Semilinear equations with dissipative time-dependent domain perturbations

LetX be a real Banach space and letA∶D(A)⊂X→X be the (linear) infinitesimal generator of the semigroupS(t) of classC ₀ (of type ω). Assume that the function (t,x)→F(t,x) is continuous, the domainD(t)=D(F(t,·)) is such thatt→D(t) is closed and for eacht∈(a,b), the operatorx→F(t,x) is dissipative. One proves that the subtangential condition (A5) is necessary and sufficient for the existence of the mild solution to the equationu′=Au+F(t,u). All previous results of this type are included here. An elementary method for proving the uniqueness is pointed out and applications to PDE are given.