ISSN:
1078-0947
eISSN:
1553-5231
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Discrete & Continuous Dynamical Systems - A
September 2006 , Volume 16 , Issue 3
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We study the effect of a zero order term on existence and optimal summability of solutions to the elliptic problem
$ -\text{div}( M(x)\nabla u)- a\frac{u}{|x|^2}=f \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega$,
with respect to the summability of $f$ and the value of the parameter $a$. Here $\Omega$ is a bounded domain in $\mathbb{R}^N$ containing the origin.
We describe as $\varepsilon \to 0$ radially symmetric sign-changing solutions to the problem
$ -\Delta u =|u|^{\frac 4{N-2} -\varepsilon} u \quad \text{in } B $
where $B$ is the unit ball in $\R^N$, $N\ge 3$, under zero Dirichlet boundary conditions. We construct radial solutions with $k$ nodal regions which resemble a superposition of "bubbles'' of different signs and blow-up orders, concentrating around the origin. A dual phenomenon is described for the slightly supercritical problem
$ -\Delta u =|u|^{\frac 4{N-2} +\varepsilon} u \quad \text{in } \R^N \setminus B $
under Dirichlet and fast vanishing-at-infinity conditions.
We study the Cauchy problem with bounded continuous initial-value functions for the differential-difference equation
$\frac{\partial u}{\partial t}= \sum$nk,j,m=1$ a_{kjm}\frac{\partial^2u}{\partial x_k\partial x_j} (x_1,...,x_{m-1},x_m+h_{kjm},x_{m+1},...,x_n,t),$
assuming that the operator on the right-hand side of the equation is strongly elliptic and the coefficients $a_{kjm}$ and $h_{kjm}$ are real. We prove that this Cauchy problem has a unique solution (in the sense of distributions) and this solution is classical in ${\mathbb R}^n \times (0,+\infty),$ find its integral representation, and construct a differential parabolic equation with constant coefficients such that the difference between its classical bounded solution satisfying the same initial-value function and the investigated solution of the differential-difference equation tends to zero as $t\to\infty$.
We prove a universal bound, independent of the initial data, for all global nonnegative solutions of the Dirichlet problem of the quasilinear parabolic equation with convection $u_t = \Delta u^m +a\cdot \nabla u^q+ u^p$ in $\Omega\times (0,\infty)$, where $\Omega$ is a smoothly bounded domain in $\mathbf{R^N}$, $a \in \mathbf {R^N}$, $ 1 \le m < p$ <$m+2/(N+1)$ and $(m+1)/2 \le q < (m+p)/2$ (or $q = (m+p)/2$ and $|a|$ is small enough). The universal bound can be obtained by showing that any solution $u$ in $\Omega\times(0,T)$ satisfies the estimate $ \||u(t)\||_{L^{\infty}(\Omega)} \le C(p,m,q,|a|, \Omega,\alpha,T)t^{-\alpha}$ in $ 0 $<$t \le T/2 $ for $\alpha $>$ (N+1 )/[(m-1)(N+1)+2]$, which describes the initial blow-up rates of solutions.
We present the necessary and sufficient conditions and a new method to study the existence of pullback attractors of nonautonomous infinite dimensional dynamical systems. For illustrating our method, we apply it to nonautonomous 2D Navier-Stokes systems. We also show that the parametrically inflated pullback attractors and uniform attractors are robust with respect to the perturbations of both cocycle mappings and driving systems. As an example, we consider the nonautonomous 2D Navier-Stokes system with rapidly oscillating external force.
It is proved that for a prescribed potential $V$ there are many quasi-periodic solutions of nonlinear wave equations $u_{t t}-u_{x x}+V(x)u\pm u^3+O(|u|^5)=0$ subject to Dirichlet boundary conditions.
In this paper we prove the persistence of elliptic lower dimensional invariant tori for nearly integrable Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition by an improved KAM iteration, and the persisting invariant tori are Gevrey smooth with respect to parameters in the sense of Whitney, with a Gevrey index depending on the Gevrey class of Hamiltonian systems and on the exponent in the Diophantine condition. Moreover the Gevrey index should be optimal for the Diophantine condition in the proof of our theorem.
We consider the following singularly perturbed Schrödinger-Maxwell system with Dirichlet boundary condition
$-\varepsilon^2\Delta v+v+\omega\phi v-
\varepsilon^{\frac{p-1}{2}} v^p=0
\quad \text{ in}\ B_1,$
$-\Delta \phi=4\pi \omega v^2
\quad \text{in}\ B_1$,
$v,\ \phi>0 \ \text{in}\ B_1
\quad \text{and}\quad v=\phi=0
\quad \text{on}\ \partial B_1$,
where $B_1$ is the unit ball in $\mathbb{R}^3,\ \omega>0$ and $\ \frac{7}{3}$<$p\leq 5$ are constants, and $\varepsilon$>$0$ is a small parameter. Using the localized energy method, we prove that for every sufficiently large integer $N$, the system has a family of radial solutions $(v_\varepsilon, \phi_\varepsilon)$ such that $v_\varepsilon$ has $N$ sharp spheres concentrating on a sphere $\{|x|=r_N\}$ as $\varepsilon\to 0$.
The seemingly straightforward stability issue in three-dimensional homogeneous continuous piecewise linear systems with two linear zones is considered. The only equilibrium at the origin, being in the separation plane of the linear zones, has two linearization matrices. For the important case where both matrices have complex eigenvalues with the whole spectrum in the left half plane, the possible counter-intuitive instability of the origin is proved. Some sufficient conditions for the global asymptotic stability of such systems are also shown.
In this paper we study an eigenvalue boundary value problem which arises when seeking radial convex solutions of the Monge-Ampère equations. We shall establish several criteria for the existence, multiplicity and nonexistence of strictly convex solutions for the boundary value problem with or without an eigenvalue parameter.
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