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March/April 2010 Asymptotic behavior of solutions of a semilinear heat equation with localized reaction
Ryuichi Suzuki
Adv. Differential Equations 15(3/4): 283-314 (March/April 2010). DOI: 10.57262/ade/1355854751

Abstract

We consider non-negative solutions to the Dirichlet problem of a semilinear heat equation with localized reaction in $\Omega$: $u_t = \Delta u +f(u(x_0(t),t))$, where $\Omega$ is a smooth bounded domain, $x_0(t)$ is a locally Hölder continuous function from $[0,\infty)$ into $\Omega$ and $f$ satisfies $f(0)=f'(0)=0$ and some blow-up condition. We show that, if $x_0(t)$ remains in some compact subset of $\Omega$ as $t\to\infty$, then all global solutions are bounded in $\Omega\times(0,\infty)$ and, if $x_0(t)$ approaches the boundary of $\Omega$ as $t\to \infty$, then some unbounded global solution (infinite time blow-up solution) exists. These results are parts of our main results on the classification of all solutions.

Citation

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Ryuichi Suzuki. "Asymptotic behavior of solutions of a semilinear heat equation with localized reaction." Adv. Differential Equations 15 (3/4) 283 - 314, March/April 2010. https://doi.org/10.57262/ade/1355854751

Information

Published: March/April 2010
First available in Project Euclid: 18 December 2012

zbMATH: 1198.35128
MathSciNet: MR2588771
Digital Object Identifier: 10.57262/ade/1355854751

Subjects:
Primary: 35B35 , 35B40 , 35B45 , 35K20 , 35K55 , 35K57

Rights: Copyright © 2010 Khayyam Publishing, Inc.

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Vol.15 • No. 3/4 • March/April 2010
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