Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 87650, 8 pages doi:10.1155/2007/87650 Research Article Extinction and Decay Estimates of Solutions for a Class of Porous Medium Equations Wenjun Liu, Mingxin Wang, and Bin Wu Received 3 April 2007; Accepted 6 September 2007 Recommended by Michel Chipot The extinction phenomenon of solutions for the homogeneous Dirichlet boundary value problem of the porous medium equation u t = Δu m + λ|u| p−1 u −βu,0 <m<1, is studied. Sufficient conditions about the extinction and decay estimates of solutions are obtained by using L p -integral model estimate methods and two crucial lemmas on differential in- equality. Copyright © 2007 Wenjun Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and main results This paper is devoted to the extinction and decay estimates for the porous medium equa- tion u t = Δu m + λ|u| p−1 u −βu, x ∈ Ω, t>0, (1.1) u(x,t) = 0, x ∈ ∂Ω, t>0, (1.2) u(x,0) = u 0 (x) ≥ 0, x ∈ Ω, (1.3) with 0 <m<1andp,λ,β>0, where Ω ⊂ R N (N>2) is a bounded domain with smooth boundary. The phenomenon of extinction is an important property of solutions for many evo- lutionary equations which have been studied extensively by many researchers. Especially, there are also some papers concerning the extinction for the porous medium equation. For instance, in [1–3], the authors studied the extinction and large-time behavior of solu- tion of (1.1)forthecaseβ = 0andλ<0; and in [4], the authors obtained conditions for the extinction of solutions of (1.1) without absorption by using sub- and supersolution 2 Journal of Inequalities and Applications methods and an eigenfunction argument. But as far as we know, few works are concerned with the decay estimates of solutions for the porous medium equation. The existence and uniqueness of nonnegative solution for problem (1.1)–(1.3)have been studied in [ 5, 6]. The purpose of the present paper is to establish sufficient condi- tions about the extinction and decay estimations of solutions for problem (1.1)–(1.3). For the proof of our result, we employ L p -integral model estimate methods and two crucial lemmas on differential inequality. Our main results read as follows. Theorem 1.1. Assume that 0 ≤ u 0 (x) ∈ L ∞ (Ω) ∩ W 1,2 0 (Ω), 0 <m= p<1,andλ 1 is the first eigenvalue of −Δψ(x) = λψ(x), ψ| ∂Ω = 0, (1.4) and ϕ 1 (x) ≥ 0 with ϕ 1 ∞ = 1 is the eigenfunction corresponding to the eigenvalue λ 1 . (1) If λ<4mλ 1 /(m +1) 2 , then the weak solution of problem (1.1)–(1.3) vanishes in the sense of · 2 as t →∞. (2) If (N − 2)/(N +2)≤ m<1 with λ<λ 1 or 0 <m<(N − 2)/(N +2)with λ<λ ∗ , then the weak solution of problem (1.1)–(1.3) vanishes in finite time, and u(·,t) m+1 ≤ u 0 1−m m+1 + C 1 β e (m−1)βt − C 1 β 1/(1−m) , N − 2 N +2 ≤ m<1, u(·,t) r+1 ≤ u 0 1−m r+1 + C 2 β e (m−1)βt − C 2 β 1/(1−m) ,0<m< N − 2 N +2 , (1.5) for t ∈ [0,T ∗ ),where 0 <T ∗ ≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ T 1 , (N − 2) (N +2) ≤ m<1, T 2 ,0<m< (N − 2) (N +2) , (1.6) r = N(1 − m) 2 − 1, λ ∗ = (r + m) 2 λ 4rm <λ 1 , (1.7) and C 1 , C 2 , T 1 ,andT 2 are