Analysis and Applied Mathematics Seminar
Fred Weissler
Universite de Paris Nord
Sign-changing solutions of the nonlinear heat equation with positive initial value
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.
Monday October 22, 2018 at 4:00 PM in 636 SEO