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

Abstract

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*} with $q\in (0,\infty )$ in a bounded noncylindrical domain $E\subset {\mathbb{R}}^{n+1}$. Further, we suppose that $x\mapsto f(x,u,\xi )$ is integrable, that $(u,\xi )\mapsto f(x,u,\xi )$ is convex, and that $f$ satisfies a $p$-growth and -coercivity condition for some $p>\max \{1,\frac{n(q+1)}{n+q+1}\}$. Merely assuming that ${\mathcal{L}}^{n+1}(\partial E)=0$, we prove the existence of variational solutions $u\in L^{\infty }(0,T;L^{q+1}(E,{\mathbb{R}}^N))$. If $E$ does not shrink too fast, we show that for the solution $u$ constructed in the first step, $|u{|}^{q-1}u$ admits a distributional time derivative. Moreover, under suitable conditions on $E$ and the stricter lower bound $p\ge \frac{(n+1)(q+1)}{n+q+1}$, $u$ is continuous with respect to time.

References in corpus (7)

• Higher integrability for doubly nonlinear parabolic systems
• Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs
• Boundary regularity for degenerate and singular parabolic equations
• Parabolic equations in time dependent domains
• Boundary regularity for the porous medium equation
• The Petrovski\uı criterion and barriers for degenerate and singular p-parabolic equations
• Existence of variational solutions to doubly nonlinear systems in nondecreasing domains