Entropy solutions to doubly nonlinear integro-differential equations. (March 2020) - Sapountzoglou, Niklas
Abstract: We consider doubly nonlinear history-dependent problems of the form ∂ t [ k ∗ ( b ( v ) − b ( v 0 ) ) ] − div a ( x, ∇ v ) = f . The kernel k satisfies certain assumptions which are, in particular, satisfied by k ( t ) = t − α Γ ( 1 − α ), i.e., the case of fractional derivatives of order α ∈ ( 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.