We consider the mixed local-nonlocal parabolic equation of the form \(\begin{aligned} \begin{array}{c} u_t-\Delta u+(-\Delta )^s u=\frac{f(x,t)}{u^{\gamma (x,t)}} \text{ in } \Omega _T:=\Omega \times (0, T), \\ u>0 \text { in } \Omega _T, \quad u=0 \text{ in } (\mathbb {R}^n \backslash \Omega ) \times (0, T), \\ u(x, 0)=u_0(x) \text{ in } \Omega ; \end{array} \end{aligned}\) where \(\begin{aligned} (-\Delta )^s u:= c_{n,s}\operatorname {P.V.}\int _{\mathbb {R}^n}\frac{u(x,t)-u(y,t)}{|x-y|^{n+2s}} d y, \end{aligned}\) under the assumptions that \(\gamma \) is a positive continuous function on \(\overline{\Omega }_T\) and \(\Omega \) is a bounded domain of class \(C^{1}\) 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, \(f\not \equiv 0\) . Here \(c_{n,s}\) is a suitable normalization constant, and \(\operatorname {P.V.}\) stands for Cauchy Principal Value. We obtain several existence and regularity results (depending on the summability of f) in the spirit of Boccardo-Orsina (Calc Var Part Differ Equ 37(3–4), 363–380, 2010). Under certain conditions on f, a new optimal summability result and asymptotic behavior of finite energy solution are also obtained.
