On strongly nonlinear parabolic equations

Abstract

The initial-value problem du dt + A(u) = ƒ; u(0) = 0 , where A is a nonlinear coercive operator mapping X into Y ∗ is considered. X and Y are two reflexive Banach spaces and A is assumed to satisfy some weak continuity properties. The results give the existence of Hopf's solution of the Navier-Stokes equations as well as solutions of equations of the form ∂u ∂t + ∑ ∥α∥,∥β∥⩽m D α(a αβ(x, t)D βu)+ u ∥ D m−1u∥=ƒ Results of Browder and of Lions on nonlinear parabolic equations are special cases of those obtained in this paper.