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Communications on Pure & Applied Analysis

2005 , Volume 4 , Issue 4

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Regularity of the attractor for symmetric regularized wave equation
Abdelghafour Atlas
2005, 4(4): 695-704 doi: 10.3934/cpaa.2005.4.695 +[Abstract](106) +[PDF](196.7KB)
In this article we study the long time behavior of the solutions to dissipative generalized symmetric regularized long wave equations with damping. We prove that the long time behavior is described by a compact attractor which captures all trajectories in $ H^1\times L^2$. We moreover establish that the attractor $\mathcal A$ is a compact set in $ H^2\times H^1$.
Weakly dissipative semilinear equations of viscoelasticity
Monica Conti and V. Pata
2005, 4(4): 705-720 doi: 10.3934/cpaa.2005.4.705 +[Abstract](251) +[PDF](258.8KB)
We consider an integro-partial differential equation of hyperbolic type with a cubic nonlinearity, in which no dissipation mechanism is present, except for the convolution term accounting for the past memory of the variable. Setting the equation in the history space framework, we prove the existence of a regular global attractor.
Global existence and blow-up to a reaction-diffusion system with nonlinear memory
Lili Du, Chunlai Mu and Zhaoyin Xiang
2005, 4(4): 721-733 doi: 10.3934/cpaa.2005.4.721 +[Abstract](111) +[PDF](214.7KB)
In this paper, we consider a reaction-diffusion system coupled by nonlinear memory. Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time. Furthermore, the blow-up rate estimates are obtained.
A variational principle for nonlinear transport equations
Nassif Ghoussoub
2005, 4(4): 735-742 doi: 10.3934/cpaa.2005.4.735 +[Abstract](102) +[PDF](185.5KB)
We verify -after appropriate modifications- an old conjecture of Brezis-Ekeland [4] concerning the feasibility of a global and variational approach to the problems of existence and uniqueness of solutions of non-linear transport equations, which do not normally fit in an Euler-Lagrange framework. Our method is based on a concept of "anti-self duality" that seems to be inherent in many problems, including gradient flows of convex energy functionals treated in [10] and other parabolic evolution equations ([7]).
Structural stability of optimal control problems
M'hamed Kesri
2005, 4(4): 743-756 doi: 10.3934/cpaa.2005.4.743 +[Abstract](119) +[PDF](208.9KB)
In this article, we prove that solutions for a class of optimal control problems we study, defined "in a less restrictive sense then usual" a dynamical system such that every state which is optimal is a sink and between every pair of sinks there is a unique source at least generically. The corresponding problem is shown to be structurally stable.
Some special solutions of the multidimensional Euler equations in $R^N$
Tianhong Li
2005, 4(4): 757-762 doi: 10.3934/cpaa.2005.4.757 +[Abstract](132) +[PDF](176.4KB)
In this paper we give some special solutions, including some blow-up solutions, to the multi-dimensional Euler equations for compressible gas dynamics. We also show that finite energy implies nonexistence of $\delta$-function blow-up for $\gamma>1$.
Global existence of the entropy solutions to the isentropic relativistic Euler equations
Yachun Li and Qiufang Shi
2005, 4(4): 763-778 doi: 10.3934/cpaa.2005.4.763 +[Abstract](130) +[PDF](170.4KB)
We study the relativistic Euler equations for isentropic fluids with a general equation of state $p=p(\rho)$ satisfying the genuine nonlinearity condition. For the $\gamma$-law case $(1<\gamma<2)$, we establish an existence theorem for global entropy solutions to the Cauchy problem using the Glimm difference scheme. For general pressure, the nonlinear elementary waves and Riemann problem are also studied.
Coupled problems on stationary non-isothermal flow of electrorheological fluids
W. G. Litvinov and R. H.W. Hoppe
2005, 4(4): 779-803 doi: 10.3934/cpaa.2005.4.779 +[Abstract](116) +[PDF](359.1KB)
We set up and investigate a coupled problem on stationary non-isothermal flow of electrorheological fluids. The problems consist in finding functions of velocity, pressure and temperature which satisfy the motion equations, the condition of incompressibility, the equation of the balance of thermal energy and boundary conditions. We consider original and regularized coupled problems. In the regularized problem the dissipation of energy is defined by the regularized velocity field which leads to a nonlocal model. We introduce the notions of generalized solutions for the original and regularized problems. The existence of the generalized solution of the regularized problem is proved by using the methods of monotonicity, compactness, and topological degree. We prove that there exists a solution of the original problem where the domain of flow is two-or three-dimensional (in the case of a three-dimensional domain an extra condition is being assumed). It is shown that the solution of the original problem is a limiting point of the set of solutions of the regularized problems in which the parameter of regularization tends to zero.
On the dimension of the attractor for a class of fluids with pressure dependent viscosities
M. Bulíček, Josef Málek and Dalibor Pražák
2005, 4(4): 805-822 doi: 10.3934/cpaa.2005.4.805 +[Abstract](199) +[PDF](301.7KB)
We consider two-dimensional flows of an incompressible non Newtonian fluid where the departure from the Navier-Stokes fluid is due to the viscosity depending on both the rate of deformation and the pressure. We assume that the resulting extra-stress is uniformly elliptic and its derivative with respect to pressure is bounded in a proper manner. Considering just the spatially-periodic setting, one can prove the global existence and uniqueness of the strong solution. Using the so-called method of trajectories, we also prove the existence of an exponential attractor and estimate its fractal dimension in terms of the data of the equation.
Stability and index of periodic solutions of a nonlinear telegraph equation
Rafael Ortega
2005, 4(4): 823-837 doi: 10.3934/cpaa.2005.4.823 +[Abstract](124) +[PDF](253.7KB)
For a class of abstract evolution equations, the topological index is sufficient to characterize the stability of periodic solutions. This fact can be applied to discuss the stability properties in the sine-Gordon equation with friction and periodic forcing.
The Boussinesq system:dynamics on the center manifold
Claudia Valls
2005, 4(4): 839-860 doi: 10.3934/cpaa.2005.4.839 +[Abstract](120) +[PDF](316.7KB)
In this paper we analyze the behavior of small amplitude solutions of the variant of the classical Boussinesq system given by

