Scholarlink

<p>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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(VMO_x\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mi>M</mi> <msub> <mi>O</mi> <mi>x</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>-type with respect to the variables (<i>x</i>,&#xa0;<i>t</i>). Assuming the existence of a strong solution <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(u_0,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(u_0\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> itself. Furthermore, we employ the Newton iteration procedure to construct an approximating sequence that converges to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(u_0\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> in the corresponding Sobolev space.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Approximation of the Solutions to Quasilinear Parabolic Problems with Perturbed \(VMO_x\) Coefficients

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2025年12月12日

查看全文
  • Rosamaria Rescigno,
  • Lubomira G. Softova

摘要

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 \(VMO_x\) V M O x -type with respect to the variables (x, t). Assuming the existence of a strong solution \(u_0,\) u 0 , 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 \(u_0\) u 0 itself. Furthermore, we employ the Newton iteration procedure to construct an approximating sequence that converges to \(u_0\) u 0 in the corresponding Sobolev space.