Extinction and decay estimates of solutions for a porous medium equation with nonlocal source and strong absorption

  • Xianghui Xu1,

    Affiliated with

    • Zhong Bo Fang2Email author and

      Affiliated with

      • Su-Cheol Yi3

        Affiliated with

        Boundary Value Problems20132013:24

        DOI: 10.1186/1687-2770-2013-24

        Received: 25 September 2012

        Accepted: 21 December 2012

        Published: 5 March 2013

        Abstract

        In this paper, we investigate extinction properties of the solutions for the initial Dirichlet boundary value problem of a porous medium equation with nonlocal source and strong absorption terms. We obtain some sufficient conditions for the extinction of nonnegative nontrivial weak solutions and the corresponding decay estimates which depend on the initial data, coefficients, and domains.

        1 Introduction

        We consider the initial Dirichlet boundary value problem for a class of porous medium equations with nonlocal source and strong absorption terms
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ1_HTML.gif
        (1)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ2_HTML.gif
        (2)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ3_HTML.gif
        (3)

        where 0 < m , k < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq1_HTML.gif, d , λ , β , q > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq2_HTML.gif, Ω R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq3_HTML.gif ( N 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq4_HTML.gif) is a bounded domain with smooth boundary, and u 0 ( x ) L ( Ω ) W 0 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq5_HTML.gif is a nonnegative function. The symbols p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq6_HTML.gif and 1 , p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq7_HTML.gif, where p 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq8_HTML.gif, denote L p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq9_HTML.gif- and W 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq10_HTML.gif-norm, respectively, and | Ω | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq11_HTML.gif denotes the measure of Ω.

        Equation (1) describes the fast diffusion of concentration of some Newtonian fluids through a porous medium or the density of some biological species in many physical phenomena and biological species theories, while nonlocal source and absorption terms cooperate and interact with each other during the diffusion. It has been known that the nonlocal source term presents a more realistic model for population dynamics; see [13]. In the nonlinear diffusion theory, obvious differences exist among the situations of slow ( m > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq12_HTML.gif), fast ( 0 < m < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq13_HTML.gif), and linear ( m = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq14_HTML.gif) diffusions. For example, there is a finite speed propagation in the slow and linear diffusion situations, whereas an infinite speed propagation exists in the fast diffusion situation.

        Recently, many scholars have been devoted to the study of blow-up and extinction properties of solutions for nonlinear parabolic equations with nonlocal terms. The blow-up rates and blow-up sets of solutions to equation (1) have been investigated when d = λ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq15_HTML.gif, m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq16_HTML.gif, and the linear absorption term is replaced with a nonlinear term with exponent (cf. [49]). Extinction is the phenomenon whereby there exists a finite time T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq17_HTML.gif such that the solution is nontrivial for 0 < t < T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq18_HTML.gif and then u ( x , t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq19_HTML.gif for all ( x , t ) Ω × [ T , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq20_HTML.gif. In this case, T is called the extinction time. It is also an important property of solutions for nonlinear parabolic equations which have been studied by many researchers. For instance, Evans and Knerr [10] investigated the extinction behaviors of solutions for the Cauchy problem of the semilinear parabolic equation
        u t ( x , t ) = Δ u ( x , t ) β ( u ( x , t ) ) , x R n , t > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equa_HTML.gif
        by constructing a suitable comparison function. Ferreira and Vazquez [11] studied the extinction phenomena of solutions for the Cauchy problem of the porous medium equation with an absorption term
        u t = ( u m ) x x u p , x R , t > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equb_HTML.gif
        by using the analysis of self-similar solutions. Li and Wu [12] considered the problem of the porous medium equation with a source term
        u t = Δ u m + λ u p , x Ω , t > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ4_HTML.gif
        (4)

        subject to (2) and (3). They obtained some conditions for the extinction and non-extinction of solutions to equation (4) and decay estimates by the upper and lower solutions method. On extinctions of solutions to the p-Laplacian equations or the doubly degenerate equations, we refer readers to [13, 14] and the references therein.

        Replacing the nonlocal term in equation (1) with a local term, Liu [15]et al. considered the initial Dirichlet boundary value problem for a class of porous medium equations
        u t = Δ u m + u q β u , x Ω , t > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equc_HTML.gif

        and obtained sufficient conditions for the extinction and non-extinction of solutions to that equation. Thereafter, Fang and Li [16] extended their results to the doubly degenerate equation in the whole dimensional space.

