Fred Weissler : Sign-changing solutions of the nonlinear heat equation with positive initial value

Posted by Jerry Bona , part of the Analysis and Applied Mathematics Seminar.

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Oct. 22, 2018, 4 p.m.

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Abstract

We consider the nonlinear heat equation $u_t - \Delta u = |u|^\alpha u$ on ${\mathbb R}^N$, where $\alpha >0$. It is well known that the Cauchy problem is locally well-posed in a variety of spaces. For instance, for every $\alpha >0$, it is well-posed in the space $C_0 ( {\mathbb R}^N )$ of continuous functions that converge to $0$ at infinity. It is also well-posed in $L^p({\mathbb R}^N )$ for $p\ge 1$, $p>\frac {N\alpha } {2}$, but not well-posed in $L^p$ for $1\le p< \frac {N\alpha } {2}$ if $\alpha >\frac {2} {N}$. In particular, for such $p$ there exist positive initial values $u_0 \in L^p$ for which there is no local in time positive solution. Also, if one considers the initial value $u_0 (x)= c |x|^{-\frac {2} {\alpha }}$ for all $x\in {\mathbb R}^N \setminus \{0\}$, with $c>0$, it is known that if $c$ is small, there exists a global in time (positive) solution with $u_0$ as initial value, and in fact this solution is self-similar. On the other hand, if $c$ is large, there is no local in time positive solution, self-similar or otherwise. We prove that in the range $0 < \alpha <\frac {4} {N-2}$, for every $c>0$, there exist infinitely many self-similar solutions to the Cauchy problem with initial value $u_0 (x)= c |x|^{-\frac {2} {\alpha }}$. Of course, these solutions are all sign-changing if $c$ is sufficiently large. Also, in the range $\frac {2} {N}< \alpha <\frac {4} {N-2}$, we prove the existence of local in time sign-changing solutions for a class of nonnegative initial values $u_0 \in L^p$, for $1\le p< \frac {N\alpha } {2}$, for which no local in time positive solution exists. This is joint work with T. Cazenave, F. Dickstein and I. Naumkin.