Logarithmic corrections in Fisher–KPP type porous medium equations

Abstract

We consider the large time behavior of solutions to the porous medium equation with a Fisher–KPP type reaction term and nonnegative, compactly supported initial function in L∞(RN)∖{0}:(⋆)ut=Δum+u−u2 in Q:=RN×R+,u(⋅,0)=u0in RN, with m>1. It is well known that the spatial support of the solution u(⋅,t) to this problem remains bounded for all time t>0 (whose boundary is called the free boundary), which is a main different feature of (⋆) to the corresponding semilinear case m=1. Similar to the corresponding semilinear case m=1, it is known that there is a minimal speed c⁎>0 such that for any c≥c⁎, the equation admits a wavefront solution Φc(r): For any ν∈SN−1, v(x,t):=Φc(x⋅ν−ct) solves vt=Δvm+v−v2. When m=1, it is well known that the long-time behavior of the solution with compact initial support can be well approximated by Φc⁎(|x|−c⁎t+N+2c⁎log⁡t+O(1)), and the term N+2c⁎log⁡t is known as the logarithmic correction term. When m>1, an analogous approximation has been an open question for N≥2. In this paper, we answer this question by showing that there exists a constant c#>0 independent of the dimension N and the initial function u0, such that for all large time, any solution of (⋆) is well approximated by Φc⁎(|x|−c⁎t+(N−1)c#log⁡t+O(1)). This is achieved by a careful analysis of the radial case, where the initial function u0 is radially symmetric, which enables us to give a formula for c# (involving integrals of Φc⁎(r)), and to replace the O(1) term by C+o(1) with C a constant depending on u0. The approximation for the general non-radial case is obtained by using the radial results and simple comparison arguments. We note that in sharp contrast to the m=1 case, when N=1, there is no logarithmic correction term for (⋆).