        For equation (1) with β = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq21_HTML.gif, q > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq22_HTML.gif, and N > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq23_HTML.gif, Han and Gao [17] showed that q = m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq24_HTML.gif is the critical exponent for the occurrence of extinction or non-extinction. When m = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq14_HTML.gif, q > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq22_HTML.gif, and N > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq23_HTML.gif, the conditions for the extinction and non-extinction of solutions and corresponding decay estimates were obtained (cf. [18]). Recently, Fang and Xu [19] considered equation (1) with k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq25_HTML.gif, when the diffusion term was replaced with a p-Laplacian operator in the whole dimensional space, and showed that the extinction of the weak solution is determined by the competition of two nonlinear terms. They also obtained the exponential decay estimates which depend on the initial data, coefficients, and domains. The extinctions of solutions to equation (1) with nonlocal source terms do not depend on the first eigenvalue of the corresponding operator, which is different from the case of local source terms. The extinction and decay estimates for solutions to the nonlocal fast diffusion equations with nonzero coefficients and strong absorption terms, like equation (1), are still being investigated.

        Motivated by the above works, we investigate whether the existence of strong absorption can change extinction behaviors for solutions to problem (1)-(3) in the whole dimensional space. The main tools we use are the integral estimate method and the Gagliardo-Nirenberg inequality. This technique has a wide application, especially for equations that do not satisfy the maximum principle (cf. [20]). Our goals are to show that the extinction of nonnegative nontrivial weak solutions to problem (1)-(3) occurs when 0 < k q < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq26_HTML.gif and to find the decay estimates depending on the initial data, coefficients, and domains.

        Our paper is organized as follows. In Section 2, we give preliminary knowledge including lemmas that are required in the proofs of our results and present the proofs for the results in Section 3.

        2 Preliminary knowledge

        Due to the singularity of equation (1), problem (1)-(3) has no classical solutions in general. To state the definition of the weak solution, we let Q T = Ω × ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq27_HTML.gif and firstly define the class of nonnegative testing functions
        F = { ξ : ξ C ( Q ¯ T ) C 2 , 1 ( Q T ) , ξ t , Δ ξ L 2 ( Q T ) ; ξ 0 , ξ | Ω × ( 0 , T ) = 0 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equd_HTML.gif
        Definition 1 A function u L ( Q T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq28_HTML.gif is called a weak subsolution (supersolution) of problem (1)-(3) in Q T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq29_HTML.gif if the following conditions hold:
        1. a.

          u ( x , 0 ) ( ) u 0 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq30_HTML.gif in Ω;

           
        2. b.

          u ( x , t ) ( ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq31_HTML.gif on Ω × ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq32_HTML.gif;

           
        3. c.
          For every t ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq33_HTML.gif and every ξ F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq34_HTML.gif,
          Ω u ( x , t ) ξ ( x , t ) d x Ω u 0 ( x ) ξ ( x , 0 ) d x + 0 T Ω { u ξ s + d u m Δ ξ + λ Ω u q ( y , s ) d y ξ ( x , s ) β u k ( x , s ) ξ ( x , s ) } d x d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Eque_HTML.gif
           

        A function u is called a locally weak solution of problem (1)-(3) if it is both a subsolution and a supersolution for some T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq17_HTML.gif.

        Remark 1 The existence and uniqueness of locally nonnegative solutions in time to problem (1)-(3) can be obtained by the standard parabolic regular theory that can be applied to get suitable estimates in the standard limiting process (cf. [2, 21, 22]). The proof is similar to the ones in the cited references, and so it is omitted here.

        Lemma 1 Let k, α be positive constants and k < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq35_HTML.gif. If y ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq36_HTML.gif is a nonnegative absolutely continuous function on [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq37_HTML.gif satisfying the problem
        d y d t + α y k 0 , t 0 ; y ( 0 ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equf_HTML.gif
        then we have the decay estimate
        y ( t ) [ y 1 k ( 0 ) α ( 1 k ) t ] 1 1 k , t [ 0 , T ) , y ( t ) = 0 , t [ T , + ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equg_HTML.gif

        where T = y 1 k ( 0 ) α ( 1 k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq38_HTML.gif.

        Proof

        By solving the initial problem
        d y d t + α y k = 0 , t > 0 ; y | t = 0 = y ( 0 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equh_HTML.gif

        and using the comparison principle, one can easily obtain the result. □

        Lemma 2 (The Gagliardo-Nirenberg inequality) [23]

        Suppose that u W 0 k , m ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq39_HTML.gif, 1 m + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq40_HTML.gif, 0 j < k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq41_HTML.gif, and 1 1 r 1 m k N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq42_HTML.gif. We then have the inequality
        D j u q C D k u m θ u r 1 θ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equi_HTML.gif

        where C is a constant depending on N, m, r, j, k, and q such that 1 q = j N + θ ( 1 m k N ) + 1 θ r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq43_HTML.gif and θ [ 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq44_HTML.gif. While if m < N k j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq45_HTML.gif, then q [ N r N + r j , N m N ( k j ) m ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq46_HTML.gif, and if m N k j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq47_HTML.gif, then q [ N r N + r j , + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq48_HTML.gif.

