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

TL;DR

This paper studies how small data perturbations affect solutions to certain quasilinear parabolic PDEs with VMO_x coefficients, using implicit function theorem and Newton iteration to approximate solutions.

Contribution

It introduces a method to approximate solutions of quasilinear parabolic problems with VMO_x coefficients via Newton iteration and perturbation analysis.

Findings

Small data perturbations lead to small changes in solutions.

Newton iteration converges to the strong solution in Sobolev space.

The approach applies locally in time for the perturbed problems.

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.