given by ( 2.18 ), (2.24), (2.20), and (2.26),respectively. Wenjun Liu et al. 3 Theorem 1.2. Let 0 <m<1, m<p. Then the weak solution of problem (1.1)–(1.3) vanishes in finite time, and u(·,t) m+1 ≤ B 1 e −α 1 t , t ∈ 0,T 01 , u(·,t) m+1 ≤ u(·,T 01 1−m m+1 + C 3 β e (m−1)β(t−T 01 ) − C 3 β 1/(1−m) , t ∈ T 01 ,T 3 , u(·,t) m+1 ≡ 0, t ∈ T 3 ,+∞ , (1.8) for (N − 2)/(N +2)≤ m<1, u(·,t) r+1 ≤ B 2 e −α 2 t , t ∈ 0,T 02 , u(·,t) r+1 ≤ u(·,T 02 1−m r+1 + C 4 β e (m−1)β(t−T 02 ) − C 4 β 1/(1−m) , t ∈ T 02 ,T 4 , u(·,t) r+1 ≡ 0, t ∈ T 4 ,+∞ , (1.9) for 0 <m<(N − 2)/(N +2),whereC 3 , C 4 , T 3 ,andT 4 are given by (2.29), (2.34), (2.31), and (2.36), respectively. To obtain the above results, we will use the follow ing lemmas which are of crucial importance in the proofs of decay estimates. Lemma 1.3 [7]. Let y(t) ≥ 0 be a solution of the differential inequality dy dt + Cy k + βy ≤ 0(t ≥ 0), y T 0 ≥ 0, (1.10) where C>0 is a constant and k ∈ (0,1). Then one has the decay estimate y(t) ≤ y T 0 1−k + C β e (k−1)β(t−T 0 ) − C β 1/(1−k) , t ∈ T 0 ,T ∗ , y(t) ≡ 0, t ∈ T ∗ ,+∞ , (1.11) where T ∗ = (1/(1 − k)β)ln(1+(β/C)y(T 0 ) 1−k ). Lemma 1.4 [8]. Let 0 <k<p,andlety(t) ≥ 0 be a solution of the differential inequality dy dt + Cy k + βy ≤ γy p (t ≥ 0), y(0) ≥ 0, (1.12) where C,γ>0 and k ∈ (0,1). Then there ex ist α>β, B>0, such that 0 ≤ y(t) ≤ Be −αt , t ≥ 0. (1.13) 4 Journal of Inequalities and Applications 2. Proofs of theorems In this section, we will give detailed proofs for our result. Let · p and · 1,p denote L p (Ω)andW 1,p (Ω) norms, respectively, 1 ≤ p ≤∞. 2.1. Proof of Theorem 1.1. (1) First of all, we show that u(·,t) ∞ ≤ u 0 (·) ∞ := M. (2.1) Multiplying (1.1)by(u − M) + and integrating over Ω,weobtain 1 2 d dt Ω (u − M) 2 + dx + A M (t) ∇u m ·∇udx ≤ λ Ω u m (u − M) + dx − β Ω u(u − M) + dx ≤ λ A M (t) u m+1 dx, (2.2) where A M (t) ={x ∈ Ω | u(x,t) >M}.Sinceλ 1 is the first eigenvalue, then we have Ω ∇u m ·∇udx ≥ 4m (m +1) 2 λ 1 Ω u m+1 dx, (2.3) for any u ∈ W 1,2 0 (Ω). We fur ther have A M (t) ∇u m ·∇udx ≥ 4m (m +1) 2 λ 1 A M (t) u m+1 dx. (2.4) Therefore, we have d dt Ω (u − M) 2 + dx ≤ 0. (2.5) Since Ω (u 0 − M) 2 + dx = 0, it follows that Ω (u − M) 2 + dx ≡ 0, ∀t ≥ 0, (2.6) which implies that u(·,t) ∞ ≤u 0 (·) ∞ . Multiplying (1.1)byu and integrating over Ω,weconcludethat 1 2 d dt Ω u 2 dx + Ω ∇u m ·∇udx ≤ λ Ω u m+1 dx − β Ω u 2 dx. (2.7) We further have 1 2 d dt Ω u 2 dx + mλ 1 − λ Ω u m+1 dx + β Ω u 2 dx ≤ 0. (2.8) Let v = u/M. Then, we have d dt Ω v 2 dx +2M m−1 mλ 1 − λ Ω v m+1 dx +2βM m−1 Ω v 2 dx ≤ 0. (2.9) Wenjun Liu et al. 5 Since 0 <m<1, we have d dt Ω v 2 dx +2M m−1 mλ 1 − λ + β Ω v 2 dx ≤ 0, (2.10) which implies that Ω v 2 dx ≤ e −2M m−1 mλ 1 −λ+β t Ω v 2 0 dx, (2.11) that is, Ω u 2 dx ≤ e −2u 0 m−1 ∞ mλ 1 −λ+β t Ω u 2 0 dx. (2.12) Therefore, we conclude that u(·,t) 2 → 0ast →∞. (2) We consider first the case (N − 2)/(N +2)≤ m<1. Multiplying (1.1)byu m and integrating over Ω,wehave[9] 1 m +1 d dt u m+1 m+1 + u m 2 1,2 = λu 2m 2m − βu m+1 m+1 . (2.13) Noticing that λ 1 = inf v∈W 1,2 0 (Ω),v=0 Ω |∇v| 2 dx/ Ω v 2 