Large time behavior of solutions to 1-dimensional bipolar quantum hydrodynamic model for semiconductors

In this article, we study the 1-dimensional bipolar quantum hydrodynamic model for semiconductors in the form of Euler-Poisson equations, which contains dispersive terms with third order derivations. We deal with this kind of model in one dimensional case for general perturbations by constructing some correction functions to delete the gaps between the original solutions and the diffusion waves in L²-space, and by using a key inequality we prove the stability of diffusion waves. As the same time, the convergence rates are also obtained.     

Large time behavior and energy relaxation time limit of the solutions to an energy transport model in semiconductors

In this paper, the global existence and the large time behavior of smooth solutions to the initial boundary value problem for the multi-dimensional energy transport model are studied. It is also proved that the solutions of the problem converge to an isothermal drift-diffusion model as energy relaxation time τ goes to 0 by compactness argument with the help of energy estimates and entropy inequality.     

Existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic model of semiconductors

We consider a one-dimensional bipolar hydrodynamic model of semiconductors. Although some results exist for the bipolar case, almost their conditions (the boundary condition, the doping profile, etc.) are far from practical application. In the present paper, under a condition appropriate for engineering, we shall prove the existence and the uniqueness of classical solutions for the stationary problem. The most difficult point is to obtain the bounded estimate and the energy estimate.     

Large time behavior of bounded solutions to a parabolic system of chemotaxis in the whole space

We consider the Cauchy problem for a parabolic system of chemotaxis in RN(N?1), and give the decay rates and asymptotic profiles of bounded solutions.     

The stationary solution of a one-dimensional bipolar quantum hydrodynamic model

In this paper, we consider the existence and uniqueness of stationary solution to the bipolar quantum hydrodynamic model in one dimensional space with general non-constant doping profile. The existence of the stationary solution is proved by Leray-Schauder fixed-point theorem and a crucial truncation technique is used to derive the positive upper and lower bounds of the stationary solution. The uniqueness of the stationary solution is shown by a delicate energy estimate.     

Large time behavior of Euler-Poisson equations for isothermal fluids with spherical symmetry

In this paper, the large time behavior of spherically symmetric weak solutions to the multi-dimensional isothermal Euler-Poisson system in an annulus is considered. When space dimension N=2, it is shown that the weak solutions converge to the unique stationary solution exponentially in time. No smallness and regularity conditions are assumed.     

Large time behavior of solutions to a degenerate parabolic equation not in divergence form

In this paper, we investigate positive solutions of the degenerate parabolic equation not in divergence form: ut=upΔu+auq−bur, subject to the null Dirichlet boundary condition. We at first discuss the existence and nonexistence of global solutions to the problem, and then study the large time behavior for the global solutions. When the positive source dominates the model, we prove that the global solutions uniformly tend to the positive steady state of the problem as t→∞. In particular, we establish the uniform asymptotic profiles for the decay solutions when the problem is governed by the nonlinear diffusion or absorption.     

Large time behavior of solutions for a system of nonlinear damped wave equations

We consider the Cauchy problem of the semilinear damped wave system:

     

Structural stability of subsonic solutions to a steady hydrodynamic model for semiconductors: From the perspective of boundary data

In this paper, we are concerned with a one-dimensional isothermal steady hydrodynamic model for semiconductors driven by boundary data. In the purely subsonic setting, we obtain the existence, uniqueness and structural stability of purely subsonic solutions. Moreover, when the boundary data range from the subsonic region to the sonic line, we further study the degenerate problem from the perspective of boundary data, and prove that there exists a unique interior subsonic solution to the degenerate problem. As a byproduct, we also establish the structural stability between purely subsonic solution and interior subsonic solution in a relatively weak sense. These results provide us with a completely new perspective to understand the singularity caused by the boundary degeneracy. A number of numerical simulations are also carried out, which confirm our theoretical results.     

Large time behavior of solutions to semilinear systems of wave equations

This paper is concerned with the initial value problem for semilinear systems of wave equations. First we show a global existence result for small amplitude solutions to the systems. Then we study asymptotic behavior of the global solution. We underline that ``modified' free profiles are obtained for all global solutions to the systems even in the case where the free profile might not exist. Moreover, we prove non–existence of any free profiles for the global solution in some cases where the effect of the nonlinearity is strong enough. The first author was partially supported by Grant-in-Aid for Science Research (14740114), JSPS.     

Global smooth solutions to the multidimensional hydrodynamic model for plasmas with insulating boundary conditions

The existence of global smooth solutions to the multidimensional hydrodynamic model for plasmas of electrons and positively charged ions with insulating boundary conditions is shown under the assumption that the initial densities are close to a constant. Furthermore it is proved that the particle densities converge exponentially fast to the constant steady state.     

Large time behavior of entropy solutions to some hyperbolic system with dissipative structure

In this paper the large time behavior of the global L∞ entropy solutions to the hyperbolic system with dissipative structure is investigated. It is proved that as t →∞ the entropy solutions tend to a constant equilibrium state in L2 norm with exponential decay even when the initial values are arbitrarily large. As an illustration, a class of 2 × 2 system is studied.     

Large time behavior of solutions to a fully parabolic chemotaxis–haptotaxis model in N dimensions

This paper deals with the Neumann problem for a fully parabolic chemotaxis–haptotaxis model of cancer invasion given by

$\{\begin{array}{c}{u}_t=\mathrm{\Delta }u?\chi \mathrm{?}?(u\mathrm{?}v)?\xi \mathrm{?}?(u\mathrm{?}w)+u(a?\mu u^{r?1}?\lambda w),x\in \mathrm{\Omega },t>0,\hfill \\ \tau v_t=\mathrm{\Delta }v?v+u,x\in \mathrm{\Omega },t>0,\hfill \\ w_t=?vw,x\in \mathrm{\Omega },t>0.\hfill \end{array}$

Here, $\mathrm{\Omega }?{\mathbb{R}}^N(N\ge 1)$ is a bounded domain with smooth boundary and $\tau >0,r>1,\lambda \ge 0$, $a\in \mathbb{R}$, $\mu ,\xi$ and χ are positive constants. It is shown that the corresponding initial–boundary value problem possesses a unique global bounded classical solution in the cases $r>2$ or $r=2$, with $\mu >{\mu }^{?}=\frac{{(N?2)}_{+}}{N}(\chi +C_{\beta })C_{\frac{N}{2}+1}^{\frac{1}{\frac{N}{2}+1}}$ for some positive constants $C_{\beta }$ and $C_{\frac{N}{2}+1}$. Furthermore, the large time behavior of solutions to the problem is also investigated. Specially speaking, when a is appropriately large, the corresponding solution of the system exponentially decays to $({(\frac{a}{\mu })}^{\frac{1}{r?1}},{(\frac{a}{\mu })}^{\frac{1}{r?1}},0)$ if μ is large enough. This result improves or extends previous results of several authors.