        3 Main results

        In this section, we give some extinction properties of nonnegative nontrivial weak solutions of problem (1)-(3) stated in the following theorems. The corresponding decay estimates to the solutions will be presented in the proofs of the theorems for brief expressions instead of in the statements.

        Theorem 1 Suppose that 0 < k < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq49_HTML.gif and 0 < m = q < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq50_HTML.gif. Then the nonnegative nontrivial weak solution of problem (1)-(3) vanishes in finite time for any nonnegative initial data provided that either | Ω | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq11_HTML.gif or λ is sufficiently small.

        Proof We first consider the case that N = 1  or  2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq51_HTML.gif. Multiplying both sides of (1) by u and integrating the result over Ω, we have
        1 2 d d t u 2 2 + 4 m d ( m + 1 ) 2 u m + 1 2 2 2 + β u k + 1 k + 1 = λ Ω u m d x Ω u d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equj_HTML.gif
        By Hölder’s inequality, we get the inequality
        Ω u m d x Ω u d x | Ω | 2 s 1 1 m s 1 u s 1 m + 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equk_HTML.gif
        In particular, if s 1 = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq52_HTML.gif, we get the inequality
        1 2 d d t u 2 2 + 4 m d ( m + 1 ) 2 u m + 1 2 2 2 + β u k + 1 k + 1 λ | Ω | 3 m 2 u 2 m + 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ5_HTML.gif
        (5)
        from the two expressions above. By using the Sobolev embedding inequality, one can show that there exists an embedding constant γ ( N , Ω ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq53_HTML.gif such that
        u m + 1 2 s 2 γ ( N , Ω ) u m + 1 2 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equl_HTML.gif
        i.e.,
        u ( m + 1 ) s 2 2 m + 1 γ ( N , Ω ) 2 u m + 1 2 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equm_HTML.gif
        In particular, if s 2 = 4 m + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq54_HTML.gif, then the inequality above turns out to be
        γ ( N , Ω ) 2 u 2 m + 1 u m + 1 2 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ6_HTML.gif
        (6)
        Since 0 < m < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq13_HTML.gif, we get 4 m + 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq55_HTML.gif, and hence, inequality (5) becomes
        1 2 d d t u 2 2 + [ 4 m d ( m + 1 ) 2 λ | Ω | 3 m 2 γ ( N , Ω ) 2 ] u m + 1 2 2 2 + β u k + 1 k + 1 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ7_HTML.gif
        (7)
        By Lemma 2, we get the inequality
        u 2 C ( N , m , k ) u k + 1 1 θ 1 u m + 1 2 2 2 θ 1 m + 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ8_HTML.gif
        (8)

        where θ 1 = m + 1 2 ( 1 k + 1 1 2 ) ( 1 N 1 2 + m + 1 2 1 k + 1 ) 1 = N ( 1 k ) ( m + 1 ) 2 [ 2 ( k + 1 ) + N ( m k ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq56_HTML.gif. Since N = 1  or  2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq51_HTML.gif, and 0 < k < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq49_HTML.gif, it can be easily seen that 0 < θ 1 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq57_HTML.gif.

        It then follows from (8) and Young’s inequality that
        u 2 k 1 C ( N , m , k ) k 1 u k + 1 k 1 ( 1 θ 1 ) u m + 1 2 2 2 k 1 θ 1 m + 1 C ( N , m , k ) k 1 ( η 1 u m + 1 2 2 2 + C ( η 1 ) u k + 1 k 1 ( 1 θ 1 ) ( m + 1 ) m + 1 k 1 θ 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ9_HTML.gif
        (9)
        where k 1 > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq58_HTML.gif and η 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq59_HTML.gif will be determined later. If we choose
        k 1 = ( k + 1 ) ( m + 1 ) ( 1 θ 1 ) ( m + 1 ) + θ 1 ( k + 1 ) = 4 ( k + 1 ) + 2 N ( m k ) 4 + N ( m k ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equn_HTML.gif
        then 1 < k 1 < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq60_HTML.gif and k 1 ( 1 θ 1 ) ( m + 1 ) m + 1 k 1 θ 1 = k + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq61_HTML.gif. From (9) we have
        C ( N , m , k ) k 1 β C ( η 1 ) u 2 k 1 η 1 β C ( η 1 ) u m + 1 2 2 2 + β u k + 1 k + 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ10_HTML.gif
        (10)
        By inequalities (7) and (10), we get the inequality
        1 2 d d t u 2 2 + [ 4 m d ( m + 1 ) 2 λ | Ω | 3 m 2 γ ( N , Ω ) 2 η 1 β C ( η 1 ) ] u m + 1 2 2 2 + C ( N , m , k ) k 1 β C ( η 1 ) u 2 k 1 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equo_HTML.gif
        Here, we can choose η 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq62_HTML.gif and λ or | Ω | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq11_HTML.gif small enough so that
        4 m d ( m + 1 ) 2 λ | Ω | 3 m 2 γ ( N , Ω ) 2 η 1 β C ( η 1 ) 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equp_HTML.gif

        Setting C 1 = C ( N , m , k ) k 1 β C ( η 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq63_HTML.gif, we have d d t u 2 + C 1 u 2 k 1 1 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq64_HTML.gif.

