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1.
This paper is concerned with the attractor for a viscous two-component generalization of the Camassa-Holm equation subject to an external force, where the viscosity term is given by a second order differential operator. The global existence of solution to the viscous two-component Camassa-Holm equation with the periodic boundary condition is studied. We obtain the compact and bounded absorbing set and the existence of the global attractor in H2×H2 for the viscous two-component Camassa-Holm equation by uniform prior estimate and many inequalities.  相似文献   

2.
The relaxation-time limit from the quantum hydrodynamic model to the quantum drift-diffusion equations in R3 is shown for solutions which are small perturbations of the steady state. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohm potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. The relaxation-time limit is performed both in the stationary and the transient model. The main assumptions are that the steady-state velocity is irrotational, that the variations of the doping profile and the velocity at infinity are sufficiently small and, in the transient case, that the initial data are sufficiently close to the steady state. As a by-product, the existence of global-in-time solutions to the quantum drift-diffusion model in R3 close to the steady-state is obtained.  相似文献   

3.
We present comparison, uniqueness and existence results for unbounded solutions of a viscous Hamilton-Jacobi or eikonal equation. The equation includes an unbounded potential term V(x) subject to a quadratic upper bound. The results are obtained through a tailor-made change of variables in combination with the Hopf-Cole transformation. An integral representation formula for the solution of the Cauchy problem is derived in the case where V(x)=ω2|x|2/2.  相似文献   

4.
We consider the dynamical behavior of the reaction-diffusion equation with nonlinear boundary condition for both autonomous and non-autonomous cases. For the autonomous case, under the assumption that the internal nonlinear term f is dissipative and the boundary nonlinear term g is non-dissipative, the asymptotic regularity of solutions is proved. For the non-autonomous case, we obtain the existence of a compact uniform attractor in H1(Ω) with dissipative internal and boundary nonlinearities.  相似文献   

5.
The L2- and H1-approximate controllability and homogenization of a semilinear elliptic boundary-value problem is studied in this paper. The principal term of the state equation has rapidly oscillating coefficients and the control region is locally distributed. The observation region is a subset of codimension 1 in the case of L2-approximate controllability or is locally distributed in the case of H1-approximate controllability. By using the classical Fenchel-Rockafellar's duality theory, the existence of an approximate control of minimal norm is established by means of a fixed point argument. We consider its asymptotic behavior as the rapidly oscillating coefficients H-converge. We prove its convergence to an approximate control of minimal norm for the homogenized problem.  相似文献   

6.
We investigate the existence of reflection formulas supported on a finite set. It is found that for solutions of the Laplace and Helmholtz equation there are no finitely supported reflection principles unless the support is a single point. This confirms that in order to construct a reflection formula that is not ‘point to point’, it is necessary to consider a continuous support. For solutions of the wave equation 2u/∂xy=0, there exist finitely supported reflection principles that can be constructed explicitly. For solutions of the telegraph equation 2u/∂xy+λ2u=0, we show that if a reflection principle is supported on less than five points then it is a point to point reflection principle.  相似文献   

7.
We establish the existence of a weak solution u of the semilinear wave equation where a(t,x) is equal to 1 outside a compact set with respect to x and a non-linear term fk which satisfies |fk(u)|≤C|u|k. For some non-trapping time-periodic perturbations a(t,x), we obtain the long time existence of a solution from little initial data.  相似文献   

8.
In this paper we study the existence of a uniform attractor for strongly damped wave equations with a time-dependent driving force. If the time-dependent function is translation compact, then in a certain parameter region, the uniform attractor of the system has a simple structure: it is the closure of all the values of the unique, bounded complete trajectory of the wave equation. And it attracts any bounded set exponentially. At the same time, we consider the strongly damped wave equations with rapidly oscillating external force gε(x,t)=g(x,t,t/ε) having the average g0(x,t) as ε0+. We prove that the Hausdorff distance between the uniform attractor Aε of the original equation and the uniform attractor A0 of the averaged equation is less than O(ε1/2). We mention, in particular, that the obtained results can be used to study the usual damped wave equations.  相似文献   

9.
The stationary Boltzmann equation for hard forces in the context of a two‐component gas is considered in the slab. An L1 existence theorem is proved when one component satisfies a given indata profile and the other component satisfies diffuse reflection at the boundaries. Weak L1 compactness is extracted from the control of the entropy production term. Trace at the boundaries are also controlled. Copyright © 2007 John Wiley & Sons, Ltd.  相似文献   

10.
The Busemann-equation is a classical equation coming from fluid dynamics. The well-posed problem and regularity of solution of Busemann-equation with nonlinear term are interesting and important. The Busemann-equation is elliptic in y>0 and is degenerate at the line y=0 in R2. With a special nonlinear absorb term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain. By means of elliptic regularization technique, a delicate prior estimate and compact argument, we show that the solution of mixed boundary value problem of Busemann-equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary on some conditions. The result is better than the result of classical boundary degenerate elliptic equation.  相似文献   

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