Lower bounds for the finite-time blow-up of solutions of a cancer invasion model

In this article, we consider non-negative solutions of the nonlinear cancer invasion mathematical model involving proliferation and growth functions with homogeneous Neumann and Robin type boundary conditions. We first obtain lower bounds for the finite time blow-up of solutions in R3 with assumed boundary conditions. Finally, we extend the blow-up results of the given system in R2 using first-order differential inequality techniques and under appropriate assumptions on data

Renormalized and Entropy Solutions of Tumor Growth Model with Nonlinear Acid Production

This paper establishes the existence of renormalized and entropy solutions for a system of nonlinear reaction-diffusion equations which describes the tumor growth along with acidification and interaction. Under the assumptions of L1 data and no growth conditions with zero Dirichlet boundary conditions, we prove the existence of renormalized and entropy solutions for the considered mathematical model

A biophysical model of tumor invasion

Three-dimensional finite element computations of a cancer invasion model with nonlinear density-dependent diffusion and haptotactic sensitivity function are presented. The nonlinear model includes three key variables, namely the cancer cell density, the extra cellular matrix (ECM) density and the matrix degrading enzymes (MDE) concentration. In order to investigate the effects of tumor growth and invasion on a realistic geometry, the interactions between the cancer cells and the host tissue are incorporated into the model. The convergence study and the validation are first performed for the proposed numerical scheme. Then the effects of nonlinear diffusion and ECM-dependent haptotaxis on tumor growth and invasion in three-dimensional geometries are presented. Finally, several numerical simulations are performed with different combinations of nonlinear diffusion and haptotaxis functions to get an insight into the tumor invasion on a realistic (breast) geometry. The proposed computational model can be used to predict the location and shape of the tumor in realistic geometries at a particular instance. (C) 2016 Elsevier B.V. All rights reserved

Renormalized and entropy solutions of nonlinear parabolic systems

In this article, we study the existence of renormalized and entropy solutions of SIR epidemic disease cross-diffusion model with Dirichlet boundary conditions. Under the assumptions of no growth conditions and integrable data, we establish that the renormalized solution is also an entropy solution

Lower bounds for the finite-time blow-up of solutions of a cancer invasion model

In this article, we consider non-negative solutions of the nonlinear cancer invasion mathematical model involving proliferation and growth functions with homogeneous Neumann and Robin type boundary conditions. We first obtain lower bounds for the finite time blow-up of solutions in $\mathbb{R}^3$ with assumed boundary conditions. Finally, we extend the blow-up results of the given system in $\mathbb{R}^2$ using first-order differential inequality techniques and under appropriate assumptions on data

Weak Solutions for Nonlinear Parabolic Equations with Variable Exponents

summary:In this work, we study the existence and uniqueness of weak solutions of fourth-order degenerate parabolic equation with variable exponent using the difference and variation methods