        By Lemma 1, we then obtain
        u 2 [ u 0 2 2 k 1 C 1 ( 2 k 1 ) t ] 1 2 k 1 , t [ 0 , T 1 ) , u 2 = 0 , t [ T 1 , + ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equq_HTML.gif

        where T 1 = u 0 2 2 k 1 C 1 ( 2 k 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq65_HTML.gif, which give the decay estimates in finite time for N = 1  or  2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq66_HTML.gif.

        Secondly, we consider the case that N > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq67_HTML.gif. If N 2 N + 2 m < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq68_HTML.gif, multiplying both sides of (1) by u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq69_HTML.gif ( s > m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq70_HTML.gif) and integrating the result over Ω, we get
        1 s + 1 d d t u s + 1 s + 1 + 4 m s d ( m + s ) 2 u m + s 2 2 2 + β u k + s k + s = λ Ω u m d x Ω u s d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equr_HTML.gif
        By Hölder’s inequality, we have the inequality
        Ω u m d x Ω u s d x | Ω | 2 s 3 m s s 3 u s 3 m + s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equs_HTML.gif
        In particular, if s 3 = s + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq71_HTML.gif, we then get the inequality
        1 s + 1 d d t u s + 1 s + 1 + 4 m s d ( m + s ) 2 u m + s 2 2 2 + β u k + s k + s λ | Ω | s m + 2 s + 1 u s + 1 m + s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ11_HTML.gif
        (11)
        by the two expressions above. By the Sobolev embedding inequality, one can see that there exists an embedding constant C 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq72_HTML.gif such that
        u m + s 2 2 N N 2 2 C 0 2 u m + s 2 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ12_HTML.gif
        (12)
        Using Hölder’s inequality again, we have the inequality
        u s + 1 m + s | Ω | m + s s + 1 N 2 N u m + s 2 2 N N 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ13_HTML.gif
        (13)
        From inequalities (11), (12), and (13), we then obtain the inequality
        1 s + 1 d d t u s + 1 s + 1 + [ 4 m s d ( m + s ) 2 λ C 0 2 | Ω | 1 + 2 N ] u m + s 2 2 2 + β u k + s k + s 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ14_HTML.gif
        (14)
        By Lemma 2, we can also have
        u s + 1 C ( N , k , s ) u k + s 1 θ 2 u m + s 2 2 2 θ 2 m + s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ15_HTML.gif
        (15)
        where θ 2 = m + s 2 ( 1 k + s 1 s + 1 ) ( 1 N 1 2 + m + s 2 1 k + s ) 1 = N ( 1 k ) ( m + s ) ( s + 1 ) [ 2 ( k + s ) + N ( m k ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq73_HTML.gif. Since N 2 N + 2 m < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq74_HTML.gif and 0 < k < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq49_HTML.gif, one can easily see that 0 < θ 2 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq75_HTML.gif. Then it follows from (15) and Young’s inequality that
        u s + 1 k 2 C ( N , k , s ) k 2 u k + s k 2 ( 1 θ 2 ) u m + s 2 2 2 k 2 θ 2 m + s C ( N , k , s ) k 2 ( η 2 u m + s 2 2 2 + C ( η 2 ) u k + s k 2 ( 1 θ 2 ) ( m + s ) m + s k 2 θ 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ16_HTML.gif
        (16)
        where k 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq76_HTML.gif and η 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq77_HTML.gif will be determined later. If we choose
        k 2 = ( k + s ) ( m + s ) ( 1 θ 2 ) ( m + s ) + θ 2 ( k + s ) = 2 ( s + 1 ) ( k + s ) + N ( s + 1 ) ( m k ) 2 ( s + 1 ) + N ( m k ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equt_HTML.gif
        then s < k 2 < s + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq78_HTML.gif and k 2 ( 1 θ 2 ) ( m + s ) m + s k 2 θ 2 = k + s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq79_HTML.gif. We then have the inequality
        C ( N , k , s ) k 2 β C ( η 2 ) u s + 1 k 2 η 2 β C ( η 2 ) u m + s 2 2 2 + β u k + s k + s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ17_HTML.gif
        (17)
        by (16). From inequalities (14) and (17), we can also obtain the inequality
        1 s + 1 d d t u s + 1 s + 1 + [ 4 m s d ( m + s ) 2 λ C 0 2 | Ω | 1 + 2 N η 2 β C ( η 2 ) ] u m + s 2 2 2 + C ( N , k , s ) k 2 β C ( η 2 ) u s + 1 k 2 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equu_HTML.gif
        Here, we can choose η 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq80_HTML.gif and λ or | Ω | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq11_HTML.gif small enough so that 4 m s d ( m + s ) 2 λ C 0 2 | Ω | 1 + 2 N η 2 β C ( η 2 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq81_HTML.gif. Setting C 2 = C ( N , k , s ) k 2 β C ( η 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq82_HTML.gif, we have d d t u s + 1 + C 2 u s + 1 k 2 s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq83_HTML.gif from the inequality above. By Lemma 1, we obtain that
        u s + 1 [ u 0 s + 1 s + 1 k 2 C 2 ( s + 1 k 2 ) t ] 1 s + 1 k 2 , t [ 0 , T 2 ) , u s + 1 = 0 , t [ T 2 , + ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equv_HTML.gif

