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

## TL;DR

This paper constructs infinitely many sign-changing self-similar solutions to a nonlinear heat equation with positive initial data, revealing complex solution behaviors in certain parameter ranges.

## Contribution

It demonstrates the existence of infinitely many sign-changing self-similar solutions for the nonlinear heat equation with positive initial values in a specific parameter range.

## Key findings

- Existence of infinitely many sign-changing solutions.
- Solutions constructed for initial data where nonnegative solutions do not exist.
- Analysis based on the inverted profile equation.

## 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.