). Publisher Full Text

14. Kong, L, Wang, L, Zheng, S: Asymptotic analysis to a parabolic equation with a weighted localized source. Appl Math Comput. 197, 819–827 (2008). Publisher Full Text

15. Chen, Y, Xie, C: Blow-up for a porous medium equation with a localized source. Appl Math Comput. 159, 79–93 (2004). Publisher Full Text

16. Du, L, Xiang, Z: A further blow-up analysis for a localized porous medium equation. Appl Math Comput. 179, 200–208 (2006). Publisher Full Text

17. Wang, J, Kong, L, Zheng, S: Asymptotic analysis for a localized nonlinear diffusion equation. Comput Math Appl. 56, 2294–2304 (2008). Publisher Full Text

18. Friedman, A, Mcleod, J: Blow-up of positive solutions of semilinear heat equations. Indiana Univ Math J. 34, 425–447 (1985). Publisher Full Text

19. Rouchon, P: Boundeness of global solutions of nonlinear diffusion equations with localized reaction term. Diff Integral Equ. 16(9), 1083–1092 (2003)

20. Li, H, Wang, M: Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete Contin Dyn Syst. 13, 683–700 (2005)