相关论文: A nonlocal model describing tumor angiogenesis
In this follow up paper, we focus on the viscous Burgers equation. There, using the Hopf-Cole transformation, we compute the long time behavior of solutions for some classes of infinite mass initial datas. We show that an enhanced…
We consider a non-local tumour growth model of phase-field type, describing the evolution of tumour cells through proliferation in presence of a nutrient. The model consists of a coupled system, incorporating a non-local Cahn-Hilliard…
We present a study of the Burgers equation in one and two dimensions $d=1,2$ following the analytic approach indicated in the previous paper I. For the problem of the initial conditions decay we consider two classes of initial condition…
In this paper, we propose a new non-isothermal Allen-Cahn (Ginzburg-Landau) model for tumor growth. After deriving it using a microforces approach, we study its well-posedness. In particular, we are able to prove the existence and…
We consider the generalised Burgers equation $$ \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2}=0,\ t \geq 0,\ x \in S^1, $$ where $f$ is strongly convex and $\nu$ is small and…
For some spatially nonlocal diffusion models with a finite range of nonlocal interactions measured by a positive parameter $\delta$, we review their formulation defined on a bounded domain subject to various conditions that correspond to…
We present a mathematical analysis of a reaction-diffusion model describing acid-mediated tumor invasion. The model describes the spatial distribution and temporal evolution of tumor cells, normal cells, and excess lactic acid…
We investigate the evolution of tumor growth relying on a nonlinear model of partial differential equations which incorporates mechanical laws for tissue compression combined with rules for nutrients availability and drug application.…
This study presents a mathematical model formulated as a system of first-order non-linear ordinary differential equations, aimed at examining the effects of different factors, classified as local and systemic factors on a wound healing…
In this work, we investigate the dynamical properties of a reaction-diffusion system arising from tumor-therapy modelling that features both nonlinear interactions and nonlocal delay. By applying the Lyapunov-Schmidt reduction, we establish…
We consider a one--spatial dimensional tumour growth model [2, 3, 4] that consists of three dependent variables of space and time: volume fraction of tumour cells, velocity of tumour cells, and nutrient concentration. The model variables…
The Burgers' equation is a one-dimensional momentum equation for a Newtonian fluid. The Cole-Hopf transformation solves the equation for a given initial and boundary condition. However, in most cases the resulting integral equation can only…
We have previously reported that a universal growth law, as proposed by West and collaborators for all living organisms, appears to be able to describe also the growth of tumors in vivo. In contrast to the assumption of a fixed power…
We consider a nonlocal aggregation equation with nonlinear diffusion which arises from the study of biological aggregation dynamics. As a degenerate parabolic problem, we prove the well-posedness, continuation criteria and smoothness of…
At the continuous level, we consider two types of tumor growth models: the cell density model, which is based on the fluid mechanical construction, is more favorable for scientific interpretation and numerical simulations; and the free…
Cancer is a disease that takes millions of lives every year. Then, to propose treatments, avoid recurrence, and improve the patient's life quality, we need to analyze this disease from a biophysical perspective with a solid mathematical…
We study lower and upper bounds for the density of a diffusion process in ${\mathbb{R}}^n$ in a small (but not asymptotic) time, say $\delta$. We assume that the diffusion coefficients $\sigma_1,\ldots,\sigma_d$ may degenerate at the…
In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes an anisotropic nonlinear diffusion term with a diffusion velocity increasing with respect to vasculature. First, we prove the existence of…
We introduce a new diffuse interface model for tumour growth in the presence of a nutrient, in which we take into account mechanical effects and reversible tissue damage. The highly nonlinear PDEs system mainly consists of a Cahn-Hilliard…
We consider a one-dimensional version of a model obtained in [C. Engwer, A. Hunt, and C. Surulescu: Effective equations for anisotropic glioma spread with proliferation: a multiscale approach and comparisons with previous settings, IMA J.…