Existence of variational solutions to doubly nonlinear systems in nondecreasing domains

For $q\in (0,\infty )$ , we consider the Cauchy-Dirichlet problem to doubly nonlinear systems of the form $\begin{array}{c}{\partial }_t({\left|u\right|}^{q-1}u)-div(D_{\xi }f(x,u,Du))=-D_{u}f(x,u,Du)\end{array}$ in a bounded noncylindrical domain $E\subset R^{n+1}$ . 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 $L^{n+1}\left(\partial E\right)=0$ , we prove the existence of variational solutions $u\in C^0([0,T];L^{q+1}(E,R^N))$ 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 ${\left|u\right|}^{q-1}u$ admits a weak time derivative in the dual ${\left(V^{p,0}\left(E\right)\right)}^{\text{'}}$ of the subspace $V^{p,0}\left(E\right)\subset L^p(0,T;W^{1,p}(\Omega ,R^N))$ that encodes zero boundary values.