The application of dissipative operators to nonlinear diffusion equations

Abstract

with U(L, ~1) positive, smooth for z > 0 and other technical conditions. The initial value problem with o(u, u,) = a(u), o(O) = 0 and g continuous has been previously studied by a different approach; see [8]. In Section 2 we prove that, under the appropriate conditions, A, the closure of Au = (a(~, u,)u,)~, may be regarded as a dissipative operator in Lp which generates a contraction semigroup there. Burnell McKissick proved this for the case a(u, u,) = u(u) in L’ (private communication). Conditions are given, in Section 3, for the existence of a positive lower bound of u(& X) = e”g. g E D(A), in sets of f > 0, --co < x < co. In such a region it is shown, in Section 4, that u(t. x) is a classical solution. The advance over (81 is that, for u(u) independent of u,, u(O) = co is allowed if