dx,weobtain 1 m +1 d dt u m+1 m+1 + 1 − λ λ 1 u m 2 1,2 + βu m+1 m+1 ≤ 0. (2.14) By the H ¨ older inequality, we have u m+1 m+1 = Ω 1 · u m · udx ≤|Ω| m/(m+1)−(N−2)/2N u m 2N/(N−2) u m+1 . (2.15) The embedding theorem g ives that u m m+1 ≤|Ω| m/(m+1)−(N−2)/2N u m 2N/(N−2) ≤ C 0 |Ω| m/(m+1)−(N−2)/2N u m 1,2 , (2.16) where C 0 is the embedding constant. By (2.14)–(2.16), we obtain the differential inequal- ity d dt u m+1 + C 1 u m m+1 + βu m+1 ≤ 0, (2.17) where C 1 = C −2 0 |Ω| (N−2)/N−2m/(m+1) 1 − λ λ 1 . (2.18) Setting y(t) =u(·,t) m+1 , y(0) =u 0 (·) m+1 ,byLemma 1.3,weobtain u m+1 ≤ u 0 1−m m+1 + C 1 β e (m−1)βt − C 1 β 1/(1−m) , t ∈ 0,T 1 , u m+1 ≡ 0, t ∈ T 1 ,+∞ , (2.19) 6 Journal of Inequalities and Applications where T 1 = 1 (1 − m)β ln 1+ β C 1 u 0 1−m m+1 . (2.20) We now turn to the case 0 <m<(N − 2)/(N +2)withλ<λ ∗ = (r + m) 2 λ/4rm < λ 1 . Multiplying (1.1)byu r (r = N(1 − m)/2 − 1) and integr ating over Ω,wehave 1 r +1 d dt u r+1 r+1 + 4rm (r + m) 2 − λ λ 1 u (r+m)/2 2 1,2 + βu r+1 r+1 ≤ 0. (2.21) By the embedding theorem and the specific choice of r,weobtain u (r+m)/2 r+1 = Ω u ((r+m)/2)·(2N/(N−2)) dx (N−2)/2N ≤ C 0 u (r+m)/2 1,2 . (2.22) Therefore, d dt u r+1 + C 2 u m r+1 + βu r+1 ≤ 0, (2.23) where C 2 = C −2 0 4rm (r + m) 2 − λ λ 1 > 0. (2.24) Setting y(t) =u(·,t) r+1 , y(0) =u 0 (·) r+1 ,byLemma 1.3,weobtain u r+1 ≤ u 0 1−m r+1 + C 2 β e (m−1)βt − C 2 β 1/(1−m) , t ∈ 0,T 2 , u r+1 ≡ 0, t ∈ T 2 ,+∞ , (2.25) where T 2 = 1 (1 − m)β ln 1+ β C 2 u 0 1−m m+1 . (2.26) 2.2. Proof of Theorem 1.2. We consider first the case p ≤ 1. When (N − 2)/(N +2)≤ m<1, multiplying (1.1)byu m , and by the embedding theorem and the H ¨ older inequality, we can easily obtain d dt u m+1 + C −2 0 Ω| (N−2)/N−2m/(m+1) u m m+1 + βu m+1 ≤ λ|Ω| 1−(m+p)/(m+1) u p m+1 . (2.27) By Lemma 1.4, there exist α 1 >β, B 1 > 0, such that 0 ≤u m+1 ≤ B 1 e −α 1 t , t ≥ 0. (2.28) Wenjun Liu et al. 7 Furthermore, there exist T 01 ,suchthat C −2 0 |Ω| (N−2)/N−2m/(m+1) − λ|Ω| 1−(m+p)/(m+1) u p−m m+1 ≥ C −2 0 |Ω| (N−2)/N−2m/(m+1) − λ|Ω| 1−(m+p)/(m+1) B 1 e −α 1 T 01 p−m := C 3 > 0 (2.29) holds for t ∈ [T 01 ,+∞). Therefore, (2.27)turnsto d dt u m+1 + C 3 u m m+1 + βu m+1 ≤ 0. (2.30) By Lemma 1.3, we can obtain the desire decay estimate for T 3 = 1 (1 − m)β ln 1+ β C 3 u · ,T 01 1−m m+1 . (2.31) For the case 0 <m<(N − 2)/(N + 2), we multiply (1.1)byu r and obtain d dt u r+1 + C −2 0 4rm (r + m) 2 u m r+1 + βu r+1 ≤ λ|Ω| 1−(r+p)/(r+1) u p r+1 . (2.32) By Lemma 1.4, there exist α 2 >β, B 2 > 0, such that 0 ≤u r+1 ≤ B 2 e −α 2 t , t ≥ 0. (2.33) Furthermore, there exist T 02 ,suchthat C −2 0 4rm (r + m) 2 − λ|Ω| 1−(r+p)/(r+1) u p−m r+1 ≥ C −2 0 4rm (r + m) 2 − λ|Ω| 1−(r+p)/(r+1) B 2 e −α 2 T 02 p−m := C 4 > 0 (2.34) holds for t ∈ [T 02 ,+∞). Therefore, (2.32)turnsto d dt u r+1 + C 4 u m r+1 + βu r+1 ≤ 0. (2.35) By Lemma 1.3, we can obtain the desire decay estimate for T 4 = 1 (1 − m)β ln 1+ β C 4 u · ,T 02 1−m r+1 . (2.36) For the case p>1, we can rewrite (2.27)and(2.32)as(e.g.,(2.27)) d dt u m+1 + C −2 0 |Ω| (N−2)/N−2m/(m+1) u m m+1 + βu m+1 ≤ λk p−1 u m+1 