        where T 2 = u 0 s + 1 s + 1 k 2 C 2 ( s + 1 k 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq84_HTML.gif, which give the decay estimates in finite time for N > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq67_HTML.gif such that N 2 N + 2 m < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq85_HTML.gif.

        If 0 < m < N 2 N + 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq86_HTML.gif, one can show that there exists an embedding constant C 00 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq87_HTML.gif such that
        u s + 1 m + s | Ω | 1 ( s + 1 ) ( N + 2 ) N ( m + s ) u N ( m + s ) N 2 m + s C 00 2 | Ω | 1 ( s + 1 ) ( N + 2 ) N ( m + s ) u m + s 2 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equw_HTML.gif
        by multiplying both sides of (1) by u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq69_HTML.gif ( s > N ( 1 m ) 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq88_HTML.gif) and integrating the result over Ω, and the Sobolev embedding inequality. By using the inequality above and a similar argument as above, the following decay estimates can be obtained:
        u s + 1 [ u 0 s + 1 s + 1 k 2 C ( N , k , s ) k 2 β C ( η 2 ( s + 1 k 2 ) t ] 1 s + 1 k 2 , t ( 0 , T 2 ) , u s + 1 = 0 , t [ T 2 , + ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equx_HTML.gif
        provided that
        4 m s d ( m + s ) 2 λ C 00 2 | Ω | 1 ( s + 1 ) ( N + 2 ) N ( m + s ) + s m + 2 s + 1 η 2 β C ( η 2 ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equy_HTML.gif

        where T 2 = C ( η 2 ) u 0 s + 1 s + 1 k 2 C ( N , k , s ) k 2 β ( s + 1 k 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq89_HTML.gif. □

        Theorem 2 If 0 < k < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq49_HTML.gif, then the nonnegative nontrivial weak solution of problem (1)-(3) vanishes in finite time provided that u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq90_HTML.gif, | Ω | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq11_HTML.gif or λ is sufficiently small, and q > 2 k ( s + 1 ) + N ( m k ) 2 ( s + 1 ) + N ( m k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq91_HTML.gif, where if N = 1  or  2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq51_HTML.gif, then s = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq92_HTML.gif, and if N > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq23_HTML.gif, then s > m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq70_HTML.gif.

        Proof Assume that q 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq93_HTML.gif. If N = 1  or  2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq51_HTML.gif, multiplying both sides of (1) by u and integrating the result over Ω, we have the equation
        1 2 d d t u 2 2 + 4 m d ( m + 1 ) 2 u m + 1 2 2 2 + β u k + 1 k + 1 = λ Ω u q d x Ω u d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ18_HTML.gif
        (18)
        By (10) and (18), and using Hölder’s inequality, we get the inequality
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equz_HTML.gif
        Choosing η 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq62_HTML.gif small enough so that 4 m d ( m + 1 ) 2 η 1 β C ( η 1 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq94_HTML.gif, we obtain the inequality
        d d t u 2 + u 2 k 1 1 [ C ( N , m , k ) k 1 β C ( η 1 ) λ | Ω | 3 q 2 u 2 q k 1 + 1 ] 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equaa_HTML.gif
        Hence, we have the inequality
        d d t u 2 + C 3 u 2 k 1 1 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equab_HTML.gif
        provided that
        u 0 2 < [ C ( N , m , k ) k 1 β C ( η 1 ) λ | Ω | 3 q 2 ] 1 q k 1 + 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equac_HTML.gif
        and
        q > k 1 1 = 4 k + N ( m k ) 4 + N ( m k ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equad_HTML.gif
        where C 3 = C ( N , m , k ) k 1 β C ( η 1 ) λ | Ω | 3 q 2 u 0 2 q k 1 + 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq95_HTML.gif, from which and a similar argument as the one used in the proof of Theorem 1, the following decay estimates can be obtained:
        u 2 [ u 0 2 2 k 1 C 3 ( 2 k 1 ) t ] 1 2 k 1 , t [ 0 , T 3 ) , u 2 = 0 , t [ T 3 , + ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equae_HTML.gif

        where T 3 = u 0 2 2 k 1 C 3 ( 2 k 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq96_HTML.gif.

