We apply functional analytical and variational methods in order to study well-posedness and qualitative properties of evolution equations on product Hilbert spaces. To this aim we introduce an algebraic formalism for matrices of sesquilinear mappings. We apply our results to parabolic problems of different nature: a coupled diffusive system arising in neurobiology, a strongly damped wave equation, and a heat equation with dynamic boundary conditions.

We consider the approximation of a mean field stochastic process by a large interacting particle system. We derive non-asymptotic large deviation bounds measuring the concentration of the empirical measure of the paths of the particles around the law of the process. The method is based on a coupling argument, strong integrability estimates on the paths in Hölder norm, and a general concentration result for the empirical measure of identically distributed independent paths.

Diffusion preserves the positivity of concentrations, therefore, multicomponent diffusion should be nonlinear if there exist non-diagonal terms. The vast variety of nonlinear multicomponent diffusion equations should be ordered and special tools are needed to provide the systematic construction of the nonlinear diffusion equations for multicomponent mixtures with significant interaction between components. We develop an approach to nonlinear multicomponent...

We consider the Cauchy problem for nonlinear parabolic equations with functional dependence represented by the Hale functional acting on the unknown function and its gradient. We prove convergence theorems for a general quasilinearization method in natural subclasses of unbounded solutions.

On considère un opérateur $L$ défini par$$Lu=\sum _{i=1}^n{D}_i{\mathbf{P}}_{i,p}\left(u\right)-\sum _{k=1}^N{D}_t[{\alpha }_{p,k}{u}_k]$$où $u$ est une application de $\bar{\Omega }\times [0,T]$ dans ${\mathbf{R}}^N$ ($\Omega$ ouvert quelconque de $R^n$), ${\mathbf{P}}_{i,p}\left(u\right)$$(1\le i\le n...$

This paper concerns the study of the numerical approximation for the following boundary value problem: $$\left\{\begin{array}{cccc}{u}_t(x,t)-u_{xx}(x,t)=-u^{-p}(x,t),\hfill & 0<x<1,t>0,u_x(0,t)=0,\hfill & u(1,t)=1,t>0,u(x,0)=u_0\left(x\right)>0,& 0\le x\le 1,\end{array}\right.$$ where $p>0$. We obtain some conditions under which the solution of a semidiscrete form of the above problem quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time. Finally, we give some numerical experiments to illustrate our analysis.