The nonlinear heat equation involving highly singular initial values and new blowup and life span results

TL;DR

This paper establishes local existence and blowup criteria for solutions to a nonlinear heat equation with highly singular initial data, extending previous results to new parameter ranges and initial conditions.

Contribution

It introduces new local existence results for highly singular initial values and develops novel blowup criteria and solutions for the nonlinear heat equation.

Findings

Proves local existence with singular initial data.

Establishes new blowup criteria for certain lpha ranges.

Constructs solutions that blow up in finite time with small initial data.

Abstract

In this paper we prove local existence of solutions to the nonlinear heat equation $u_t = \Delta u +a |u|^\alpha u, \; t\in(0,T),\; x=(x_1,\,\cdots,\, x_N)\in {\mathbb R}^N,\; a = \pm 1,\; \alpha>0;$ with initial value $u(0)\in L^1_{\rm{loc}}\left({\mathbb R}^N\setminus\{0\}\right)$, anti-symmetric with respect to $x_1,\; x_2,\; \cdots,\; x_m$ and $|u(0)|\leq C(-1)^m\partial_{1}\partial_{2}\cdot \cdot \cdot \partial_{m}(|x|^{-\gamma})$ for $x_1>0,\; \cdots,\; x_m>0,$ where $C>0$ is a constant, $m\in \{1,\; 2,\; \cdots,\; N\},$ $0<\gamma<N$ and $0<\alpha<2/(\gamma+m).$ This gives a local existence result with highly singular initial values. As an application, for $a=1,$ we establish new blowup criteria for $0<\alpha\leq 2/(\gamma+m)$, including the case $m=0.$ Moreover, if $(N-4)\alpha<2,$ we prove the existence of initial values $u_0 = \lambda f,$ for which the resulting solution blows up in finite time $T_{\max}(\lambda f),$ if $\lambda>0$ is sufficiently small. We also construct blowing up solutions with initial data $\lambda_n f$ such that $\lambda_n^{[({1\over \alpha}-{\gamma+m\over 2})^{-1}]}T_{\max}(\lambda_n f)$ has different finite limits along different sequences $\lambda_n\to 0$. Our result extends the known "small lambda" blow up results for new values of $\alpha$ and a new class of initial data.