Abstract
This paper investigates the existence and uniqueness of solutions for Kirchhoff type parabolic equations with reaction terms in bounded domain, subject to Cauchy-Dirichlet boundary conditions. We target two types of nonlocal diffusion coefficient: diffusion coefficient which is nonlocal only in space, most often considered in the literature, and diffusion coefficient which is nonlocal both in space and time, representing a memory term of the model. Under suitable Lipschitz continuity assumptions on the reaction term and on the nonlocal diffusion coefficient, we establish the existence of a unique solution using a fixed point approach based on Banach’s contraction principle. Under weaker conditions, we also prove the existence of solutions using compactness arguments and Darbo’s fixed point theorem. Our analysis relies on specific function spaces and properties of the solution operator associated with the classical parabolic equation. We provide concrete examples of nonlocal diffusion coefficients with physical meaning, highlighting the applicability of the results.
Authors
David Brumar
Faculty of Mathematics and Computer Science, Babes-Bolyai, University, Cluj-Napoca, Romania
Radu Precup
Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babes-Bolyai, University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania
Keywords
Kirchhoff type parabolic equation, Sobolev space, Weak solution, Operator method, Fixed point theorem, Existence and uniqueness of solution
Paper coordinates
D. Brumar, R. Precup: Kirchhoff type parabolic equations with nonlocal in space and time diffusion coefficients. Rend. Circ. Mat. Palermo, II. Ser 75, 27 (2026), 26 pp. https://doi.org/10.1007/s12215-025-01345-y
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