Dr Robert Laister Robert.Laister@uwe.ac.uk
Senior Lecturer
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
Authors
James C. Robinson
Mikolaj Sierzega
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
Journal Article Type | Article |
---|---|
Publication Date | Jul 25, 2013 |
Publicly Available Date | Jun 7, 2019 |
Journal | arXiv |
Peer Reviewed | Not Peer Reviewed |
Keywords | semilinear heat equation, Dirichlet problem, non-existence, instantaneous blow-up, Osgood condition, Dirichlet heat kernel |
Public URL | https://uwe-repository.worktribe.com/output/929699 |
Publisher URL | http://arxiv.org/abs/1307.6688v1 |
Additional Information | Additional Information : Imported from arXiv |
Files
1307.6688v1.pdf
(200 Kb)
PDF
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