Open Access

Asymptotic analysis for reaction-diffusion equations with absorption

Boundary Value Problems20122012:84

DOI: 10.1186/1687-2770-2012-84

Received: 31 May 2012

Accepted: 20 July 2012

Published: 2 August 2012

Abstract

In this paper, we study the blow-up and nonextinction phenomenon of reaction-diffusion equations with absorption under the null Dirichlet boundary condition. We at first discuss the existence and nonexistence of global solutions to the problem, and then give the blow-up rate estimates for the nonglobal solutions. In addition, the nonextinction of solutions is also concerned.

MSC:35B33, 35K55, 35K60.

Keywords

reaction-diffusion absorption blow-up blow-up rate non-extinction

1 Introduction

In this paper, we consider the reaction-diffusion equations with absorption
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ1_HTML.gif
(1.1)

where m > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq1_HTML.gif, p > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq2_HTML.gif, q 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq3_HTML.gif, p q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq4_HTML.gif, Ω R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq5_HTML.gif is a bounded domain with smooth boundary Ω, and u 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq6_HTML.gif is a nontrivial, nonnegative, bounded, and appropriately smooth function. Parabolic equations like (1.1) appear in population dynamics, chemical reactions, heat transfer, and so on. We refer to [2, 8, 9] for details on physical models involving more general reaction-diffusion equations.

The semilinear case ( m = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq7_HTML.gif) of (1.1) has been investigated by Bedjaoui and Souplet [3]. They obtained that the solutions exist globally if either p < max { q , 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq8_HTML.gif or p = max { q , 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq9_HTML.gif, and the solutions may blow up in finite time for large initial value if p > max { q , 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq10_HTML.gif. Recently, Xiang et al. [11] considered the blow-up rate estimates for nonglobal solutions of (1.1) ( m = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq7_HTML.gif) with p > max { q , 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq10_HTML.gif, and obtained that (i) max Ω u ( x , t ) c ( T t ) 1 p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq11_HTML.gif; (ii)  max Ω u ( x , t ) C ( T t ) 1 p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq12_HTML.gif if p 1 + 2 N + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq13_HTML.gif, where c , C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq14_HTML.gif are positive constants. Liu et al. [7] studied the extinction phenomenon of solutions of (1.1) for the case 0 < m < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq15_HTML.gif with q = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq16_HTML.gif and obtained some sufficient conditions about the extinction in finite time and decay estimates of solutions in Ω R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq5_HTML.gif ( N > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq17_HTML.gif).

Recently, Zhou et al. [10] investigated positive solutions of the degenerate parabolic equation not in divergence form
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ2_HTML.gif
(1.2)

where p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq18_HTML.gif, q , a , b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq19_HTML.gif, r > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq20_HTML.gif. They at first gave some conditions about the existence and nonexistence of global solutions to (1.2), and then studied the large time behavior for the global solutions.

Motivated by the above mentioned works, the aim of this paper is threefold. First, we determine optimal conditions for the existence and nonexistence of global solutions to (1.1). Secondly, by using the scaling arguments we establish the exact blow-up rate estimates for solutions which blow up in a finite time. Finally, we prove that every solution to (1.1) is nonextinction.

As it is well known that degenerate equations need not possess classical solutions, we give a precise definition of a weak solution to (1.1).

Definition 1.1 Let T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq21_HTML.gif and Q T = Ω × ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq22_HTML.gif, E = { u L 2 p ( Q T ) L 2 q ( Q T ) ; u t , u L 2 ( Q T ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq23_HTML.gif, E 0 = { u E ; u = 0  on  Ω } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq24_HTML.gif, a nonnegative function u ( x , t ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq25_HTML.gif is called a weak upper (or lower) solution to (1.1) in Q T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq26_HTML.gif if for any nonnegative function φ E 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq27_HTML.gif, one has
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equa_HTML.gif

In particular, u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq28_HTML.gif is called a weak solution of (1.1) if it is both a weak upper and a weak lower solution. For every T < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq29_HTML.gif, if u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq28_HTML.gif is a weak solution of (1.1) in Q T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq26_HTML.gif, we say that u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq28_HTML.gif is global. The local in time existence of nonnegative weak solutions have been established (see the survey [1]), and the weak comparison principle is stated and proved in the Appendix in this paper.

