Gaussian lower bounds on the Dirichlet heat kernel and non-existence of local solutions for semilinear heat equations of Osgood type

Abstract

We give a simple proof of a lower bound for the Dirichlet heat kernel in terms of the Gaussian heat kernel. Using this we establish a non-existence result for semilinear heat equations with zero Dirichlet boundary conditions and initial data in $L^q(\Omega )$ when the source term $f$ is non-decreasing and ${\mathrm{limsup}}_{s\to \infty }{s}^{-\gamma }f(s)=\infty$ for some $\gamma >q(1+2/n)$. This allows us to construct a locally Lipschitz $f$ satisfying the Osgood condition $\int_{1}^{\infty}1/f(s)\ \,\d s =\infty$, which ensures global existence for bounded initial data, such that for every $q$ with $1\le q<\infty$ there is an initial condition $u_0\in L^q(\Om)$ for which the corresponding semilinear problem has no local-in-time solution.

References in corpus (1)

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

Cited by in corpus (1)

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