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1.
This paper is concerned with the interactions of δ-shock waves and the vacuum states between the two contact discontinuities for the transport equations. The solutions are obtained constructively when the initial data are three piecewise constant states. The global structure and large time-asymptotic behaviors of the solutions are analyzed case by case. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with initial data by studying the limits of the solutions when the perturbed parameter ε tends to zero. 相似文献
2.
In this paper we study the asymptotic behavior of global classical solutions to the Cauchy problem with initial data given on a semi-bounded axis for quasilinear hyperbolic systems. Based on the existence result on the global classical solution, we prove that, when t tends to the infinity, the solution approaches a combination of C1 travelling wave solutions with the algebraic rate (1 + t)^-u, provided that the initial data decay with the rate (1 + x)^-(l+u) (resp. (1 - x)^-(1+u)) as x tends to +∞ (resp. -∞), where u is a positive constant. 相似文献
3.
In this article, we study the interactions of delta shock waves for the one-dimensional Euler equations for Chaplygin gas with split delta functions. We constructively obtain the solutions when the initial data are three piecewise constant states. The global structure and large time-asymptotic behaviors of the solutions are analyzed case by case. Moreover, we obtain the stability of solutions by letting perturbed parameter ? tend to zero. 相似文献
4.
Zhang Linghai 《数学年刊B辑(英文版)》1998,19(1):35-58
UNIFORMSTABILITYANDASYMPTOTICBEHAVIOROFSOLUTIONSOF2-DIMENSIONALMAGNETOHYDRODYNAMICSEQUATIONSZHANGLINGHAIManuscriptreceivedJu... 相似文献
5.
Interactions of delta shock waves for the relativistic Chaplygin Euler equations with split delta functions
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In this article, we are concerned with the interactions of delta shock waves with contact discontinuities for the relativistic Euler equations for Chaplygin gas by using split delta functions method. The solutions are obtained constructively and globally when the initial data consists of three piecewise constant states. The global structure and large time‐asymptotic behaviors of the solutions are analyzed case by case. During the process of the interaction, the strengths of delta shock waves are computed completely. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with special initial data by letting perturbed parameter ε tends to zero. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
6.
研究一类Korteweg型不可压流体模型的强解问题.针对粘性系数依赖于密度的情形,当初始值满足兼容性条件(9)对,证明了强解的局部存在性和唯一性.我们在这指出,本文允许初始真空存在. 相似文献
7.
Global dynamics of a reaction and diffusion model for an HTLV-I infection with mitotic division of actively infected cells
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This paper is concerned with the global dynamics of a reaction and diffusion model for an HTLV-I infection with mitotic division of actively infected cells and CTL immune response. The well posedness of the proposed model is investigated. In the case of a bounded spatial domain, we establish the threshold dynamics in terms of the basic reproduction number $\mathcal{R}_0$ for the spatially heterogeneous model. Also, by means of different Lyapunov functions, the global asymptotic properties of the steady states for the spatially homogeneous model are studied. In the case of an unbounded spatial domain, there are no travelling wave solutions connecting the infection-free steady state with itself when $\mathcal{R}_0 < 1$. Finally, numerical simulations and conclusions are given. 相似文献
8.
Consideration is given to initial value problem for systems of two evolution equations of generalized BBM-type coupled through nonlinearity described in (1.3). It is shown that the problem is always locally well-posed in the $L_2$-based Sobolev spaces $H^s(\mathbb{R}) \times H^s(\mathbb{R})$ for $s \ge 0$. Under exact conditions on $A, \cdots, F,$ the local well-posedness theory extends globally, and bounds for the growth in time of relevant norms of solutions corresponding to very general auxiliary data are derived. 相似文献
9.
