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

TL;DR

This paper establishes the existence and boundedness of weak solutions for a class of time-fractional degenerate nonlinear equations, including fractional derivatives, using novel compactness criteria and De Giorgi iteration.

Contribution

It introduces a new compactness criterion for integro-differential equations and proves existence, boundedness, and uniqueness of solutions for time-fractional porous medium type problems.

Findings

Existence of bounded weak solutions for degenerate quasilinear subdiffusion problems.

Development of a new Aubin-Lions type compactness criterion involving integro-differential operators.

Proof of solution boundedness and uniqueness under certain conditions.

Abstract

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 ${\cal PC}$ 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.