Sign-changing self-similar 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 $\alpha >0$ and $N\ge 1$. We prove that in the range $0<\alpha <\frac{4}{N-2}$, for every $\mu >0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0(x)=\mu |x{|}^{-\frac{2}{\alpha }}$. 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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