$ \partial_t u = -\partial_x v - \alpha \partial_{x x x}v - \partial_x(u v), \quad \partial_t v = - \partial_x u - v \partial_x v,$

For $\alpha \leq 1$, this equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every $\alpha$, it admits solutions that are quasiperiodic in time. The proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the Boussinesq system to the center manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem and studying the Birkhoff normal form.

Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition
Hongwei Zhang and Qingying Hu
2005, 4(4): 861-869 doi: 10.3934/cpaa.2005.4.861 +[Abstract](130) +[PDF](198.7KB)
In this paper the asymptotic behavior and nonexistence of solution for the wave equation $u_{t t}-\Delta u=0$ with nonlinear boundary condition $\frac {\partial u}{\partial \nu}=-|u_t|^{m-2}u_t+|u|^{p-2}u$ are given.
Structure of positive radial solutions including singular solutions to Matukuma's equation
Hiroshi Morishita, Eiji Yanagida and Shoji Yotsutani
2005, 4(4): 871-888 doi: 10.3934/cpaa.2005.4.871 +[Abstract](150) +[PDF](237.8KB)
We consider positive solutions of Matukuma's equation, which is described by a nonlinear elliptic equation with a weight. Any radially symmetric solution of this equation is said to be regular or singular according to its behavior near the origin and infinity. We investigate the structure of positive radial regular and singular solutions.
On a nonlinear parabolic system-modeling chemical reactions in rivers
Wenxiong Chen, Congming Li and Eric S. Wright
2005, 4(4): 889-899 doi: 10.3934/cpaa.2005.4.889 +[Abstract](118) +[PDF](217.0KB)
We study the global existence and qualitative properties of the solutions of nonlinear parabolic systems. Such systems commonly arise in situations pertaining to reactive transport. Particular examples include the modeling of chemical reactions in rivers or in blood streams.
In this paper, we first establish a maximum principle that generates a-priori bounds for solutions to a broad class of parabolic systems. Afterward, we develop an alternative technique for establishing global bounds on solutions to a specific system of three equations that belong to a different class of parabolic systems. Finally, we prove that the only bounded traveling wave solutions to this system are constants.

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