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

Laister, Robert; Robinson, James C.; Sierzega, Mikolaj

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 $\limsup_{s\to\infty}s^{-\gamma}f(s)=\infty$ 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

Citation

Laister, R., Robinson, J. C., & Sierzega, M. (2013). Gaussian lower bounds on the Dirichlet heat kernel and non-existence of local solutions for semilinear heat equations of Osgood type

• Journal Article Type: Article
• Publication Date: Jul 25, 2013
• Journal: arXiv
• Peer Reviewed: Not Peer Reviewed
• Keywords: semilinear heat equation, Dirichlet problem, non-existence,; instantaneous blow-up, Osgood condition, Dirichlet heat kernel
• Publisher URL: http://arxiv.org/abs/1307.6688v1
• Additional Information: Additional Information : Imported from arXiv