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-Δu+(-Δ)^s u&=\frac{f(x,t)}{u^{γ(x,t)}} \text { in } Ω_T:=Ω\times(0, T), \\ u&=0 \text { in }(\mathbb{R}^n \backslash Ω) \times(0, T), \\ u(x, 0)&=u_0(x) \text { in } Ω; \end{split} \end{eqnarray} where \begin{equation*} (-Δ)^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 ${\mathbb{R}}^n$, $n>2$, $s\in (0,1)$, $0<T<+\infty$, $f\ge 0$, $u_0\ge 0$, $f$ and $u_0$ belongs to suitable Lebesgue spaces. Here $c_{n,s}$ is a suitable normalization constant, and $P.V.$ stands for Cauchy Principal Value.