The behavior of the weak solutions is determined by the interactions among the multinonlinear mechanisms in the nonlinear diffusion equations in (1.1). We divide the ( m , p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq30_HTML.gif-parameter region into three classes: (i) p < max { m , q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq31_HTML.gif; (ii) p = max { m , q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq32_HTML.gif; (iii) p > max { m , q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq33_HTML.gif.

Theorem 1.1 If p < max { m , q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq34_HTML.gif, then all solutions of (1.1) are bounded.

Let ϕ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq35_HTML.gif be the first eigenfunction of
Δ ϕ ( x ) = λ ϕ ( x ) in  Ω , ϕ ( x ) = 0 in  Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ3_HTML.gif
(1.3)

with the first eigenvalue λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq36_HTML.gif, normalized by ϕ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq37_HTML.gif, then λ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq38_HTML.gif and ϕ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq39_HTML.gif in Ω.

Theorem 1.2 Assume that p = max { m , q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq40_HTML.gif. Then all solutions are global if λ 1 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq41_HTML.gif, and there exist both global and nonglobal solutions if λ 1 < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq42_HTML.gif.

Theorem 1.3 If p > max { m , q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq33_HTML.gif, then there exist both global and nonglobal solutions to (1.1).

To obtain the blow-up rate of blow-up solutions to (1.1), we need an extra assumption that Ω = B R ( 0 ) = { x R N : | x | < R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq43_HTML.gif and u 0 = u 0 ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq44_HTML.gif, u 0 ( r ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq45_HTML.gif, here r = | x | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq46_HTML.gif. By the assumption and comparison principle, we know that u is radially decreasing in r with max Ω u ( x , t ) = u ( 0 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq47_HTML.gif.

Theorem 1.4 Suppose that p > max { m , q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq48_HTML.gif. If the solution u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq28_HTML.gif of (1.1) blows up in finite time T, then there exists a positive constant c such that
max Ω u ( x , t ) c ( T t ) 1 p 1 as t T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equb_HTML.gif
Furthermore, if p > m q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq49_HTML.gif, then we have also the upper estimate, that is, there exists a positive constant C such that
max Ω u ( x , t ) C ( T t ) 1 p 1 as t T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equc_HTML.gif
We remark that in Ω = R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq50_HTML.gif, Liang [6] studied the blow up rate of blow-up solutions to the following Cauchy problem
u t = Δ u m + u p , ( x , t ) R N × ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ4_HTML.gif
(1.4)

with the bounded initial function, 1 < m < p < m N + 2 ( N 2 ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq51_HTML.gif, and obtained that u L ( R N ) < C ( T t ) 1 p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq52_HTML.gif for t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq53_HTML.gif. By using the same scaling arguments in this paper, we can find that Theorem 1.4 is correct for (1.4) with p > m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq54_HTML.gif.

Now, we pay attention to the nonextinction property of solutions and have the following result.

Theorem 1.5 Any solution of (1.1) does not go extinct in finite time for any nontrivial and nonnegative initial value u 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq6_HTML.gif with meas { x Ω ; u 0 ( x ) > 0 } > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq55_HTML.gif.

The rest of this paper is organized as follows. In the next section, we discuss the global existence and nonexistence of solutions, and prove Theorems 1.1-1.3. Subsequently, in Sects. 3 and 4, we consider the estimate of the blow-up rate and study the nonextinction phenomenon for the problem (1.1). The weak comparison principle is stated and proved in the Appendix.

2 Global existence and nonexistence

Proof of Theorem 1.1 If m q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq56_HTML.gif, that is p < m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq57_HTML.gif, then by the comparison principle, we have u w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq58_HTML.gif, where w satisfies
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ5_HTML.gif
(2.1)

We know from [4, 5] that w is bounded.