m+1 (2.37) since kϕ 1/m 1 (x)isasupersolutionofproblem(1.1 )–(1.3), where ϕ 1 (x)isgiveninTheorem 1.1. The above argument can also be applied, and hence we omit it. 8 Journal of Inequalities and Applications Acknowledgments This work was supported by the National NSF of China (10471022), the NSF of Jiangsu Province (BK2006088), the NSF of Jiangsu Province Education Department (07KJD510133), and the Science Research Foundation of NUIST. References [1] R. Ferreira and J. L. Vazquez, “Extinction behaviour for fast diffusion equations with absorp- tion,” Nonlinear Analysis: Theory, Methods & Applications, vol. 43, no. 8, pp. 943–985, 2001. [2] G. Leoni, “A very singular solution for the porous media equation u t = Δ(u m ) − u p when 0 < m<1,” Journal of Differential Equations, vol. 132, no. 2, pp. 353–376, 1996. [3] L. A. Peletier and J. N. Zhao, “Source-ty pe solutions of the porous media equation with absorp- tion: the fast diffusion case,” Nonlinear Analysis: Theory, Methods & Applications, vol. 14, no. 2, pp. 107–121, 1990. [4] Y. Li and J. Wu, “Extinction for fast diffusion equations with nonlinear sources,” Electronic Jour- nal of Differential Equations, vol. 2005, no. 23, pp. 1–7, 2005. [5] J. R. Anderson, “Local existence and uniqueness of solutions of degenerate parabolic equations,” Communications in Partial Differential Equations, vol. 16, no. 1, pp. 105–143, 1991. [6] J. R. Anderson, “Necessary and sufficient conditions for the unique solvability of a nonlinear reaction-diffusion model,” Journal of Mathematical Analysis and Applications, vol. 228, no. 2, pp. 483–494, 1998. [7] S. L. Chen, “The extinction behavior of the solutions for a class of reaction-diffusion equations,” Applied Mathematics and Mechanics, vol. 22, no. 11, pp. 1352–1356, 2001. [8] S. L. Chen, “The extinction behavior of solutions for a reaction-diffusion equation,” Journal of Mathematical Research and Exposition, vol. 18, no. 4, pp. 583–586, 1998 (Chinese). [9] W. Liu, “Periodic solutions of evolution m-laplacian equations with a nonlinear convec- tion term,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 27368, 10 pages, 2007. Wenjun Liu: Department of Mathematics, Southeast University, Nanjing 210096, China; College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China Email address: lwjboy@126.com Mingxin Wang: Department of Mathematics, Southeast University, Nanjing 210096, China Email address: mxwang@seu.edu.cn Bin Wu: College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China Email address: wubing790831@126.com . Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 87650, 8 pages doi:10.1155/2007/87650 Research Article Extinction and Decay Estimates of Solutions. Journal of Inequalities and Applications methods and an eigenfunction argument. But as far as we know, few works are concerned with the decay estimates of solutions for the porous medium equation. The. with a nonlinear convec- tion term,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 27368, 10 pages, 2007. Wenjun Liu: Department of Mathematics, Southeast