Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems.
Global existence and convergence of solutions to a cross-diffusion cubic predator-prey system with stage structure for the prey.
Global existence and estimates of solutions for a quasilinear parabolic system.
Global existence for degenerate quadratic reaction-diffusion systems
Homogenization of a carcinogenesis model with different scalings with the homogenization parameter
In the context of periodic homogenization based on two-scale convergence, we homogenize a linear system of four coupled reaction-diffusion equations, two of which are defined on a manifold. The system describes the most important subprocesses modeling the carcinogenesis of a human cell caused by Benzo-[a]-pyrene molecules. These molecules are activated to carcinogens in a series of chemical reactions at the surface of the endoplasmic reticulum, which constitutes a fine structure inside the cell....
Influence of diffusion on interactions between malignant gliomas and immune system
We analyse the influence of diffusion and space distribution of cells in a simple model of interactions between an activated immune system and malignant gliomas, among which the most aggressive one is GBM Glioblastoma Multiforme. It turns out that diffusion cannot affect stability of spatially homogeneous steady states. This suggests that there are two possible outcomes-the solution is either attracted by the positive steady state or by the semitrivial one. The semitrivial steady state describes...
Instability of Turing type for a reaction-diffusion system with unilateral obstacles modeled by variational inequalities
We consider a reaction-diffusion system of activator-inhibitor type which is subject to Turing's diffusion-driven instability. It is shown that unilateral obstacles of various type for the inhibitor, modeled by variational inequalities, lead to instability of the trivial solution in a parameter domain where it would be stable otherwise. The result is based on a previous joint work with I.-S. Kim, but a refinement of the underlying theoretical tool is developed. Moreover, a different regime of parameters...
Long time behaviour of a Cahn-Hilliard system coupled with viscoelasticity
The long-time behaviour of a unique regular solution to the Cahn-Hilliard system coupled with viscoelasticity is studied. The system arises as a model of the phase separation process in a binary deformable alloy. It is proved that for a sufficiently regular initial data the trajectory of the solution converges to the ω-limit set of these data. Moreover, it is shown that every element of the ω-limit set is a solution of the corresponding stationary problem.
Mathematical models of tumor growth systems
We study a class of parabolic-ODE systems modeling tumor growth, its mathematical modeling and the global in time existence of the solution obtained by the method of Lyapunov functions.
Non-constant positive steady states for a predator-prey cross-diffusion model with Beddington-DeAngelis functional response.
Non-simultaneous blow-up for a reaction-diffusion system with absorption and coupled boundary flux.
On a Caginalp phase-field system with a logarithmic nonlinearity
We consider a phase field system based on the Maxwell Cattaneo heat conduction law, with a logarithmic nonlinearity, associated with Dirichlet boundary conditions. In particular, we prove, in one and two space dimensions, the existence of a solution which is strictly separated from the singularities of the nonlinear term and that the problem possesses a finite-dimensional global attractor in terms of exponential attractors.
On gradient estimates and other qualitative properties of solutions of nonlinear non autonomous parabolic systems.
Periodic solutions of -Laplacian systems with a nonlinear convection term.
Phase field model for mode III crack growth in two dimensional elasticity
A phase field model for anti-plane shear crack growth in two dimensional isotropic elastic material is proposed. We introduce a phase field to represent the shape of the crack with a regularization parameter and we approximate the Francfort–Marigo type energy using the idea of Ambrosio and Tortorelli. The phase field model is derived as a gradient flow of this regularized energy. We show several numerical examples of the crack growth computed with an adaptive mesh finite element method.
Quenching for a reaction-diffusion system with coupled inner singular absorption terms.
Regularity and uniqueness in quasilinear parabolic systems
Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma.
Self-similar solutions for the two-dimensional Nernst-Planck-Debye system
We investigate the two-component Nernst-Planck-Debye system by a numerical study of self-similar solutions using the Runge-Kutta method of order four and comparing the results obtained with the solutions of a one-component system. Properties of the solutions indicated by numerical simulations are proved and an existence result is established based on comparison arguments for singular ordinary differential equations.
Semi-analytical approach to initial problems for systems of nonlinear partial differential equations with constant delay
This paper deals with the differential transform method for solving of an initial value problem for a system of two nonlinear functional partial differential equations of parabolic type. We consider non-delayed as well as delayed types of coupling and the different variety of initial functions are thought over. The convergence of solutions and the error estimation to the presented procedure is studied. Two numerical examples for non-delayed and delayed systems are included.
Solutions to a model for compressible immiscible two phase flow in porous media.