Existence of variational solutions to doubly nonlinear systems in nondecreasing domains

For q∈(0,∞), we consider the Cauchy–Dirichlet problem to doubly nonlinear systems of the form (Formula presented.) in a bounded noncylindrical domain E⊂Rn+1. We assume that x↦f(x,u,ξ) is integrable, that (u,ξ)↦f(x,u,ξ) is convex, and that f satisfies a p-coercivity condition for some p∈(1,∞). However, we do not impose any specific growth condition from above on f. For nondecreasing domains that merely satisfy Ln+1(∂E)=0, we prove the existence of variational solutions u∈C0([0,T];Lq+1(E,RN)) 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-1u admits a weak time derivative in the dual (Vp,0(E))′ of the subspace Vp,0(E)⊂Lp(0,T;W1,p(Ω,RN)) that encodes zero boundary values.