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<p>We demonstrate the existence in the sense of sequences of solutions for some integro-differential type problems involving the drift term and the square of the Laplace operator, on the whole real line or on a finite interval with periodic boundary conditions in the corresponding <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^{4}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation> spaces. Our argument is based on the fixed point technique when the elliptic equations contain fourth order differential operators with and without the Fredholm property. It is established that, under the reasonable technical conditions, the convergence in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^{1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> of the integral kernels yields the existence and convergence in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(H^{4}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation> of the solutions.</p>

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Solvability of Some Integro-Differential Equations with the Bi-Laplacian and Transport

  • 期刊论文

  • 发表日期：

    2026年2月4日

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  • Vitali Vougalter

摘要

We demonstrate the existence in the sense of sequences of solutions for some integro-differential type problems involving the drift term and the square of the Laplace operator, on the whole real line or on a finite interval with periodic boundary conditions in the corresponding \(H^{4}\) H 4 spaces. Our argument is based on the fixed point technique when the elliptic equations contain fourth order differential operators with and without the Fredholm property. It is established that, under the reasonable technical conditions, the convergence in \(L^{1}\) L 1 of the integral kernels yields the existence and convergence in \(H^{4}\) H 4 of the solutions.