Approximation of the Solutions to Quasilinear Parabolic Problems with Perturbed $VMO_x$ Coefficients
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915663060140032 |
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| author | Rescigno, Rosamaria Softova, Lubomira |
| author_facet | Rescigno, Rosamaria Softova, Lubomira |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_16510 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximation of the Solutions to Quasilinear Parabolic Problems with Perturbed $VMO_x$ Coefficients Rescigno, Rosamaria Softova, Lubomira Analysis of PDEs Functional Analysis 35K55, 35K61, 47J07 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. |
| title | Approximation of the Solutions to Quasilinear Parabolic Problems with Perturbed $VMO_x$ Coefficients |
| topic | Analysis of PDEs Functional Analysis 35K55, 35K61, 47J07 |
| url | https://arxiv.org/abs/2505.16510 |