Existence of variational solutions to doubly nonlinear systems in nondecreasing domains

Abstract

For $q\in (0,\infty )$, we consider the Cauchy-Dirichlet problem to doubly nonlinear systems of the form \begin{align*} \partial_t \big( |u|^{q-1}u \big) - \operatorname{div} \big( D_ξf(x,u,Du) \big) = - D_u f(x,u,Du) \end{align*} in a bounded noncylindrical domain $E\subset {\mathbb{R}}^{n+1}$. We assume that $x\mapsto f(x,u,\xi )$ is integrable, that $(u,\xi )\mapsto f(x,u,\xi )$ is convex, and that $f$ satisfies a $p$-coercivity condition for some $p\in (1,\infty )$. However, we do not impose any specific growth condition from above on $f$. For nondecreasing domains that merely satisfy ${\mathcal{L}}^{n+1}(\partial E)=0$, we prove the existence of variational solutions $u\in C^0([0,T];L^{q+1}(E,{\mathbb{R}}^N))$ via a nonlinear version of the method of minimizing movements. Moreover, under additional assumptions on $E$ and a $p$-growth condition on $f$, we show that $|u{|}^{q-1}u$ admits a weak time derivative in the dual $(V^{p,0}(E){)}^{\prime }$ of the subspace $V^{p,0}(E)\subset L^p(0,T;W^{1,p}(\Omega ,{\mathbb{R}}^N))$ that encodes zero boundary values.

This version is the Author Accepted Manuscript

References in corpus (3)

• Higher integrability for doubly nonlinear parabolic systems
• Parabolic equations in time dependent domains
• Boundary regularity for the porous medium equation

Cited by in corpus (1)

• Existence of variational solutions to doubly nonlinear systems in general noncylindrical domains