Existence and qualitative features of entire solutions for delayed reaction diffusion system: the monostable case
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Yanling Meng Weiguo Zhang Shengqiang Zhang 《Journal of Applied Analysis & Computation》2019,9(5):1769-1800
The paper is concerned with the existence and qualitative features of entire solutions for delayed reaction diffusion monostable systems. Here the entire solutions mean solutions defined on the $ (x,t)\in\mathbb{R}^{N+1} $. We first establish the comparison principles, construct appropriate upper and lower solutions and some upper estimates for the systems with quasimonotone nonlinearities. Then, some new types of entire solutions are constructed by mixing any finite number of traveling wave fronts with different speeds $ c\geq c_* $ and propagation directions and a spatially independent solution, where $c_*>0$ is the critical wave speed. Furthermore, various qualitative properties of entire solutions are investigated. In particularly, the relationship between the entire solution, the traveling wave fronts and a spatially independent solution are considered, respectively. At last, for the nonquasimonotone nonlinearity case, some new types of entire solutions are also investigated by introducing two auxiliary quasimonotone controlled systems and establishing some comparison theorems for Cauchy problems of the three systems. 相似文献
10.
Ryo Ikehata 《Mathematical Methods in the Applied Sciences》2001,24(9):659-670
This paper is concerned with some uniform energy decay estimates of solutions to the linear wave equations with strong dissipation in the exterior domain case. We shall derive the decay rate such as $(1+t)E(t)\le C$\nopagenumbers\end for some kinds of weighted initial data, where E(t) represents the total energy. Our method is based on the combination of the argument due to Ikehata–Matsuyama with the Hardy inequality, which is an improvement of Morawetz method. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
11.
Weihong Mao 《Journal of Applied Analysis & Computation》2020,10(1):210-222
By using the method of dynamical systems to Mikhailov-Novikov-Wang Equation, through qualitative analysis, we obtain bifurcations of phase portraits of the traveling system of the derivative $\phi(\xi)$ of the wave function $\psi(\xi)$. Under different parameter conditions, for $\phi(\xi)$, exact explicit solitary wave solutions, periodic peakon and anti-peakon solutions are obtained. By integrating known $\phi(\xi)$, nine exact explicit traveling wave solutions of $\psi(\xi)$ are given. 相似文献
12.
This paper is devoted to proving the sharpness on the lower bound of the lifespan of classical solutions to general nonlinear
wave equations with small initial data in the case n = 2 and cubic nonlinearity (see the results of T. T. Li and Y. M. Chen in 1992). For this purpose, the authors consider the
following Cauchy problem:
$\left\{ \begin{gathered}
\square u = \left( {u_t } \right)^3 , n = 2, \hfill \\
t = 0: u = 0, u_t = \varepsilon g\left( x \right), x \in \mathbb{R}^2 , \hfill \\
\end{gathered} \right.$\left\{ \begin{gathered}
\square u = \left( {u_t } \right)^3 , n = 2, \hfill \\
t = 0: u = 0, u_t = \varepsilon g\left( x \right), x \in \mathbb{R}^2 , \hfill \\
\end{gathered} \right. 相似文献
13.
ZHOU Yi 《数学年刊A辑(中文版)》2003,(3):293-302
This paper considers the following
Cauchy problem for semilinear wave equations in $n$ space
dimensions
$$\align
\square\p &=F(\partial\p ),\\p (0,x)&=f(x),\quad \partial_t\p (0,x)=g(x),
\endalign$$
where $\square =\partial_t^2-\triangle$ is the wave operator, $F$ is
quadratic in $\partial\p$ with
$\partial =(\partial_t,\partial_{x_1},\cdots ,\partial_{x_n})$.
The minimal value of $s$ is determined such that the above
Cauchy problem is locally well-posed in $H^s$. It turns out that
for the general equation $s$ must satisfy
$$s>\max\Big(\frac{n}{2}, \frac{n+5}{4}\Big).$$
This is due to Ponce and Sideris (when $n=3$) and Tataru (when $n\ge
5$). The purpose of this paper is to supplement with a proof in the
case $n=2,4$. 相似文献
14.
