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

We consider the Cauchy-Dirichlet problem for second-order quasilinear non-divergence form operators of parabolic type. The data are Cara-thé-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.

💡 Research Summary

The paper studies the Cauchy‑Dirichlet problem for second‑order quasilinear parabolic equations in non‑divergence form whose principal coefficients belong to the class VMO(_x) (vanishing mean oscillation with respect to the spatial variable). The model problem is

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