If m < q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq59_HTML.gif, we have p < q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq60_HTML.gif. It is obvious that u ¯ = max { 1 , u 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq61_HTML.gif is a time-independent upper solution to (1.1). □

Proof of Theorem 1.2 Since p q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq4_HTML.gif and p = max { m , q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq40_HTML.gif imply p = m > q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq62_HTML.gif. Due to the fact that the solution of (2.1) is an upper solution of (1.1), the conclusions for λ 1 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq41_HTML.gif is obvious true; see [4, 5].

Now consider λ 1 < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq42_HTML.gif with small initial data. Let ψ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq63_HTML.gif be the unique solution of
Δ ψ ( x ) = 1 in  Ω , ψ ( x ) = 0 on  Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ6_HTML.gif
(2.2)
and h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq64_HTML.gif solves h ( t ) = δ h ( t ) m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq65_HTML.gif with h ( 0 ) = h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq66_HTML.gif, where 0 < δ ψ 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq67_HTML.gif. Set u ¯ = h ( t ) ψ 1 m ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq68_HTML.gif. Then
u ¯ t Δ u ¯ m u ¯ m + u ¯ q = δ h m ψ 1 m + h m h m ψ + h q ψ q m = h m ( 1 δ ψ 1 m ) + h q ψ q m ( 1 h m q ψ m q m ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equd_HTML.gif

provided h 0 m q ψ m q m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq69_HTML.gif. Thus, u ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq70_HTML.gif is an upper solution of (1.1), and consequently, u u ¯ = h ( t ) ψ 1 m ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq71_HTML.gif as t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq72_HTML.gif.

If λ 1 < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq42_HTML.gif with large initial data, we first introduce some transformations. Let v = u m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq73_HTML.gif and τ = m t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq74_HTML.gif, then (1.1) becomes the following equations not in divergence form:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Eque_HTML.gif

where r = m 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq75_HTML.gif, s = q m < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq76_HTML.gif and v 0 ( x ) = u 0 m ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq77_HTML.gif.

Let J ( τ ) = 1 1 r Ω v 1 r ϕ d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq78_HTML.gif, where ϕ is given in (1.3). Then we have
J ( τ ) = Ω ( Δ v + v v s ) ϕ ( x ) d x = ( 1 λ 1 ) Ω v ϕ d x Ω v s ϕ d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ7_HTML.gif
(2.3)
By using Hölder’s inequality, we discover
Ω v s ϕ d x ( Ω v ϕ d x ) s ( Ω ϕ d x ) 1 s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ8_HTML.gif
(2.4)
and
Ω v 1 r ϕ d x ( Ω v ϕ d x ) 1 r ( Ω ϕ d x ) r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equf_HTML.gif
i.e.,
Ω v ϕ d x [ ( 1 r ) J ( τ ) ( Ω ϕ d x ) r ] 1 1 r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ9_HTML.gif
(2.5)
Inserting (2.4) into (2.3), we have
J ( τ ) ( 1 λ 1 ) Ω v ϕ d x ( Ω v ϕ d x ) s ( Ω ϕ d x ) 1 s = ( Ω v ϕ d x ) s [ ( 1 λ 1 ) ( Ω v ϕ d x ) 1 s ( Ω ϕ d x ) 1 s ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ10_HTML.gif
(2.6)
According to (2.5), (2.6), we obtain
J ( τ ) ( 1 λ 1 ) 2 [ ( 1 r ) ( Ω ϕ d x ) r ] 1 1 r J ( τ ) 1 1 r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ11_HTML.gif
(2.7)
as long as
J ( τ ) 1 1 r ( 2 1 λ 1 ) 1 r 1 s ( Ω ϕ d x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equg_HTML.gif
Hence, if u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq79_HTML.gif satisfies
J ( 0 ) 1 1 r ( 2 1 λ 1 ) 1 r 1 s ( Ω ϕ d x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equh_HTML.gif

we then follow from (2.7) that J ( τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq80_HTML.gif, and consequently u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq28_HTML.gif, blows up in finite time since J ( τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq80_HTML.gif is increasing and 1 1 r = m > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq81_HTML.gif. □

