Long Time Behavior of Solutions to P-Laplacian Equation with Absorption

Abstract

In this paper, we study the long time behavior of solutions to the Cauchy problem of $u_t = \mbox{div} ( |\nabla u|^{p-2} \nabla u ) -u^q$ in $R^n \times (0, \infty)$, with nonnegative initial value $ u(x,0) = \phi(x)$ in $R^n$, where $ (2n)/(n+1) < p < 2$ and q > 1. For initial data of various decay rates, especially the critical decay $\phi=O(|x|^{-\mu})$ with $\mu=p/(q+1-p)$, we show that the solution converges as $ t \rightarrow \infty$ to a self-similar solution. This extends the recent result of Escobedo, Kavian, and Matano for the semilinear case of p = 2. Here an essential role is played by singular and very singular self-similar solutions established in our previous works [X. Chen, Y. Qi, and M. Wang, J. Differential Equations, 190 (2003), pp. 1--15; X. Chen, Y. Qi, and M. Wang, preprint, Department of Mathematics, HKUST, Hong Kong, 1998].