A class of strongly coupled reaction-diffusion systems is studied. First, under some conditions, it is shown that a nonnegative solution exists globally in time. After that, asymptotic behavior of the nonnegative global solution is considered. Especially, when the solution uniformly converges to a steady state with a polynomial rate as time goes to infinity, large-time approximation of the solution is investigated. By the energy method and analytic semigroup theory, it is proved that a global solution for the corresponding system of ordinary differential equations has the role of an asymptotic solution for the reaction-diffusion system and that the spatial average of the global solution to the reaction-diffusion system gives an asymptotic description.

We investigate the connection between the geometry of an unbounded domain $\Omega$ and the existence and qualitative behaviour of solutions to a degenerate doubly non linear parabolic equation posed in $\Omega$. The domain $\Omega$ is assumed to be ``narrowing'' at infinity in a suitable sense, so that it has infinite volume. On the boundary of $\Omega$ we prescribe a homogeneous Neumann condition. Among other results, we prove sharp estimates for the finite speed of propagation of the support of positive solutions originating from initial data with bounded support. This is done by means of a new approach, which is flexible enough to be applied to the geometry at hand, and to cover the case of initial data measures. We also show that even if the initial datum has finite mass, the solution need not be globally bounded over $\Omega$ for a fixed positive time. We provide sharp estimates for such solutions. Our main tool is a new embedding inequality connected with the geometry of $\Omega$.

In this paper we consider the existence and decay of moments of the solutions to the Navier-Stokes equations in the whole space ${{\bf R}^n}$; $n \geq 2$ for existence, $2 \leq n \leq 5$ for decay. The decay obtained is algebraic, of order \[ \int_{{{\bf R}^n}} |x|^k |u|^2\,dx \leq C(t+1)^{-2\mu(1-k/n)} \] for $0\leq k\leq n$, for solutions $u$ of appropriate data for which the $L^2$ norm decays at a rate of order $\mu$. That is for solutions that satisfy $||u(t)||_2 \leq C(t+1) ^{-\mu}$. Where $\mu >1/2$. Such solutions are easy to obtain as for example it suffices for $n=3$ that the data $u_0$ is in $L^2 \cap L^1$.

We study concentrated positive bound states of the following nonlinear Schr\"odinger equation: \[ h^2 \Delta u - V(x) u + u^p=0,\ \ \ u>0, \ \ x \in R^N , \] where $ p$ is subcritical. We prove that, at a local maximum point $x_0$ of the potential function $V(x)$ and for arbitrary positive integer $K (K>1)$, there always exist solutions with $K$ interacting bumps concentrating near $x_0$. We also prove that at a nondegenerate local minimum point of $V(x) $ such solutions do not exist.

This paper is the fourth in a series that undertakes a systematic investigation of Riemann solutions of systems of two conservation laws in one spatial dimension. In this paper, three degeneracies that can occur only in transitional wave groups, together with the degeneracies that pair with them, are studied in detail. Conditions for a codimension-one degeneracy are identified in each case, as are conditions for folding of the Riemann solution surface.

We define a Thomas-Fermi-von Weizsäcker model for polymers and thin films through a thermodynamic limit process. Our argument makes use of standard techniques for elliptic PDEs, such as maximum principles or supersolution methods. In the course of our work, we establish some existence and uniqueness results for a system of nonlinear PDEs.

Let $\Omega\subset\Bbb{R}^2$ be a bounded domain and $H\in\Bbb{R}^3$. The Landau-Lifshitz equation with external field $H$ and boundary data $\gamma\in C^{\infty}(\partial\Omega;\Bbb{S}^2)$ is the following: $$\Delta u+|\nabla u|^2u-(H,u)u+H=0\quad\text{in $\Omega$},\quad u=\gamma\quad\text{on $\partial\Omega$}.$$ Here $u\in C^{\infty}(\Omega;\Bbb{S}^2)$. We study the asymptotic behavior of the solutions of this equation as $H\to 0$. We show that the "large solutions" obtained by Hong and Lemaire blow up only when $\gamma\equiv\text{const.}$ and in such a case blow-up occurs only at a single point in $\Omega$. We characterize the blow-up point as a critical point of a certain function defined in $\Omega$. We also give the asymptotic value estimate of $\|\nabla u\|_{{L^{\infty}(\Omega)}}$ as $H\to 0$.

An extension of the Hardy-Littlewood inequality for rearrangements is established. It is used for giving several conditions of existence of a minimum for nonweakly-lower-semicontinuous functionals of the form $J(v) \! = \! \int_0^1f(x,v(x),v'(x)) dx$ with constraints on $v$ and $v'$.

Reaction-diffusion and coupled oscillator systems are considered. They possess inertial manifolds and, moreover, in a sense, their inertial dynamics can be "controlled" by some system parameters $\mathcal P$: the inertial dynamics can be specified to within an arbitrarily small error by adjusting of $\mathcal P$. Due to the classical persistence hyperbolic set theorem, this property of their inertial forms yields that any hyperbolic local attractors and invariant sets can be embedded into the global attractors of these systems. The method of the proof is based on recent results from the neural network theory and the ideas connected with the dynamics of localized modes in singular perturbed systems. This approach can be considered as a development of the method of the realization of vector fields pioneering by P. Poláčik and has a physical interpretation. It is shown, in particular, that the fundamental models of neural network type (the Hopfield systems) can generate any structurally stable (persistent) large-time behaviour.

In this paper we prove monotonicity and symmetry properties of positive solutions of the equation $ - div (|Du|^{p-2}Du)=f(u)$, $1 <p <2$, in a smooth bounded domain ${\Omega} $ satisfying the boundary condition $u=0$ on ${\partial} {\Omega} $. We assume $f$ locally Lipschitz continuous only in $(0, \infty) $ and either $f \geq 0 $ in $[0, \infty] $ or $f$ satisfying a growth condition near zero. In particular we can treat the case of $f(s) = s^{{\alpha}} -c\, s^q $, ${\alpha} >0 $, $ c \geq 0 $, $ q \geq p-1 $. As a consequence we get an extension to the $p$--Laplacian case of a symmetry theorem of Serrin for an overdetermined problem in bounded domains. Finally we apply the results obtained to the problem of finding the best constants for the classical isoperimetric inequality and for some Sobolev embeddings.