On a mixed local-nonlocal evolution equation with singular nonlinearity

Abstract

We will prove several existence and regularity results for the mixed local-nonlocal parabolic equation of the form \begin{eqnarray} \begin{split} u_t-\Delta u+(-\Delta)^s u&=\frac{f(x,t)}{u^{\gamma(x,t)}} \text { in } \Omega_T:=\Omega \times(0, T), \\ u&=0 \text { in }(\mathbb{R}^n \backslash \Omega) \times(0, T), \\ u(x, 0)&=u_0(x) \text { in } \Omega ; \end{split} \end{eqnarray} where \begin{equation*} (-\Delta )^s u= c_{n,s}\operatorname{P.V.}\int_{\mathbb{R}^n}\frac{u(x,t)-u(y,t)}{|x-y|^{n+2s}} d y. \end{equation*} Under the assumptions that $\gamma$ is a positive continuous function on $\overline{\Omega}_T$ and $\Omega$ is a bounded domain %of class $\mathcal{C}^{1,1}$ with Lipschitz boundary in $\mathbb{R}^{n}$, $n> 2$, $s\in(0,1)$, $0<T<+\infty$, $f\geq 0$, $u_0\geq 0$, $f$ and $u_0$ belongs to suitable Lebesgue spaces. Here $c_{n,s}$ is a suitable normalization constant, and $\operatorname{P.V.}$ stands for Cauchy Principal Value.