Non-existence of local solutions for semilinear heat equations of Osgood type

TL;DR

This paper proves that the Osgood condition, which guarantees global solutions for certain ODEs and bounded initial data, does not ensure local solutions for semilinear heat equations with less regular initial data.

Contribution

It demonstrates the non-existence of local solutions for semilinear heat equations under Osgood conditions with initial data in $L^q$ spaces, extending understanding of solution behavior.

Findings

Osgood condition guarantees global solutions for bounded initial data.

Existence of initial conditions with no local solutions in $L^1_{loc}$.

Counterexample showing Osgood condition is not sufficient for $L^q$ initial data.

Abstract

We establish non-existence results for the Cauchy problem of some semilinear heat equations with non-negative initial data and locally Lipschitz, nonnegative source term $f$. Global (in time) solutions of the scalar ODE $\dot v=f(v)$ exist for $v(0)>0$ if and only if the Osgood-type condition $\int_{1}^{\infty}\frac{\dee s}{f(s)} =\infty$ holds; by comparison this ensures the existence of global classical solutions of $u_t=\Delta u+f(u)$ for bounded initial data $u_0\in L^{\infty}(\R^n)$. It is natural to ask whether the Osgood condition is sufficient to ensure that the problem still admits global solutions if the initial data is in $L^q(\R^n)$ for some $1\le q<\infty$. Here we answer this question in the negative, and in fact show that there are initial conditions for which there exists no local solution in $L^1_{\rm loc}(\R^n)$ for $t>0$.