Extinction and decay estimates of solutions for a polytropic filtration equation with the nonlocal source and interior absorption

This paper deals with the extinction properties of solutions for the homogeneous Dirichlet boundary value problem with the nonlocal source and interior absorption where m,λ,k,q > 0, 0 < m(p − 1) < 1, r ≤ 1, and . By using Lp-integral norm estimate method, we obtain the sufficient conditions of extinction solutions. Moreover, we also give the precise decay estimates of the extinction solutions. Copyright © 2012 John Wiley & Sons, Ltd.

[1]  Wenjie Gao,et al.  Extinction for a fast diffusion equation with a nonlinear nonlocal source , 2011 .

[2]  A. S. Kalashnikov.  The propagation of disturbances in problems of non-linear heat conduction with absorption , 1974 .

[3]  Shijie Dong,et al.  Periodic solutions for a delayed predator-prey system with dispersal and impulses , 2005 .

[4]  M. Nakao,et al.  Global Existence and Gradient Estimates for the Quasilinear Parabolic Equations of m-Laplacian Type with a Nonlinear Convection Term , 2000 .

[5]  Andrey Shishkov,et al.  Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential , 2009, 0903.4351.

[6]  Yoshie Sugiyama,et al.  Extinction, decay and blow-up for Keller–Segel systems of fast diffusion type , 2011 .

[7]  Chunhua Jin,et al.  Critical extinction and blow‐up exponents for fast diffusive p‐Laplacian with sources , 2007 .

[8]  Zhong Bo Fang,et al.  Extinction behavior of solutions for the p-Laplacian equations with nonlocal sources , 2012 .

[9]  À. Calsina.  NON-LOCAL REACTION-DIFFUSIO N EQUATIONS MODELLIN G PREDATOR-PREY COEVOLUTIO N , 1994 .

[10]  M. Grinfeld,et al.  Local vs. non-local interactions in population dynamics , 1989 .

[11]  Chunlai Mu,et al.  Extinction and non-extinction for a p-Laplacian equation with nonlinear source ☆ , 2008 .

[12]  Wenjie Gao,et al.  Extinction and non-extinction for a polytropic filtration equation with a nonlocal source , 2013 .

[13]  Jesús Ildefonso Díaz Díaz,et al.  Energy Methods for Free Boundary Problems , 2002 .

[14]  Chunhua Jin,et al.  Critical extinction and blow-up exponents for fast diffusive polytropic filtration equation with sources , 2009, Proceedings of the Edinburgh Mathematical Society.

[15]  Bin Wu,et al.  A note on extinction for fast diffusive p-Laplacian with sources , 2008 .

[16]  Andrey Shishkov,et al.  Long-time extinction of solutions of some semilinear parabolic equations , 2007, 0805.2314.

[17]  Bin Wu,et al.  Extinction and Decay Estimates of Solutions for a Class of Porous Medium Equations , 2007 .

[18]  G. Fragnelli,et al.  Coexistence and optimal control problems for a degenerate predator–prey model , 2011 .

[19]  A. Friedman,et al.  Extinction properties of semilinear heat equations with strong absorption , 1987 .

[20]  Chunlai Mu,et al.  CRITICAL BLOW-UP AND EXTINCTION EXPONENTS FOR NON-NEWTON POLYTROPIC FILTRATION EQUATION WITH SOURCE , 2009 .

[21]  Sergey Shmarev,et al.  Vanishing solutions of anisotropic parabolic equations with variable nonlinearity , 2010 .

[22]  Gao Wen-jie,et al.  Extinction and positivity for the evolution p-Laplacian equation in Rn , 2005 .

[23]  Wenjun Liu,et al.  Extinction properties of solutions for a class of fast diffusive p -Laplacian equations , 2011 .