An existence theorem for parabolic equations on \(\mathbb R^N\) with discontinuous nonlinearity (Q2781084)

The paper deals with the initial value problem NEWLINE\[NEWLINE(1)\quad \partial_tu= \Delta u+f(u),\qquad (2)\quad u(x,0)=\alpha(x),NEWLINE\]NEWLINE where \(x\in\mathbb{R}^N\), \(t>0\), 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).