        If N > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq67_HTML.gif and 0 < m < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq13_HTML.gif, multiplying both sides of (1) by u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq69_HTML.gif ( s > m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq70_HTML.gif) and integrating the result over Ω, we get the equation
        1 s + 1 d d t u s + 1 s + 1 + 4 m s d ( m + s ) 2 u m + s 2 2 2 + β u k + s k + s = λ Ω u q d x Ω u s d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ19_HTML.gif
        (19)
        By (17) and (19), and using Hölder’s inequality, we obtain the inequality
        1 s + 1 d d t u s + 1 s + 1 + [ 4 m s d ( m + s ) 2 η 2 β C ( η 2 ) ] u m + s 2 2 2 + C ( N , k , s ) k 2 β C ( η 2 ) u s + 1 k 2 λ | Ω | s q + 2 s + 1 u s + 1 q + s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equaf_HTML.gif
        Choosing η 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq80_HTML.gif small enough so that 4 m s d ( m + s ) 2 η 2 β C ( η 2 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq97_HTML.gif, we have
        d d t u s + 1 + u s + 1 k 2 s [ C ( N , k , s ) k 2 β C ( η 2 ) λ | Ω | s q + 2 s + 1 u s + 1 q k 2 + s ] 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equag_HTML.gif
        Therefore, we obtain the inequality
        d d t u s + 1 + C 4 u s + 1 k 2 s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equah_HTML.gif
        provided that
        u 0 s + 1 < [ C ( N , k , s ) k 2 β C ( η 2 ) λ | Ω | s q + 2 s + 1 ] 1 q k 2 + s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equai_HTML.gif
        and
        q > k 2 s = 2 k ( s + 1 ) + N ( m k ) 2 ( s + 1 ) + N ( m k ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equaj_HTML.gif
        where C 4 = C ( N , k , s ) k 2 β C ( η 2 ) λ | Ω | s q + 2 s + 1 u 0 s + 1 q k 2 + s > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq98_HTML.gif, which yields the following decay estimates:
        u s + 1 [ u 0 s + 1 s + 1 k 2 C 4 ( s + 1 k 2 ) t ] 1 s + 1 k 2 , t [ 0 , T 4 ) , u s + 1 = 0 , t [ T 4 , + ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equak_HTML.gif

        where T 4 = u 0 s + 1 s + 1 k 2 C 4 ( s + 1 k 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq99_HTML.gif.

        Since s > m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq70_HTML.gif, we have 2 ( s + 1 ) > 2 m + 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq100_HTML.gif, and hence, if k m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq101_HTML.gif, then q > k 2 s m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq102_HTML.gif.

        Assume that q > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq103_HTML.gif. If λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq104_HTML.gif is the first eigenvalue of the boundary problem
        Δ ψ ( x ) = λ ψ ( x ) , x Ω ; ψ ( x ) = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equal_HTML.gif
        and φ 1 ( x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq105_HTML.gif, φ 1 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq106_HTML.gif, is an eigenfunction corresponding to the eigenvalue λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq104_HTML.gif, then for sufficiently small a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq107_HTML.gif, it can be easily shown that a φ 1 1 m ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq108_HTML.gif is an upper solution of problem (1)-(3) provided that u 0 ( x ) a φ 1 1 m ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq109_HTML.gif, x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq110_HTML.gif. We then have u ( x , t ) a φ 1 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq111_HTML.gif for t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq112_HTML.gif by the comparison principle. Therefore, from equation (19), we can obtain the inequality
        1 s + 1 d d t u s + 1 s + 1 + [ 4 m s d ( m + s ) 2 λ a q m C 0 2 | Ω | 1 + 2 N η 2 β C ( η 2 ) ] u m + s 2 2 2 + C ( N , k , s ) k 2 β C ( η 2 ) u s + 1 k 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equam_HTML.gif
        from which the following decay estimates can be obtained:
        u s + 1 [ u 0 s + 1 s + 1 k 2 C ( N , k , s ) k 2 β C ( η 2 ( s + 1 k 2 ) t ] 1 s + 1 k 2 , t ( 0 , T 4 ) , u s + 1 = 0 , t [ T 4 , + ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equan_HTML.gif
        provided that
        4 m s d ( m + s ) 2 λ a q m C 0 2 | Ω | 1 + 2 N η 2 β C ( η 2 ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equao_HTML.gif

        where T 4 = C ( η 2 ) u 0 s + 1 s + 1 k 2 C ( N , k , s ) k 2 β ( s + 1 k 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq113_HTML.gif. □

        Remark 2 Since the Sobolev embedding inequality cannot be used in the proof of Theorem 2, it is not necessary to consider the cases that N 2 N + 2 m < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq74_HTML.gif and 0 < m < N 2 N + 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq86_HTML.gif, when N > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq23_HTML.gif. In addition, if k m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq114_HTML.gif, the conditions in Theorem 2 imply that q > m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq115_HTML.gif.

