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}: <br/> (*) ut = Δum + u − u2 in Q := RN × R+, u(·,0) = u0 in RN, (*) <br/> 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+2 c* log t + O(1)), and the term N+2 c* 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 (*).