| Autor: |
Laister, Robert, Robinson, James C., Sierzega, Mikolaj |
| Rok vydání: |
2013 |
| Předmět: |
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| Druh dokumentu: |
Working Paper |
| Popis: |
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 $1\le q<\infty$ there is an initial condition $u_0\in L^q(\Om)$ for which the corresponding semilinear problem has no local-in-time solution. |
| Databáze: |
arXiv |
| Externí odkaz: |
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