A doubly nonlinear fractional diffusive equation

Abstract

This thesis focuses on the `doubly nonlinear fractional diffusive equation', a doubly nonlinear nonlocal parabolic initial boundary value problem driven by the fractional p-Laplacian equipped with homogeneous Dirichlet boundary conditions on a domain in Euclidean space and composed with a power-like function. We also include a Lipschitz perturbation and a forcing term depending on space and time. We first generalize the nonlinear term u^m, replacing this by a continuous, strictly increasing function. Here we establish well-posedness in L1 in the sense of mild solutions and a comparison principle. For domains with finite measure and with restricted initial data we obtain that mild solutions of the inhomogeneous evolution problem are strong and distributional. We then consider the power-like case where we obtain further regularity properties. In particular, we have an Ll-L∞ regularizing effect for mild solutions (and therefore also for strong solutions), also known as ultracontractivity. We further obtain derivative and energy estimates for this problem. Using these, we extend the previous strong regularity result to obtain strong distributional solutions on general open domains with initial data in L1. Moreover, we prove local and global Hölder continuity results in restricted cases as well as a comparison principle that yields extinction in finite time of mild solutions to the homogeneous evolution equation. We finally restrict to the doubly nonlinear fractional diffusive equation without forcing terms, where we investigate self-similarity properties and, in particular, the asymptotic behaviour of solutions for large times. The main result in this case is the existence of Barenblatt solutions. However, in finding these we also prove an Aleksandrov symmetry principle for solutions and estimate solutions by global bounding functions which are integrable in space.