Bounded weak solutions of time-fractional porous medium type and more general nonlinear and degenerate evolutionary integro-differential equations

We prove existence of a bounded weak solution to a degenerate quasilinear subdiffusion problem with bounded measurable coefficients that may explicitly depend on time. The kernel in the involved integro-differential operator w. r .t. time belongs to the large class of PC kernels. In particular, the case of a fractional time derivative of order less than 1 is included. A key ingredient in the proof is a new compactness criterion of Aubin-Lions type which involves function spaces defined in terms of the integro-differential operator in time. Boundedness of the solution is obtained by the De Giorgi iteration technique. Sufficiently regular solutions are shown to be unique by means of an L-1-contraction estimate. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The paper proves the existence of a bounded weak solution to a degenerate quasilinear subdiffusion problem with bounded measurable coefficients that may explicitly depend on time. A key ingredient in the proof is a new compactness criterion of Aubin-Lions type, and boundedness of the solution is obtained by the De Giorgi iteration technique. Regular solutions are shown to be unique through an L-1-contraction estimate.