Two asexual density-dependent population dynamics models with age-dependence and child care are presented. One of them includes the random diffusion while in the other the population is assumed to be non-dispersing. The population consists of the young (under maternal care), juvenile, and adult classes. Death moduli of the juvenile and adult classes in both models are decomposed into the sum of two terms. The first presents death rate by the natural causes while the other describes the environmental...

This paper deals with a quasilinear parabolic-parabolic-elliptic attraction-repulsion chemotaxis system. Boundedness, stabilization and blow-up in this system of the fully parabolic and parabolic-elliptic-elliptic versions have already been proved. The purpose of this paper is to derive boundedness and stabilization in the parabolic-parabolic-elliptic version.

We consider the following reaction-diffusion equation:$$\left(\mathrm{KS}\right)\left\{\begin{array}{cccc}{u}_t=\nabla ·\left(\nabla u^m-u^{q-1}\nabla v\right),\hfill & x\in {ℝ}^N,0<t<\infty ,0=\Delta v-v+u,\hfill & x\in {ℝ}^N,0<t<\infty ,u(x,0)=u_0\left(x\right),\hfill & x\in {ℝ}^N,\hfill \end{array}\right.$$ where $N\ge 1,m>1,q\ge max\{m+\frac{2}{N},2\}$.
In [Sugiyama, Nonlinear Anal.63 (2005) 1051–1062; Submitted; J. Differential Equations (in press)] it was shown that in the case of $q\ge max\{m+\frac{2}{N},2\}$, the above problem (KS) is solvable globally in time for “small $L^{\frac{N(q-m)}{2}}$ data”. Moreover, the decay of the solution (u,v) in $L^p\left({ℝ}^N\right)$ was proved. In this paper, we consider the case of “$q\ge max\{m+\frac{2}{N},2\}$ and small $L^{\ell }$ data” with any fixed $\ell \ge \frac{N(q-m)}{2}$ and show that (i) there exists a time global solution (u,v) of (KS) and it decays to...

We study the large time asymptotic behavior of solutions of the doubly degenerate parabolic equation $u_t=\mathrm{div}(u^{m-1}{\left|Du\right|}^{p-2}Du)-u^q$ with an initial condition $u(x,0)=u_0\left(x\right)$. Here the exponents $m$, $p$ and $q$ satisfy $m+p\ge 3$, $p>1$ and $q>m+p-2$.

Large time behavior of the solution to the nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance is studied. Furthermore, the rate of convergence is given. Initial-boundary value problem with mixed boundary conditions is considered.

% We study the large time behaviour of entropy solutions of the Cauchy problem for a possibly degenerate nonlinear diffusion equation with a nonlinear convection term. The initial function is assumed to have bounded total variation. We prove the convergence of the solution to the entropy solution of a Riemann problem for the corresponding first order conservation law.

In this talk we shall present some joint work with A. Grigory’an. Upper and lower estimates on the rate of decay of the heat kernel on a complete non-compact riemannian manifold have recently been obtained in terms of the geometry at infinity of the manifold, more precisely in terms of a kind of $L^2$ isoperimetric profile. The main point is to connect the decay of the $L^1-L^{\infty }$ norm of the heat semigroup with some adapted Nash or Faber-Krahn inequalities, which is done by functional analytic methods. We shall...

This note is devoted to the study of the long time behaviour of solutions to the heat and the porous medium equations in the presence of an external source term, using entropy methods and self-similar variables. Intermediate asymptotics and convergence results are shown using interpolation inequalities, Gagliardo-Nirenberg-Sobolev inequalities and Csiszár-Kullback type estimates.

Dans ce papier on étudie l’existence et le comportement asymptotique des solutions de type ondes progressives à propagations finies de l’équation $U_t=A{\left({\left|U_x\right|}^{p-2}{U}_x\right)}_x+K{U}^q$. On prouve que ces solutions existent si et seulement si $q\&lt;1$ et $c\&lt;0$ ou bien $q\le p-1$ et $c\&gt;0$. On donne aussi le comportement asymptotique de ces solutions.