Gema Duro

Large time behavior for convection-diffusion equations in IRN with asymptotically constant diffusion

We describe the large time behavior of solutions of the scalar convection diffusion for some positive constants C and δ.

Well posedness of an integrodifferential kinetic model of Fokker–Planck type for angiogenesis

Tumor induced angiogenesis processes including the effect of stochastic motion and branching of blood vessels can be described coupling a (nonlocal in time) integrodifferential kinetic equation of Fokker–Planck type with a diffusion equation for the tumor induced ingiogenic factor. The chemotactic force field depends on the flux of blood vessels through the angiogenic factor. We develop an existence and uniqueness theory for this system under natural assumptions on the initial data. The proof combines the construction of fundamental solutions for associated linearized problems with comparison principles, sharp estimates of the velocity integrals and compactness results for this type of kinetic and parabolic operators

Constructing solutions for a kinetic model of angiogenesis in annular domains

Abstract We present an iterative technique to construct stable solutions for an angiogenesis model set in an annular region. Branching, anastomosis and extension of blood vessel tips is described by an integrodifferential kinetic equation of Fokker–Planck type supplemented with nonlocal boundary conditions and coupled to a diffusion problem with Neumann boundary conditions through the force field created by the tumor induced angiogenic factor and the flux of vessel tips. Convergence proofs exploit balance equations, estimates of velocity decay and compactness results for kinetic operators, combined with gradient estimates of heat kernels for Neumann problems in non convex domains.

Well posedness of an angiogenesis related integrodifferential diffusion model

Abstract We prove existence and uniqueness of nonnegative solutions for a nonlocal in time integrodifferential diffusion system related to angiogenesis descriptions. Fundamental solutions of appropriately chosen parabolic operators with bounded coefficients allow us to generate sequences of approximate solutions. Comparison principles and integral equations provide uniform bounds ensuring some convergence properties for iterative schemes and providing stability bounds. Uniqueness follows from chained integral inequalities.

Asymptotic profiles for covection–diffusion equations with variable diffusion

Abstract We investigate the large time behavior of solutions of the convection-diffusion equation u t − div (a(x) ∇ u)=d· ∇ (|u| q−1 u) d ∈ R N , in (0,∞)× R N with integrable initial data u0(x). We take a(x)=1+b(x)>0 with b smooth and decaying to zero fast enough as x→∞. When q>1+ 1 N , it is known that the solutions behave, in a first approximation, like the solutions of the head equation taking the same initial data as t→∞. We show here the influence of the nonlinear term and the variable diffusion in the large time behavior by obtaining the second term in the asymptotic development of solutions as t→∞.

Instability and Collapse in Discrete Wave Equations

Abstract Unstable growth phenomena in spatially discrete wave equations are studied. We characterize sets of initial states leading to instability and collapse and obtain analytical predictions for the blow-up time. The theoretical predictions are con- trasted with the numerical solutions computed by a variety of schemes. The behavior of the systems in the continuum limit and the impact of discreteness and friction are discussed.

A convergent numerical scheme for integrodifferential kinetic models of angiogenesis

Abstract We study a robust finite difference scheme for integrodifferential kinetic systems of Fokker–Planck type modeling tumor driven blood vessel growth. The scheme is of order one and enjoys positivity features. We analyze stability and convergence properties, and show that soliton-like asymptotic solutions are correctly captured. We also find good agreement with the solution of the original stochastic model from which the deterministic kinetic equations are derived working with ensemble averages. A numerical study clarifies the influence of velocity cut-offs on the solutions for exponentially decaying data.

Large Time Asymptotics in Contaminant Transport in Porous Media with Variable Diffusion

This paper is devoted to the study of the large time behavior of solutions of the equations modelling contaminant transport in porous media with variable diffusion for integrable initial data.