Periodic Solutions of a Nonlinear Parabolic Equation, II1

Based on a strict coercivity estimate [8] for a class of nonlinear elliptic operators A : u 7→ ∇·a(·, |∇u|)∇u, it is shown that (with suitable consistency conditions on the data) the problem: u̇+ Au = f, a∇u · ~n = φ, time periodicity has a unique solution depending continuously on f, φ and also on the nonlinear form of the diffusion coefficient a(··). This is then used to obtain existence by a fixpoint argument for the more general problem with a, f of the form a = a(·, u, |∇u|), f = f(·, u,∇u).