Approximation of the Solutions to Quasilinear Parabolic Problems with Perturbed $VMO_x$ Coefficients

Abstract

We consider the Cauchy-Dirichlet problem for second-order quasilinear non-divergence form operators of parabolic type. The data are Cara\-th\'e\-o\-dory functions, and the principal part is of $VMO_x$-type with respect to the variables $ (x,t).$ Assuming the existence of a strong solution $u_0,$ we apply the Implicit Function Theorem in a small domain of this solution to show that small bounded perturbations of the data, locally in time, lead to small perturbations of the solution $u_0$. Additionally, we apply the Newton Iteration Procedure to construct an approximating sequence converging to the solution $u_0$ in the corresponding Sobolev space.