Initial traces and solvability for a semilinear heat equation on a half space of ${\mathbb R}^N$
Abstract
We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of ${\mathbb R}^N$ under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the corresponding Cauchy--Dirichlet problem. Our necessary conditions and sufficient conditions are sharp and enable us to find optimal singularities of initial data for the solvability of the Cauchy--Dirichlet problem.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2022
- DOI:
- 10.48550/arXiv.2209.06398
- arXiv:
- arXiv:2209.06398
- Bibcode:
- 2022arXiv220906398H
- Keywords:
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- Mathematics - Analysis of PDEs;
- 35K58;
- 35A01;
- 35A21;
- 35K20