Proof of Theorem 1.3 Let h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq64_HTML.gif solves h ( t ) = h ( t ) p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq82_HTML.gif with h ( 0 ) = h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq66_HTML.gif, and set u ¯ = h ( t ) ψ 1 m ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq68_HTML.gif, where ψ is defined in (2.2). Then
u ¯ t Δ u ¯ m u ¯ p + u ¯ q = h p ψ 1 m + h m h p ψ p m + h q ψ q m = h m ( 1 h p m ψ 1 m ) + h q ψ q m ( 1 h p q ψ p q m ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equi_HTML.gif

Since p > max { m , q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq33_HTML.gif, we can choose h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq83_HTML.gif small enough such that u ¯ t Δ u ¯ m u ¯ p + u ¯ q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq84_HTML.gif. Thus, u ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq70_HTML.gif is an upper solution of (1.1) provided u 0 ( x ) h 0 ψ 1 m ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq85_HTML.gif, and consequently, u u ¯ = h ( t ) ψ 1 m ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq86_HTML.gif as t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq72_HTML.gif.

Now deal with the nonexistence of global solutions, we seek a blow-up self-similar lower solution of the problem (1.1). Without loss of generality, we may assume that Ω contains the origin. Since p > max { m , q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq33_HTML.gif, we can choose constant α such that
1 p 1 < α < min { 1 m 1 , 1 q 1 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equj_HTML.gif
and consider the function
u ̲ ( x , t ) = ( T t ) α f ( ξ ) , ξ = | x | ( T t ) β , β = 1 ( m 1 ) α 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ12_HTML.gif
(2.8)

where f ( ξ ) = ( a 2 ξ 2 ) + 1 m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq87_HTML.gif. Note that the support of u ̲ ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq88_HTML.gif is contained in B ( 0 , a T β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq89_HTML.gif, which is included in Ω if T is sufficiently small.

After some computations, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equk_HTML.gif
It will be obtained from the above equalities that
u ̲ t Δ u ̲ m u ̲ p + u ̲ q 0 , in  Ω × ( 0 , T ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equl_HTML.gif
if f ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq90_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ13_HTML.gif
(2.9)
It is easy to see that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equm_HTML.gif
To satisfy (2.9), we distinguish the two zones 0 < ξ θ a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq91_HTML.gif and θ a < ξ < a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq92_HTML.gif, where
θ = α + 2 m N m 1 α + 2 m N m 1 + 4 m ( m 1 ) 2 < 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ14_HTML.gif
(2.10)
For θ a < ξ < a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq92_HTML.gif, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equn_HTML.gif
then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equo_HTML.gif

For 0 < ξ θ a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq91_HTML.gif, we have f ( ξ ) ( 1 θ 2 ) 1 m 1 a 2 m 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq93_HTML.gif. It follow from p α > α + 1 > q α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq94_HTML.gif that (2.9) is satisfied for 0 < ξ θ a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq95_HTML.gif, θ a < ξ < a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq92_HTML.gif if T is sufficiently small. Therefore, u ̲ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq96_HTML.gif given by (2.8) is a blow-up lower solution of the problem (1.1) with appropriately large u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq79_HTML.gif. And consequently, there exist nonglobal solutions to (1.1). □

3 Blow-up rate

In this section, we study the speeds at which the solutions to (1.1) blow up. Assume that Ω = B R ( 0 ) = { x R N : | x | < R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq43_HTML.gif and u 0 = u 0 ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq44_HTML.gif, u 0 ( r ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq45_HTML.gif, here r = | x | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq46_HTML.gif. Then we know from the assumption and comparison principle that u is radially decreasing in r with max Ω u ( x , t ) = u ( 0 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq47_HTML.gif. In this section, denote by T the blow-up time for the nonglobal solutions to (1.1).

