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An existence theorem for parabolic equations on N with discontinuous nonlinearity. (English) Zbl 1056.35082

The paper deals with the initial value problem

(1) t u=Δu+f(u),(2)u(x,0)=α(x),

where x N , t>0, here α 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 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).

35K57Reaction-diffusion equations
35K15Second order parabolic equations, initial value problems
35D05Existence of generalized solutions of PDE (MSC2000)