Strong solutions for semilinear problems with almost sectorial operators

In this paper. we study a semilinear parabolic problem u(t) + Au = f(t, u), t > tau; u(tau) = u(0) is an element of X, in a Banach space X, where A : D(A) subset of X -> X is an almost sectorial operator. This problem is locally well-posed in the sense of mild solutions. By exploring properties of the semigroup of growth 1 - alpha generated by -A, we prove that the local mild solution is actually strong solution for the equation. This is done without requiring any extra regularity for the initial condition u(0) is an element of X and under suitable assumptions on the nonlinearity f . We apply the results for a reaction-diffusion equation in a domain with handle where the nonlinearity f satisfies a polynomial growth vertical bar f(t, u) - f(t, v)vertical bar <= C vertical bar u - v vertical bar(1 + vertical bar u vertical bar(rho-1) + vertical bar v vertical bar(rho-1)), and we establish values of rho for which the problem still have strong solution.