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Existence of variational solutions to doubly nonlinear systems in nondecreasing domains (2505.00148v1)

Published 30 Apr 2025 in math.AP

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_\xi f(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 $V{p,0}(E) \subset Lp(0,T;W{1,p}(\Omega,\mathbb{R}N))$ that encodes zero boundary values.

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