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arXiv · 1706.01403

Sign-changing self-similar solutions of the nonlinear heat equation with positive initial value

Thierry Cazenave·Flávio Dickstein·Ivan Naumkin·Fred B. Weissler
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Abstract

We consider the nonlinear heat equation $u_t - Δu = |u|^αu$ on ${\mathbb R}^N$, where $α>0$ and $N\ge 1$. We prove that in the range $0 < α<\frac {4} {N-2}$, for every $μ>0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0 (x)= μ|x|^{-\frac {2} {α}}$. The construction is based on the analysis of the related inverted profile equation. In particular, we construct (sign-changing) self-similar solutions for positive initial values for which it is known that there does not exist any local, nonnegative solution.

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Thierry Cazenave, Flávio Dickstein, Ivan Naumkin, Fred B. Weissler. 2020-05-28. Sign-changing self-similar solutions of the nonlinear heat equation with positive initial value. https://doi.org/10.1353/ajm.2020.0037

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