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

Abstract We consider the Cauchy-Dirichlet problem for second-order quasilinear operators of parabolic type in non-divergence form. The data are Carathéodory functions, and the principal part is of $$VMO_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mi>M</mml:mi> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:mrow> </mml:math> -type with respect to the variables ( x , t ). Assuming the existence of a strong solution $$u_0,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> we apply the Implicit Function Theorem in a neighbourhood of this solution to show that small bounded perturbations of the data lead to small perturbations of the solution $$u_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> itself. Furthermore, we employ the Newton iteration procedure to construct an approximating sequence that converges to $$u_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> in the corresponding Sobolev space.