LONG-TIME BEHAVIOR OF SOLUTIONS OF THE FAST DIFFUSION EQUATIONS WITH CRITICAL ABSORPTION TERMS

Abstract

This paper is devoted to the long-time behavior of solutions to the Cauchy problem of the porous medium equation $u_t=\Delta(u^m)-u^p$ in $\mathbb{R}^n\times (0,\infty)$ with $(1-2/n)_{+}<m<1$ and the critical exponent $p=m+2/n$. For the strictly positive initial data $u(x,0)=O(1+|x|)^{-k}$ with $n+mn(2-n+nm)/(2[2-m+mn(1-m)])\leq k<2/(1-m)$, we prove that the solution of the above Cauchy problem converges to a fundamental solution of $u_t=\Delta(u^m)$ with an additional logarithmic anomalous decay exponent in time as $t\to \infty$.