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 uₜ- u+ (-) ˢ u split eqnarray where equation* (-) ˢ u= c₍, ₒP. V. ₑ䂞u (x, t) -u (y, t) |x-y|^{n+2s} d y. equation* Under the assumptions that is a positive continuous function on T and is a bounded domain %of class C^1, 1 with Lipschitz boundary in R^n, n> 2, s (0, 1), 0<T<+, f 0, u₀ 0, f and u₀ belongs to suitable Lebesgue spaces. Here c₍, ₒ is a suitable normalization constant, and P. V. stands for Cauchy Principal Value.