Existence and comparison theorems for nonlinear diffusion systems

AbstractIn this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v1(x), v2(x),…, vm(x)) from Ω ⊂ Rn into Rm which satisfy ψi(x, t) ⩽ vi(x) ⩽ θi(x, t) for all x ϵ Ω and 1 ⩽ i ⩽ m, where ψiand θi are extended real-valued functions on \̄gW × [0, T). We find conditions which will ensure that a solution U(x, t) ≡ (u1(x, t), u2(x, t),…, um(x, t)) which satisfies U(x, 0) ϵ K(0) will also satisfy U(x, t) ϵ K(t) for all 0 ⩽ t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem