Quasilinear Cauchy-Dirichlet problem for parabolic equations with $VMO_x$ coefficients

Abstract

We study the strong solvability of the Cauchy-Dirichlet problem for parabolic quasilinear equations with discontinuous data. The principal coefficients depend on the point $(x,t)$ and on the solution u, the dependence on x is of VMO type while these are only measurable with respect to t. Assuming suitable structural conditions on the nonlinear terms, we prove existence and uniqueness of the strong solution, which turns out to be also Holder continuous.