ZHOU Yi 《数学年刊B辑(英文版)》2003,24(3):293-302
This paper considers the following Cauchy problem for semilinear wave equations in n space dimensions □φ=F(δφ),φ(0,x)=f(x),δtφ(0,x)=g(x),whte □=δt^2-△ is the wave operator,F is quadratic in δεφ with δ=(δt,δx1,…,δxn).The minimal value of s is determined such that the above Cauchy problem is locally wellposed in H^s.It turns out that for the general equation s must satisfy s>max(n/2,n+5/4).This is due to Ponce and Sideris (when n=3)and Tataru (when n≥5).The purpose of this paper is to supplement with a proof in the case n=2,4. 相似文献
15.
Hongqiu Chen 《数学研究》2016,49(2):111-131
The BBM equation posed on $\mathbb{R}$ and $\mathbb{R}^+$ is revisited. Improving on earlier
results, global well-posedness and bounds for the growth in time of relevant norms of
solutions corresponding to very general auxiliary data are derived. 相似文献
16.
Hildebrando M. Rodrigues Jianhong Wu Marcio Gameiro 《Journal of Applied Analysis & Computation》2018,8(2):413-426
The purpose of this paper is to study the behavior of the solutions of two synchronized chaotic systems when the solutions switch from the first to the second system and vice-versa. The initial condition is chosen in the first system and the solutions travels for time $t \in [0, h]$, where $h>0$. The value of the solution at time $h$ is then chosen as the initial condition for the solution of the second system and this solution travels for time $t \in [h, 2h]$. The value of the solution at time $2h$ is then chosen as the initial value for the solution of the first system and so on. The first system is composed of two subsystems, Master and Slave that are synchronized. We present applications using the Lorenz, Chua and Chen systems. Some simulations using Matlab are presented. 相似文献
17.
Fujita-Kato Theorem for the Inhomogeneous Incompressible Navier-Stokes Equations with Nonnegative Density
![]() Jianzhong Zhang & Hongmei Cao 《数学研究通讯:英文版》2023,39(1):79-106
In this paper, we prove the global existence and uniqueness of solutions for the inhomogeneous Navier-Stokes equations with the initial data $(\rho_0,u_0)\in L^∞\times H^s_0$, $s>\frac{1}{2}$ and $||u_0||_{H^s_0}\leq \varepsilon_0$ in bounded domain $\Omega \subset \mathbb{R}^3$, in which the density is assumed to be nonnegative. The regularity of initial data is weaker than the previous $(\rho_0,u_0)\in (W^{1,\gamma}∩L^∞)\times H^1_0$ in [13] and $(\rho_0,u_0)\in L^∞\times H^1_0$ in [7], which constitutes a positive answer to the question raised by Danchin and Mucha in [7]. The methods used in this paper are mainly the classical time weighted energy estimate and Lagrangian approach, and the continuity argument and shift of integrability method are applied to complete our proof. 相似文献
18.
ONTHEBOUNDEDANDUNBOUNDEDSOLUTIONSOFONEDIMENSIONALNONLINEARREACTION-DIFFUSIONPROBLEM¥GEWEIGAOR.O.WEBERAbstract:Theexistenceofb... 相似文献
19.
EXISTENCEANDUNIQUENESSOFTHEENTROPYSOLUTIONTOANONLINEARHYPERBOLICEQUATION¥R.EYMARD;T.GALLOUET;R.HERBIN(LaboratoireCentraldesPo... 相似文献
20.
Interactions of delta shock waves for a class of nonstrictly hyperbolic system of conservation laws
![]() In this paper, we study the perturbed Riemann problem for a class of nonstrictly hyperbolic system of conservation laws, and focuse on the interactions of delta shock waves with the shock waves and the rarefaction waves. The global solutions are constructed completely with the method of splitting delta function. In solutions, we find a new kind of nonclassical wave, which is called delta contact discontinuity with Dirac delta function in both components. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. Moreover, by letting perturbed parameter $\varepsilon$ tend to zero, we analyze the stability of Riemann solutions. 相似文献
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