Quasilinear parabolic equations with first order terms and $L^1$-data in moving domains

Abstract

The global existence of weak solutions to a class of quasilinear parabolic equations with nonlinearities depending on first order terms and integrable data in a moving domain is investigated. The class includes the p-Laplace equation as a special case. Weak solutions are shown to be global by obtaining appropriate estimates on the gradient as well as a suitable version of Aubin-Lions lemma in moving domains. CONTENTS 1. Introduction 1 2. Uniform estimates 5 2.1. Uniform bounds of approximate solutions 6 2.2. Uniform bounds of the nonlinearity 13 3. Proof of Theorem 1.1 14 Appendix A. Existence of approximate solutions 22 Appendix B. An Aubin-Lions lemma in moving domains 26 References 27 2010 Mathematics Subject Classification. 35K59, 35K90, 35K92.

Cahn-Hilliard equations on an evolving surface

“…These are also challenging problems from the point of view of mathematics. Indeed, equations on non-cylindrical domains have been considered as early as in [Lio57,Bai65], and more recently in [Nae15] with applications to the Stokes problem and in [LSTT20] for general quasilinear parabolic problems. Many other examples can be found in the references of these two articles.…”