Existence of variational solutions to doubly nonlinear systems in nondecreasing domains
Abstract
Abstract For $$q \in (0, \infty )$$ q ∈ ( 0 , ∞ ) , we consider the Cauchy–Dirichlet problem to doubly nonlinear systems of the form $$\begin{aligned} \partial _t \big ( |u|^{q-1}u \big ) - \operatorname {div} \big ( D_\xi f(x,u,Du) \big ) = - D_u f(x,u,Du) \end{aligned}$$ ∂ t ( | u | q - 1 u ) - div ( D ξ f ( x , u , D u ) ) = - D u f ( x , u , D u ) in a bounded noncylindrical domain $$E \subset \mathbb {R}^{n+1}$$ E ⊂ R n + 1 . We assume that $$x \mapsto f(x,u,\xi )$$ x ↦ f ( x , u , ξ ) is integrable, that $$(u,\xi ) \mapsto f(x,u,\xi )$$ ( u , ξ ) ↦ f ( x , u , ξ ) is convex, and that f satisfies a p -coercivity condition for some $$p \in (1,\infty )$$ p ∈ ( 1 , ∞ ) . However, we do not impose any specific growth condition from above on f . For nondecreasing domains that merely satisfy $$\mathcal {L}^{n+1}(\partial E) = 0$$ L
Fields of Study & Subjects
Nonlinear Partial Differential Equations
Nonlinear Differential Equations Analysis
Optimization and Variational Analysis
Publication details
- Journal / source
- Journal of Evolution Equations
- Publisher
- Springer Science and Business Media LLC
- Volume
- 26
- Issue
- 1
- Pages
- 6–6
- Publication date
- 16 January 2026
- Language
- EN