LOCAL EXISTENCE FOR PARABOLIC PROBLEMS WITH FULLY NONLINEAR BOUNDARY-CONDITION - AN LP-APPROACH

We study the problem (a(i) = a(i)(x, t, u, p)) of variational form and arises in heat conduction) in the Sobolev Space W(p)2,1 (OMEGA x x (0, T)), OMEGA subset-of R(n), p > n + 2. Existence of a local (in time) solution is proved under a natural compatibility condition between the data (Theorem 2. 1). This solution is also globally unique. An outlook to similar problems for parabolic systems is given (section 4). Our method also applies to quasilinear equations with conormal b.c. (cf. (P), end of section 2).