Quasilinear parabolic variational-hemivariational inequalities in ${\mathbb{R}}^N\times (0,\tau )$ under bilateral constraints

In this paper, we considered quasilinear variational-hemivariational inequalities in the unbounded cylindrical domain $\mathbb{Q}={\mathbb{R}}^N\times (0,\tau )$ of the form: Find $u\in K\subset X$ with $u(\cdot ,0)=0$ satisfying

$$\begin{array}{r}⟨u_t-\mathrm{d}\mathrm{i}\mathrm{v}A(x,t,\mathrm{\nabla }u),v-u⟩+{\displaystyle {\int }_{\mathbb{Q}}a(x,t)j^o(x,t,u;v-u)dxdt\ge 0,\mathrm{\forall }v\in K,}\end{array}$$

where $K\subset X$ represents the bilateral constraints in $X=L^p(0,\tau ;D^{1,p}({\mathbb{R}}^N))$ with $D^{1,p}({\mathbb{R}}^N)$ denoting the Beppo-Levi space (or homogeneous Sobolev space), and $j^o(x,t,s;\varrho )$ denoting Clarke's generalized directional derivative of the locally Lipschitz function $s\mapsto j(x,t,s)$ at $s$ in the direction $\varrho$. The main goal and the novelty of this paper was to prove existence results without assuming coercivity conditions on the time-dependent elliptic operators involved, and without supposing the existence of sub-supersolutions. Further difficulties arise in the treatment of the problem under consideration due to the lack of compact embedding of $D^{1,p}({\mathbb{R}}^N))$ into Lebesgue spaces $L^{\sigma }({\mathbb{R}}^N)$, and the fact that the constraint $K$ has an empty interior, which prevents us from applying recent results on evolutionary variational inequalities. Instead our approach was based on an appropriately designed penalty technique and the use of weighted Lebesgue spaces as well as multi-valued pseudomontone operator theory.