LONG-TIME BEHAVIOR OF SOLUTIONS OF THE FAST DIFFUSION EQUATIONS WITH CRITICAL ABSORPTION TERMS
Peihu Shi, Mingxin Wang · Journal of the London Mathematical Society · 2006
This paper is devoted to the long-time behavior of solutions to the Cauchy problem of the porous medium equation ut = Δ(um) − up in Rn × (0,∞) 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)]) ⩽ k < 2/(1 − m), we prove that the solution of the above Cauchy problem converges to a fundamental solution of ut = Δ(um) with an additional logarithmic anomalous decay exponent in time as t → ∞.