Existence of a weak solutions to a class of nonlinear parabolic problems via topological degree method

Abstract

In this paper, we study the existence of "weak solutions" for a class of nonlinear parabolic problems of the type

^{∂ u}⁄_{∂ t} - div Α (x, t, ∇ u) = Φ (x, t) + div B(x, t, u, ∇ u).

Using Berkovits and Mustonen's topological degree theory, we demonstrate the existence of a weak solutions to the problems under consideration in the space L^p(0, T, W₀^{1, p}(Ω)), where Ω is a bounded domain in R^N, N ≥ 2 and p ≥ 2.