Nonlocal evolution equations with p[u(x, t)]-Laplacian and lower-order terms

Abstract

We study the homogeneous Dirichlet problem for a class of nonlocal singular parabolic equations $$\begin{aligned} u_t-{\text {div}}\left( |\nabla u|^{p[u]-2}\nabla u\right) =f((x,t),u,l(u)) \quad \text {in} \, Q_T={\varOmega }\times (0,T), \end{aligned}$$where $${\varOmega }\subset {\mathbb {R}}^{d}$$, $$d\ge 2$$, is a smooth domain, $$p[u]=p(l(u))$$ is a given function with values in the interval $$[p^-,p^+]\subset (\frac{2d}{d+2},2)$$, and $$l(u)=\displaystyle \int _{{\varOmega }}|u(x,t)|^{\alpha }\,dx$$, $$\alpha \in [1,2]$$, is a functional of the unknown solution. We prove the existence of a strong solution such that $$\begin{aligned} \begin{aligned}uW^{1,2}_0({\varOmega })), \quad |D^{2}_{ij}u|^{p[u]}\in L^{1}(Q_T). \end{aligned} \end{aligned}$$Conditions of uniqueness of strong solutions are obtained.