Solvability of some integro-differential equations with the bi-Laplacian and transport

2025/09/15 by Vitali Vougalter, Vougalter, Vitali
Computer Science · Mathematics · #35J30 #35K57 #35P30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2509.11515

openalex publication_date 2025/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 H⁴ 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 L¹ of the integral kernels yields the existence and convergence in H⁴ of the solutions.

Citations

• On the well-posedness of a certain model with the bi-Laplacian appearing in the Mathematical Biology
• Diffusion-driven blow-up for a non-local Fisher-KPP type model
• Solvability Conditions for Some non Fredholm Operators

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