Entropy solutions to doubly nonlinear integro-differential equations. (English) Zbl 1447.45012
Summary: We consider doubly nonlinear history-dependent problems of the form \(\partial_t [ k \ast (b(v) - b(v_0))] - \operatorname{div} a(x, \nabla v) = f\). The kernel \(k\) satisfies certain assumptions which are, in particular, satisfied by \(k(t) = \frac{ t^{- \alpha}}{ \varGamma(1 - \alpha)}\), i.e., the case of fractional derivatives of order \(\alpha \in(0, 1)\) is included. We show existence of entropy solutions in the case of a nondecreasing \(b\). An existence result in the case of a strictly increasing \(b\) is used to get entropy solutions of approximate problems. Kruzhkov’s method of doubling variables, a comparison principle and the diagonal principle are used to obtain a.e. convergence for approximate solutions. A uniqueness result has been shown in a previous work.
MSC:
| 45K05 | Integro-partial differential equations |
| 47J35 | Nonlinear evolution equations |
| 45D05 | Volterra integral equations |
| 35D99 | Generalized solutions to partial differential equations |
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