English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We give a simple proof of a lower bound for the Dirichlet heat kernel interms of the Gaussian heat kernel. Using this we establish a non-existenceresult for semilinear heat equations with zero Dirichlet boundary conditionsand initial data in $L^q(\Omega)$ when the source term $f$ is non-decreasingand $\limsup_{s\to\infty}s^{-\gamma}f(s)=\infty$ for some $\gamma>q(1+2/n)$.This allows us to construct a locally Lipschitz $f$ satisfying the Osgoodcondition $\int_{1}^{\infty}1/f(s)\ \,\d s =\infty$, which ensures globalexistence for bounded initial data, such that for every $q$ with $1\leq<\infty$ there is an initial condition $u_0\in L^q(\Om)$ for which thecorresponding semilinear problem has no local-in-time solution.

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