Sign-changing solutions of the nonlinear heat equation with persistent singularities

Abstract

We study the existence of sign-changing solutions to the nonlinear heat equation ${\partial }_{t}u=\Delta u+|u{|}^{\alpha }u$ on ${\mathbb{R}}^N$, $N\ge 3$, with $\frac{2}{N-2}<\alpha <{\alpha }_0$, where ${\alpha }_0=\frac{4}{N-4+2\sqrt{N-1}}\in (\frac{2}{N-2},\frac{4}{N-2})$, which are singular at $x=0$ on an interval of time. In particular, for certain $\mu >0$ that can be arbitrarily large, we prove that for any $u_0\in {\mathrm{L}}_{\mathrm{loc}}^{\infty }({\mathbb{R}}^N\setminus \{0\})$ which is bounded at infinity and equals $\mu |x{|}^{-\frac{2}{\alpha }}$ in a neighborhood of $0$, there exists a local (in time) solution $u$ of the nonlinear heat equation with initial value $u_0$, which is sign-changing, bounded at infinity and has the singularity $\beta |x{|}^{-\frac{2}{\alpha }}$ at the origin in the sense that for $t>0$, $|x{|}^{\frac{2}{\alpha }}u(t,x)\to \beta$ as $|x|\to 0$, where $\beta =\frac{2}{\alpha }(N-2-\frac{2}{\alpha })$. These solutions in general are neither stationary nor self-similar.