In this paper we study a variational inequality related to a linear differential operator of elliptic type. We give a pointwise bound for the rearrangement of the solution u, and an estimate for the L2-norm of the gradient of u.

The paper deals with a weakly coupled system of functional-differential equations ${\partial }_t{u}_i(t,x)=f_i(t,x,u(t,x),u,{\partial }_x{u}_i(t,x),{\partial }_{xx}{u}_i(t,x))$, i ∈ S, where (t,x) = (t,x₁,...,xₙ) ∈ (0,a) × G, $u={u_i}_{i\in S}$ and S is an arbitrary set of indices. Initial boundary conditions are considered and the following questions are discussed: estimates of solutions, criteria of uniqueness, continuous dependence of solutions on given functions. The right hand sides of the equations satisfy nonlinear estimates of the Perron type with respect to the unknown functions. The results are...

In questa Nota gli autori presentano alcuni risultati riguardanti il comportamento alla frontiera di domini non cilindrici delle soluzioni positive dell'equazione del calore. Una conseguenza è che due soluzioni positive qualunque, che si annullano su una parte della frontiera laterale, tendono a zero con lo stesso ordine.

In the present paper, the existence of a weak time-periodic solution to the nonlinear telegraph equation $U_{tt}+d{U}_t-\sigma {(x,t,U_x)}_x+aU=f(x,t,U_x,U_t,U)$ with the Dirichlet boundary conditions is proved. No “smallness” assumptions are made concerning the function $f$. The main idea of the proof relies on the compensated compactness theory.

Intra-specific competition in population dynamics can be described by integro-differential equations where the integral term corresponds to nonlocal consumption of resources by individuals of the same population. Already the single integro-differential equation can show the emergence of nonhomogeneous in space stationary structures and can be used to model the process of speciation, in particular, the emergence of biological species during evolution [S. Genieys et al., Math. Model. Nat. Phenom....

Le but de cet article est l’étude de la compétition Réaction-Diffusion pour un problème de type ${\displaystyle \beta {\left(w\right)}_t-d_{\epsilon }diva(x,Dw)+r_{\epsilon }g\left(x,\beta \left(w\right)\right)=f,}$ où $\mathbf{a}$ est un opérateur de Lerray-Lions, $\beta$ est une fonction continue croissante et la réaction $g$ est une fonction croissante qui dépend de l’espace $x$. On suppose que les coefficients de diffusion $d_{\epsilon }$ et de Réaction $r_{\epsilon }$ dépendent du paramètre $\epsilon$ avec $d_{\epsilon }$ et/ou $r_{\epsilon }$ tends vers $+\infty$ lorsque ${\displaystyle \epsilon \to 0}$. Dans le cas où, le coefficient de réaction est très rapide, nous étudions le comportement asymptotique lorsque ${\displaystyle t\to \infty }$ de la solution...