Consider a one dimensional nonlinear reaction-diffusion equation (KPP equation) with non-homogeneous second order term, discontinuous initial condition and small parameter. For points ahead of the Freidlin-KPP front, the solution tends to 0 and we obtain sharp asymptotics (i.e. non logarithmic). Our study follows the work of Ben Arous and Rouault who solved this problem in the homogeneous case. Our proof is probabilistic, and is based on the Feynman-Kac formula and the large deviation principle...

We consider the derivative NLS equation with general quadratic nonlinearities. In [2] the first author has proved a sharp small data local well-posedness result in Sobolev spaces with a decay structure at infinity in dimension $n=2$. Here we prove a similar result for large initial data in all dimensions $n\ge 2$.

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 $q>m+p-2$.