Proof of Theorem 1.4 Fix t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq53_HTML.gif such that M ( t ) = max Ω u ( x , t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq97_HTML.gif, and let
a = M p m 2 , b = M 1 p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equp_HTML.gif
and define the function ψ M ( y , s ) = 1 M ( t ) u ( a y , b s + t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq98_HTML.gif in B M p m 2 R ( 0 ) × ( 0 , S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq99_HTML.gif, where S = M p 1 ( T t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq100_HTML.gif. ψ M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq101_HTML.gif blows up at s = S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq102_HTML.gif, moreover, it is a solution of the following problem:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ15_HTML.gif
(3.1)
We now construct an upper solution for this problem. Set
w ¯ ( y , s ) = ( S 1 s ) α ( L + δ ( L ξ ) + ) 1 m 1 , ξ = | y | ( S 1 s ) β , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equq_HTML.gif
where α = 1 p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq103_HTML.gif, β = p m 2 ( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq104_HTML.gif, and
0 < L < α m 1 p 1 2 p + 2 m 3 p 1 , S 1 α L 1 m 1 > 1 , 0 < δ < min { 1 , ( m 1 ) α 4 β , m 1 2 α L m } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equr_HTML.gif
After a direct computation, for 0 < ξ < L https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq105_HTML.gif, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equs_HTML.gif
Then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equt_HTML.gif
Clearly, w ¯ s Δ w ¯ m w ¯ p + M q p w ¯ q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq106_HTML.gif for ξ > L https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq107_HTML.gif, and w ¯ ( y , s ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq108_HTML.gif on B M p m 2 R ( 0 ) × ( 0 , S 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq109_HTML.gif, w ¯ ( y , 0 ) ψ M ( y , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq110_HTML.gif in B M p m 2 R ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq111_HTML.gif. We have an upper solution independent of M, for all M large enough. Therefore, the blow-up time of ψ M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq101_HTML.gif is greater than S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq112_HTML.gif, that is M p 1 ( T t ) S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq113_HTML.gif. This implies
max Ω u ( x , t ) c ( T t ) 1 p 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equu_HTML.gif

and the lower estimate is obtained.

In order to obtained the upper estimates for the blow-up rate, we look for a lower solution to (3.1) with M ( t ) > M 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq114_HTML.gif. Set
w ̲ ( y , s ) = ( S 2 s ) α f ( ξ ) , ξ = | y | ( S 2 s ) β , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equv_HTML.gif
where α = 1 p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq103_HTML.gif, β = p m 2 ( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq104_HTML.gif, f ( ξ ) = ( a 2 ξ 2 ) + 1 m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq87_HTML.gif, a 2 ( p 1 ) m 1 > ( 1 θ 2 ) 1 p m 1 ( α + 2 m N m 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq115_HTML.gif and θ is given in (2.10). Let M 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq116_HTML.gif satisfies
M 0 max { ( m 1 2 β θ 2 S 2 ( p q ) α a 2 ( q 1 ) m 1 ) 1 p q , ( a S 2 β R 1 ) 2 p m , ( 2 μ S 2 ( q 1 ) α a 2 ( q 1 ) m 1 ) 1 p q , ( ( ( 1 θ 2 ) p 1 m 1 a 2 ( p 1 ) m 1 ( α + 2 m N m 1 ) ) 1 S 2 ( p q ) α a 2 ( q 1 ) m 1 ) 1 p q } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equw_HTML.gif
where S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq117_HTML.gif, μ are to be determined later. Clearly, w ̲ ( y , s ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq118_HTML.gif on B M p m 2 R ( 0 ) × ( 0 , S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq119_HTML.gif. As the same arguments in the proof of Theorem 1.3, we have for θ a ξ < a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq120_HTML.gif that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equx_HTML.gif
For 0 < ξ θ a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq91_HTML.gif, we have that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equy_HTML.gif
Now, in order to deal with the initial data, consider the function
z ( y , s ) = S 2 α ( a 2 | y | 2 S 2 2 β s λ ) + 1 m 1 s μ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equz_HTML.gif

where λ = 1 ( m 1 ) N + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq121_HTML.gif, μ = 1 λ m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq122_HTML.gif, and S 2 = 4 m ( m 1 ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq123_HTML.gif.

