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Zbl 1056.35082
Hofbauer, Josef; Simon, Peter L.
An existence theorem for parabolic equations on $\Bbb R^N$ with discontinuous nonlinearity. (English)
[J] Electron. J. Qual. Theory Differ. Equ. 2001, Paper No. 8, 9 p., electronic onnly (2001). ISSN 1417-3875/e

The paper deals with the initial value problem $$(1)\quad \partial_tu= \Delta u+f(u),\qquad (2)\quad u(x,0)=\alpha(x),$$ where $x\in\bbfR^N$, $t>0$, here $\alpha$ is a bounded uniformly continuous function and $r$ is bounded, measurable but generally a non-continuous one. The problem is motivated by the model of best response dynamics arising in game theory [see the first author, Ann. Oper Res. 89, 233--251 (1999; Zbl 0942.91018)]. The theorem proved by the authors claims that (1), (2) has a generalized (in the sense derived in the paper) solution. The proof is based on solving the problem (1), (2) with $f$ replaced by $f_n$, where $(f_n)$ is a sequence of $C^\infty$ functions approximating $f$. Then the Arzela-Ascoli theorem is applied to the corresponding sequence $(u_n$ of solutions and the uniform limit of a subsequence of $(u_n)$ is the desired solution of (1), (2).
[Hanna Marcinkowska (Wrocław)]

MSC 2000:: *35K57 Reaction-diffusion equations
35K15 Second order parabolic equations, initial value problems
35D05 Existence of generalized solutions of PDE

Keywords: initial value problem; best response dynamics; Arzela-Ascoli theorem

Citations: Zbl 0942.91018

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