Flat blow-up in one-dimensional semilinear heat equations

Abstract

Consider the Cauchy problem ut=uxx+up, x∈R, t>0, u(x,0)=u0(x), x∈R, where p>1 and u0(x) is continuous, nonnegative and bounded. Assume that u(x,t) blows up at x=0, t=T and set u(x,t)=(T−t)−1/(p−1)φ(y,τ), y=x/T−t−−−−√, τ=−ln(T−t). Here we show that there exist initial values u0(x) for which the corresponding solution is such that two maxima collapse at x=0, t=T. One then has that φ(y,τ)=(p−1)1/(p−1)−C1e−τH4(y)+o(e−τ)asτ→∞,(1) with C1>0, H4(y)=c4H˜4(y/2), where c4=(23(4π)1/4)−1, H˜4(s) is the standard 4th Hermite polynomial, and convergence in (1) takes place in Ck,αloc for any k≥1 and some α∈(0,1). We also show that in this case, limt↑T(T−t)1/(p−1)u(ξ(T−t)1/4,t)=(p−1)(1+C1c4ξn)−1/(p−1),(2) where the convergence is uniform on sets |ξ|≤R with R>0. This asymptotic behaviour is different (and flatter) than that corresponding to solutions spreading from data u0(x) having a single maximum, in which case (3)φ(y,τ)=(p−1)−1/(p−1)−(4π)1/4(p−1)−1/(p−1)2√p⋅H2(y)τ+o(1τ) as τ→∞, and limt↑T(T−t)1/(p−1)u(ξ(T−t)1/2|ln(T−t)|1/2,t)=(p−1)−1/(p−1)(1+(p−1)4pξ2)−1/(p−1).(4)