After a direct computation, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equaa_HTML.gif
Then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equab_HTML.gif

Furthermore, z ( y , s ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq124_HTML.gif on B M p m 2 R ( 0 ) × ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq125_HTML.gif. In addition, z ( y , 0 ) = lim s 0 z ( y , s ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq126_HTML.gif a.e. in B M p m 2 R ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq111_HTML.gif. Therefore, by the comparison principle, we have that ψ M ( y , s ) z ( y , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq127_HTML.gif for 0 s 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq128_HTML.gif. By the virtue of w ̲ ( y , 0 ) = z ( y , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq129_HTML.gif, we have ψ M ( y , s + 1 ) w ̲ ( y , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq130_HTML.gif.

We have a lower solution independent of M, for all M > M 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq131_HTML.gif. Therefore, the blow-up time of ψ M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq101_HTML.gif is less than S 2 + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq132_HTML.gif, that is M p 1 ( T t ) S 2 + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq133_HTML.gif. This implies
max Ω u ( x , t ) C ( T t ) 1 p 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equac_HTML.gif

and the upper estimate is obtained. □

4 Nonextinction

We discuss the nonextinction of the solution to the problem (1.1) in this section. For p < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq134_HTML.gif, the uniqueness of the weak solution to (1.1) may not hold. In this case, we only consider the maximal solution, which can be obtained by standard regularized approximation methods. Clearly, the comparison principle is valid for the maximal solution.

Proof of Theorem 1.5 For meas { x Ω ; u 0 ( x ) > 0 } > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq55_HTML.gif, there exists a region Ω 0 Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq135_HTML.gif and ϵ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq136_HTML.gif such that u 0 ( x ) ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq137_HTML.gif a.e. in Ω 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq138_HTML.gif. λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq139_HTML.gif is the first Dirichlet eigenvalue of −Δ on Ω 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq138_HTML.gif with corresponding eigenfunction ϕ 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq140_HTML.gif, normalized by ϕ 0 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq141_HTML.gif, and prolong solution ϕ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq142_HTML.gif by 0 in Ω Ω 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq143_HTML.gif. We treat the five subcases for the proof.
  1. (a)
    For p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq144_HTML.gif, set u ̲ = h ( t ) ϕ 0 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq145_HTML.gif, where
    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equad_HTML.gif
     
Then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equae_HTML.gif
By the comparison principle, we have u u ̲ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq146_HTML.gif in Ω 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq138_HTML.gif.
  1. (b)
    For p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq147_HTML.gif, 1 q m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq148_HTML.gif, we let u ̲ = h ( t ) ϕ 0 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq145_HTML.gif, h ( t ) = ( 1 + λ 0 ) h q ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq149_HTML.gif with h ( 0 ) = h 0 < ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq150_HTML.gif. Then
    u ̲ t Δ u ̲ m u ̲ p + u ̲ q ( 1 + λ 0 ) h q ( t ) ϕ 0 1 m + λ 0 h m ϕ 0 + h q ϕ 0 q m = h q ( t ) ϕ 0 1 m ( 1 + λ 0 λ 0 h m q ϕ 0 m q m ϕ 0 q 1 m ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equaf_HTML.gif
     
Then we know by the comparison principle that u u ̲ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq146_HTML.gif in Ω 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq138_HTML.gif.
  1. (c)
    For 1 < p < m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq151_HTML.gif, q > m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq152_HTML.gif, we let u ̲ = h ( t ) ϕ 0 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq145_HTML.gif, and
    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equag_HTML.gif
     
where δ < min { ϵ , ( 1 1 + λ 0 ) 1 m p } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq153_HTML.gif. It is easy to see that h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq64_HTML.gif is nonincreasing and h ( t ) M 1 p m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq154_HTML.gif as t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq72_HTML.gif.
u ̲ t Δ u ̲ m u ̲ p + u ̲ q = h p ( t ) ( 1 M h m p ) ϕ 0 1 m + λ 0 h m ϕ 0 ( x ) h p ϕ 0 p m + h q ϕ 0 q m = h m ϕ 0 1 m ( M λ 0 ϕ 0 m 1 m h q m ϕ 0 q 1 m ) + h p ϕ 0 1 m ( 1 ϕ 0 p 1 m ) h m ϕ 0 1 m ( M ( 1 + λ 0 ) ϕ 0 p 1 m ) + h p ϕ 0 1 m ( 1 ϕ 0 p 1 m ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equah_HTML.gif
And consequently, u u ̲ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq146_HTML.gif in Ω 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq138_HTML.gif.
  1. (d)
    For p = m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq155_HTML.gif, q > m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq152_HTML.gif, we let u ̲ = h ( t ) ϕ 0 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq145_HTML.gif, where
    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equai_HTML.gif
     