        Theorem 3 Suppose that 0 < k < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq116_HTML.gif and m > q k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq117_HTML.gif. Then the nonnegative nontrivial weak solution of problem (1)-(3) vanishes in finite time for any nonnegative initial data provided that β is sufficiently large.

        Proof We first consider the case that N = 1  or  2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq51_HTML.gif. Multiplying both sides of (1) by u and integrating the result over Ω, and using Hölder’s inequality, we get
        1 2 d d t u 2 2 + 4 m d ( m + 1 ) 2 u m + 1 2 2 2 + β u k + 1 k + 1 = λ Ω u q d x Ω u d x λ | Ω | u q + 1 q + 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equap_HTML.gif
        By Lemma 2, we have the inequality
        u q + 1 C ( N , k , q ) u k + 1 1 θ 3 u m + 1 2 2 2 θ 3 m + 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ20_HTML.gif
        (20)
        where θ 3 = m + 1 2 ( 1 k + 1 1 q + 1 ) ( 1 N 1 2 + m + 1 2 1 k + 1 ) 1 = N ( q k ) ( m + 1 ) ( q + 1 ) [ 2 ( k + 1 ) + N ( m k ) ] [ 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq118_HTML.gif. Since q < m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq119_HTML.gif, we have m + 1 ( q + 1 ) θ 3 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq120_HTML.gif. It then follows from (20) and Young’s inequality that
        λ | Ω | u q + 1 q + 1 λ | Ω | C ( N , k , q ) q + 1 u k + 1 ( q + 1 ) ( 1 θ 3 ) u m + 1 2 2 2 ( q + 1 ) θ 3 m + 1 λ | Ω | C ( N , k , q ) q + 1 ( η 3 u m + 1 2 2 2 + C ( η 3 ) u k + 1 ( q + 1 ) ( 1 θ 3 ) ( m + 1 ) m + 1 ( q + 1 ) θ 3 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ21_HTML.gif
        (21)
        where η 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq121_HTML.gif will be determined later. From (18) and (21), one can see that
        1 2 d d t u 2 2 + [ 4 m d ( m + 1 ) 2 η 3 λ | Ω | C ( N , k , q ) q + 1 ] u m + 1 2 2 2 + β u k + 1 k + 1 C ( η 3 ) λ | Ω | C ( N , k , q ) q + 1 u k + 1 ( q + 1 ) ( 1 θ 3 ) ( m + 1 ) m + 1 ( q + 1 ) θ 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equaq_HTML.gif
        We then obtain the inequality
        1 2 d d t u 2 2 + [ 4 m d ( m + 1 ) 2 η 3 λ | Ω | C ( N , k , q ) q + 1 ] γ 2 u 2 m + 1 + u k + 1 k + 1 [ β C ( η 3 ) λ | Ω | C ( N , k , q ) q + 1 u k + 1 α 1 ] 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equar_HTML.gif
        by (6) and the inequality above, where α 1 = ( q + 1 ) ( 1 θ 3 ) ( m + 1 ) m + 1 ( q + 1 ) θ 3 ( k + 1 ) = 2 ( q k ) ( k + 1 ) 2 ( k + 1 ) + N ( m q ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq122_HTML.gif. We can choose η 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq121_HTML.gif small enough so that C 5 = 4 m d ( m + 1 ) 2 η 3 λ | Ω | C ( N , k , q ) q + 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq123_HTML.gif. Once η 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq121_HTML.gif is fixed, we may choose β large enough so that
        β C ( η 3 ) λ | Ω | C ( N , k , q ) q + 1 u k + 1 α 1 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equas_HTML.gif
        Hence, we have the inequality
        d d t u 2 + C 5 u 2 m 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equat_HTML.gif
        from which the following decay estimates can be obtained by a similar argument as the one used in the proof of Theorem 1:
        u 2 [ u 0 2 1 m C 5 ( 1 m ) t ] 1 1 m , t [ 0 , T 5 ) , u 2 = 0 , t [ T 5 , + ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equau_HTML.gif

        where T 5 = u 0 2 1 m C 5 ( 1 m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq124_HTML.gif.

