The linear problem, that is, f(x,t)=f(x), where according to the summability of the datum f and the parameter λ we give the summability of the solution u.
The problem with a nonlinear term for t>0. In this case, existence and regularity will depend on the value of σ and on the summability of h.
Looking for optimal results we will need a weak Harnack inequality for elliptic operators with singular coefficients that seems to be new.