A nonlocal model describing tumor angiogenesis. (February 2023)

• Record Type: Journal Article
• Authors: Granero-Belinchón, Rafael
• Is Part Of: Nonlinear analysis. Volume 227(2023)
• Publication Date: 2023-02
• Subjects: Burgers equation -- Dispersive equation -- Angiogenesis; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.113180
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we derive and study a new mathematical model that describes the onset of angiogenesis. This new model takes the form of a nonlocal Burgers equation with both diffusive and dispersive terms. For a particular value of the parameters, the equation reduces to ∂ t p − 1 2 ( − Δ ) ( α − 1 ) / 2 H ∂ t p = − 1 2 ( − Δ ) α / 2 p + p ∂ x p − ∂ x p, where H denotes the Hilbert transform. In addition to the derivation of the new model, the main novelty of the present paper is that we also prove a number of well-posedness results. Finally, some preliminary numerical results are shown. These numerical results suggest that the dynamics of the equation is rich enough to have solutions that blow up in finite time. Highlights: In this paper, we derive and study a new mathematical model that describes the onset of angiogenesis. This new model takes the form of a nonlocal Burgers equation with both diffusive and dispersive terms. In addition to the derivation of the new model, we also prove a number of well-posedness results.