Numerical experiments suggest interesting properties in the several fields of fluid dynamics, plasma physics and population dynamics. Among such properties, we may observe the interesting phenomena; that is, the repeated appearance and disappearance phenomena of the region penetrated by the fluid in the flow through a porous media with absorption. The model equation in two dimensional space is written in the form of the initial-boundary value problem for a nonlinear diffusion equation with the effect...

This paper is mainly concerned with the blow-up and global existence profile for the Cauchy problem of a class of fully nonlinear degenerate parabolic equations with reaction sources.

This paper is concerned with the initial boundary value problem for a nonlocal p-Laplacian evolution equation with critical initial energy. In the framework of the energy method, we construct an unstable set and establish its invariance. Finally, the finite time blow-up of solutions is derived by a combination of the unstable set and the concavity method.

This paper deals with the blow-up properties of the non-Newtonian polytropic filtration equation $u_t-div\left(\right|\nabla u^m{|}^{p-2}\nabla u^m)=f\left(u\right)$ with homogeneous Dirichlet boundary conditions. The blow-up conditions, upper and lower bounds of the blow-up time, and the blow-up rate are established by using the energy method and differential inequality techniques.

We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$-Laplacian equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part $S_T$ of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular...

We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions $u$: either the space derivative $u_x$ blows up in finite time (with $u$ itself remaining bounded), or $u$ is global and converges in $C^1$ norm to the unique steady state. The main difficulty is to prove $C^1$ boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out the method of...

We consider a quasilinear parabolic system which has the structure of Patlak-Keller-Segel model of chemotaxis and contains a class of models with degenerate diffusion. A cell population is described in terms of volume fraction or density. In the latter case, it is assumed that there is a threshold value which the density of cells cannot exceed. Existence and uniqueness of solutions to the corresponding initial-boundary value problem and existence of space inhomogeneous stationary solutions are discussed....

On étudie la classification des solutions du problème elliptique$${\left({\left|u^{\prime }\right|}^{p-2}{u}^{\prime }\right)}^{\prime }\left(t\right)+{\left({\left|u\right|}^{q-1}u\right)}^{\prime }\left(t\right)-f\left(t\right){\left|u\right|}^{m-1}u\left(t\right)=0,t\>0,$$où $q\>1,p\ge m+1\>2$et $f$ une fonction changeant de signe. En utilisant une méthode de tire, On montre qu’en partant avec une dérivée initiale nulle toutes les solutions sont globales. De plus si $p\>m+1$ et $q\>(p-1)(m+1)/p$ l’ensemble des solutions est constitué d’une seule solution à support compact et de deux familles de solutions ; celles qui sont strictement positives et celles qui changent de signes. On montre aussi que ces deux familles tendent vers l’infini quand...

We compare dewetting characteristics of a thin nonwetting solid film in the absence of stress, for two models of a wetting potential: the exponential and the algebraic. The exponential model is a one-parameter (r) model, and the algebraic model is a two-parameter (r, m) model, where r is the ratio of the characteristic wetting length to the height of the unperturbed film, and m is the exponent of h (film height) in a smooth function that interpolates the system's surface energy above and below...

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...