        Secondly, we consider the case that N > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq67_HTML.gif. If N 2 N + 2 m < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq125_HTML.gif, multiplying both sides of (1) by u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq69_HTML.gif ( s > m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq70_HTML.gif) and integrating the result over Ω, and then using Hölder’s inequality, we get
        1 s + 1 d d t u s + 1 s + 1 + 4 m s d ( m + s ) 2 u m + s 2 2 2 + β u k + s k + s = λ Ω u q d x Ω u s d x λ | Ω | u q + s q + s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equav_HTML.gif
        By Lemma 2, it can be shown that
        u q + s C ( N , k , q , s ) u k + s 1 θ 4 u m + s 2 2 2 θ 4 m + s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ22_HTML.gif
        (22)
        where θ 4 = m + s 2 ( 1 k + s 1 q + s ) ( 1 N 1 2 + m + s 2 1 k + s ) 1 = N ( q k ) ( m + s ) ( q + s ) [ 2 ( k + s ) + N ( m k ) ] [ 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq126_HTML.gif. Since q < m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq127_HTML.gif, we have m + s ( q + s ) θ 4 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq128_HTML.gif. It then follows from (22) and Young’s inequality that
        λ | Ω | u q + s q + s λ | Ω | C ( N , k , q , s ) q + s u k + s ( q + s ) ( 1 θ 4 ) u m + s 2 2 2 ( q + s ) θ 4 m + s λ | Ω | C ( N , k , q , s ) q + s ( η 4 u m + s 2 2 2 + C ( η 4 ) u k + s ( q + s ) ( 1 θ 4 ) ( m + s m + s ( q + s ) θ 4 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equ23_HTML.gif
        (23)
        where η 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq129_HTML.gif will be determined later. From (19) and (23), one can see that
        1 s + 1 d d t u s + 1 s + 1 + [ 4 m s d ( m + s ) 2 η 4 λ | Ω | C ( N , k , q , s ) q + s ] u m + s 2 2 2 + β u k + s k + s C ( η 4 ) λ | Ω | C ( N , k , q , s ) q + s u k + s ( q + s ) ( 1 θ 4 ) ( m + s ) m + s ( q + s ) θ 4 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equaw_HTML.gif
        By (12), (13), and the inequality above, we can obtain the inequality
        1 s + 1 d d t u s + 1 s + 1 + [ 4 m s d ( m + s ) 2 η 4 λ | Ω | C ( N , k , q , s ) q + s ] C 0 2 | Ω | N 2 N m + s s + 1 u s + 1 m + s + u k + s k + s [ β C ( η 4 ) λ | Ω | C ( N , k , q , s ) q + s u k + s α 2 ] 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equax_HTML.gif
        where α 2 = ( q + s ) ( 1 θ 4 ) ( m + s ) m + s ( q + s ) θ 4 ( k + s ) = 2 ( q k ) ( k + s ) 2 ( k + s ) + N ( m q ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq130_HTML.gif. We can choose η 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq129_HTML.gif small enough so that C 6 = [ 4 m s d ( m + s ) 2 η 4 λ | Ω | C ( N , k , q , s ) q + s ] C 0 2 | Ω | N 2 N m + s s + 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq131_HTML.gif. Once η 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq129_HTML.gif is fixed, we can choose β large enough so that
        β C ( η 4 ) λ | Ω | C ( N , k , q , s ) q + s u k + s α 2 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equay_HTML.gif
        Hence, we can obtain the inequality
        d d t u s + 1 + C 6 u s + 1 m 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equaz_HTML.gif
        from which the following decay estimates can be obtained:
        u s + 1 [ u 0 s + 1 1 m C 6 ( 1 m ) t ] 1 1 m , t [ 0 , T 6 ) , u s + 1 = 0 , t [ T 6 , + ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equba_HTML.gif

        where T 6 = u 0 s + 1 1 m C 6 ( 1 m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq132_HTML.gif.

        Similarly, one can obtain the following decay estimates for 0 < m < N 2 N + 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq86_HTML.gif:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_Equbb_HTML.gif

        where T 6 = u 0 s + 1 1 m [ 4 m s d ( m + s ) 2 η 4 λ | Ω | C ( N , k , q , s ) q + s ] C 00 2 | Ω | 1 ( s + 1 ) ( N + 2 ) N ( m + s ) ( 1 m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq133_HTML.gif. □

        Remark 3 One can see from Theorems 1-3 that the extinction of nonnegative nontrivial weak solutions to problem (1)-(3) occurs when 0 < k q < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq134_HTML.gif.

        Remark 4 Theorems 1-3 all require | Ω | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq11_HTML.gif, λ, or u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-24/MediaObjects/13661_2012_Article_281_IEq90_HTML.gif to be sufficiently small or β to be sufficiently large.

        Declarations

        Acknowledgements

        The second and third authors were supported by the National Science Foundation of Shandong Province of China (ZR2012AM018) and Changwon National University in 2012, respectively. The authors would like to express their sincere gratitude to the anonymous reviewers for their insightful and constructive comments.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Pusan National University
        (2)
        School of Mathematical Sciences, Ocean University of China
        (3)
        Department of Mathematics, Changwon National University

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