Thresholds on growth of nonlinearities and singularity of initial functions for semilinear heat equations

TL;DR

This paper investigates the conditions under which solutions exist or do not exist for semilinear heat equations with nonlinearities and initial functions that may be singular, providing sharp criteria and classifications.

Contribution

It establishes sharp integrability conditions for initial data and classifies existence and nonexistence for a broad class of nonlinearities, including critical and doubly critical cases.

Findings

Sharp integrability conditions for initial data in critical cases

Complete classification for nonlinearities of the form u^{1+2/N} (log(u+e))^{β}

Extension of results to the closure of bounded uniformly continuous functions in L^r_{ul}( abla)

Abstract

Let $N\ge 1$ and let $f\in C[0,\infty)$ be a nonnegative nondecreasing function and $u_0$ be a possibly singular nonnegative initial function. We are concerned with existence and nonexistence of a local in time nonnegative solution in a uniformly local Lebesgue space of a semilinear heat equation \[ \begin{cases} \partial_tu=\Delta u+f(u) & \textrm{in}\ \mathbb{R}^N\times(0,T),\\ u(x,0)=u_0(x) & \textrm{in}\ \mathbb{R}^N \end{cases} \] under mild assumptions on $f$. A relationship between a growth of $f$ and an integrability of $u_0$ is studied in detail. Our existence theorem gives a sharp integrability condition on $u_0$ in a critical and subcritical cases, and it can be applied to a regularly or rapidly varying function $f$. In a doubly critical case existence and nonexistence of a nonnegative solution can be determined by special treatment. When $f(u)=u^{1+2/N}[\log(u+e)]^{\beta}$, a complete classification of existence and nonexistence of a nonnegative solution is obtained. We also show that the same characterization as in Laister et. al. [11] is still valid in the closure of the space of bounded uniformly continuous functions in the space $L^r_{\rm ul}(\mathbb{R}^N)$. Main technical tools are a monotone iterative method, $L^p$-$L^q$ estimates, Jensen's inequality and differential inequalities.