On a mixed local-nonlocal evolution equation with singular nonlinearity

We will prove several existence and regularity results for the mixed local-nonlocal parabolic equation of the form {eqnarray} {split} u_t-Δ u+(-Δ)^s u&={f(x,t)}{uγ(x,t)} in Ω_T:=Ω ×(0, T), \\ u&=0 in (R^n Ω) ×(0, T), \\ u(x, 0)&=u_0(x) in Ω ; {split} {eqnarray} where {equation*} (-Δ )^s u= cn,sP.V.∫R^n{u(x,t)-u(y,t)}{|x-y|ⁿ⁺²ˢ} d y. {equation*} Under the assumptions that γ is a positive continuous function on Ω̄T and Ω is a bounded domain %of class C1,1 with Lipschitz boundary in Rⁿ, $n> 2$, s∈(0,1), 0<T<+∞, f≥ 0, u₀≥ 0, f and u₀ belongs to suitable Lebesgue spaces. Here cn,s is a suitable normalization constant, and P.V. stands for Cauchy Principal Value.