Obviously, h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq64_HTML.gif is nonincreasing and h ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq156_HTML.gif as t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq72_HTML.gif.
u ̲ t Δ u ̲ m u ̲ p + u ̲ q = ( 1 + λ 0 ) h m ϕ 0 1 m + λ 0 h m ϕ 0 h p ϕ 0 p m + h q ϕ 0 q m h m ϕ 0 1 m ( 1 + λ 0 λ 0 ϕ 0 m 1 m h q m ϕ 0 q 1 m ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equaj_HTML.gif
Thus, we have u u ̲ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq146_HTML.gif in Ω 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq138_HTML.gif.
  1. (e)
    For p > m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq54_HTML.gif, q > m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq152_HTML.gif, we let u ̲ = h ( t ) ϕ 0 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq145_HTML.gif, and
    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equak_HTML.gif
     
where c satisfies 1 + λ 0 < c < 1 + λ 0 + ϵ p m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq157_HTML.gif. It is easy to see that h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq64_HTML.gif is nonincreasing and h ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq156_HTML.gif as t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq72_HTML.gif.
u ̲ t Δ u ̲ m u ̲ p + u ̲ q = h m ( t ) ( h p m c ) ϕ 0 1 m + λ 0 h m ϕ 0 h p ϕ 0 p m ( x ) + h q ϕ 0 q m = h p ϕ 0 1 m ( 1 ϕ 0 p 1 m ) + h m ( t ) ϕ 0 1 m ( λ 0 ϕ 0 m 1 m + h q m ϕ 0 q 1 m c ) h p ϕ 0 1 m + h m ( t ) ϕ 0 1 m ( 1 + λ 0 c ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equal_HTML.gif

By the comparison principle, we have u u ̲ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq146_HTML.gif in Ω 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq138_HTML.gif.

 □

Appendix

Theorem A.1 (Comparison principle)

Let u ̲ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq96_HTML.gif and u ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq70_HTML.gif are a weak lower and a weak upper solutions of (1.1) in Q T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq26_HTML.gif. If p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq18_HTML.gif or u ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq70_HTML.gif has a positive lower bound, then u ̲ u ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq158_HTML.gif a.e. in Q T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq26_HTML.gif.

Proof From the definition of weak upper and lower solutions, for any 0 φ E 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq159_HTML.gif, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equam_HTML.gif
Let Q t = Ω × ( 0 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq160_HTML.gif for t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq53_HTML.gif. Choose φ = χ [ 0 , t ] ( u ̲ u ¯ ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq161_HTML.gif, where χ [ 0 , t ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq162_HTML.gif is the characteristic function defined on [ 0 , t ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq163_HTML.gif, Then we arrive at
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equan_HTML.gif
By a simple calculation, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ16_HTML.gif
(A.1)
Noticing
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equao_HTML.gif
we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equ17_HTML.gif
(A.2)
where L is a positive constant. By (A.1), (A.2), we have
Ω ( u ̲ u ¯ ) + 2 d x 2 L Q t ( u ̲ u ¯ ) + 2 d x d τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equap_HTML.gif
It follows immediately by using the Gronwall’s inequality that
Ω ( u ̲ u ¯ ) + 2 d x = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_Equaq_HTML.gif

for almost all t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq53_HTML.gif, and hence u ̲ u ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq158_HTML.gif a.e. in Ω × ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-84/MediaObjects/13661_2012_Article_211_IEq164_HTML.gif. □

Declarations

Acknowledgements

This work was partially supported by Projects Supported by Scientific Research Fund of Sichuan Provincial Education Department (09ZA119).

Authors’ Affiliations

(1)
College of Mathematic and Information, China West Normal University

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Copyright

© Du and Li; licensee Springer 2012

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