On the velocity of separation between two successive traveling waves in the asymptotics of the solution to the Cauchy problem for a Burgers-type equation

An upper bound on the distance between the centers of two successive traveling waves occurring in the asymptotics of the solution to the Cauchy problem for a Burgers-type equation is established under generic conditions. Taking into account a previously established lower bound, an asymptotically sharper estimate is derived.     

拟线性双曲型方程组Cauchy问题行波解的稳定性

当拟线性双曲系统线性退化时,其Cauchy问题最左族和最右族行波解是稳定的.而其中间族行波解未必稳定.我们在弱线性退化条件下,证明了拟线性双曲系统Cauchy问题适当小的W~(1,1)∩L~∞范数适当小的行波解是稳定的,并将此稳定性应用于可对角化的拟线性双曲系统和Chaplygin气体动力学方程组.     

Blow-up solution near the traveling waves for the second-order Camassa–Holm equation

This paper studies the blow-up solution and its blow-up rate near the traveling waves of the second-order Camassa–Holm equation. The sufficient condition for the existence of blow-up solution is obtained by a rather ingenious method. Applying the extended pseudo-conformal transformation, an equivalent proposition of the solution breaking in finite time near the traveling waves is constructed. The relation is established between the blow-up time and rate of the solution and the residual’s.     

Exponential decay rate of solutions toward traveling waves for the Cauchy problem of generalized Benjamin-Bona-Mahony-Burgers equations

In this paper, we investigate the exponential time decay rate of solutions toward traveling waves for the Cauchy problem of generalized Benjamin-Bona-Mahony-Burgers equations

(E)     

Stability of Rarefaction Waves to the 1D Compressible Navier–Stokes Equations with Density-Dependent Viscosity

In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic Navier–Stokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is constructed by approximating the system and regularizing the initial values which may contain vacuum states. Then some global in time estimates on the weak solution are obtained. Based on these uniform estimates, the vacuum states are shown to vanish in finite time and the weak solution we constructed becomes a unique strong one. Consequently, the stability of the rarefaction wave is proved in a weak sense. The theory holds for large-amplitudes rarefaction waves and arbitrary initial perturbations.     

Convergence rate of solutions toward traveling waves for the Cauchy problem of generalized Korteweg-de Vries-Burgers equations

In this paper we investigate the exponential time decay rate of solutions toward traveling waves for the Cauchy problem of generalized Korteweg-de Vries-Burgers equations

(E)     

Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow

The paper studies the existence, both locally and globally in time, stability, decay estimates and blowup of solutions to the Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow. Under the assumption that the nonlinear term of the equations is of polynomial growth order, say α, it proves that when α>1, the Cauchy problem admits a unique local solution, which is stable and can be continued to a global solution under rather mild conditions; when α?5 and the initial data is small enough, the Cauchy problem admits a unique global solution and its norm in L1,p(R) decays at the rate for 2<p?10. And if the initial energy is negative, then under a suitable condition on the nonlinear term, the local solutions of the Cauchy problem blow up in finite time.     

具常重特征的非齐次拟线性双曲组的整体弱间断解

This paper considers the Cauchy problem with a kind of non-smooth initial data for general inhomogeneous quasilinear hyperbolic systems with characteristics with constant multiplicity. Under the matching condition, based on the refined fomulas on the decomposition of waves, we obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution to the Cauchy problem.     

Solution to the boundary value problem for an arbitrary elliptic operator subject to a radiation condition

It is shown that the generalized Fourier transform can be extended to an arbitrary elliptic operator in a cylindrical domain with a Robin boundary condition. In this case, the existence of the Fourier image is a completely correct radiation condition determining a solution to the problem that is a superposition of waves traveling away from the source.     

GLOBAL EXISTENCE AND Lp ESTIMATES FOR SOLUTIONS OF DAMPED WAVE EQUATION WITH NONLINEAR CONVECTION IN MULTI-DIMENSIONAL SPACE＊

In this article,the author studies the Cauchy problem of the damped wave equation with a nonlinear convection term in multi-dimensions.The author shows that a classical solution to the Cauchy problem e...     

Cauchy–Gelfand problem and the inverse problem for a first-order quasilinear equation

Gelfand’s problem on the large time asymptotics of the solution of the Cauchy problem for a first-order quasilinear equation with initial conditions of the Riemann type is considered. Exact asymptotics in the Cauchy–Gelfand problem are obtained and the initial data parameters responsible for the localization of shock waves are described on the basis of the vanishing viscosity method with uniform estimates without the a priori monotonicity assumption for the initial data.     

Solution to Nonlinear Wave Equation with Delta Initial Data and Its Singularity Structure

In this paper we discuss a Cauchy problem for nonlinear wave equation with delta initial data, including delta impulse and/or delta displacement. The solution of the Cauchy problem in appropriate sense is given. Meanwhile, the singularity structure of the solution is also described.     

Divergent Solution to the Nonlinear Schr\"{o}dinger Equation with the Combined Power-Type Nonlinearities

In this paper, we consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation with combined power-type nonlinearities, which is mass-critical/supercr-itical, and energy-subcritical. Combing Du, Wu and Zhang'' argument with the variational method, we prove that if the energy of the initial data is negative (or under some more general condition), then the $H^1$-norm of the solution to the Cauchy problem will go to infinity in some finite time or infinite time.     

Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity

We show that a solution of the Cauchy problem for the KdV equation, has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity at x = 0. It is shown that for () data satisfying the condition the solution is analytic in both space and time variable. The above condition allows us to take as initial data the Dirac measure or the Cauchy principal value of 1/x. The argument is based on the recent progress on the well-posedness result by Bourgain [2] and Kenig-Ponce-Vega [20] and a systematic use of the dilation generator . Received 22 March 1999     

The condition on the stability of stationary lines in a curvature flow in the whole plane

The long time behavior of a curve in the whole plane moving by a curvature flow is studied. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the x-axis. Here the initial curves are given by bounded functions on the x-axis. It is proved that the solution converges uniformly to the solution of the Cauchy problem of the heat equation with the same initial value. The difference is of order O(t−1/2) as time goes to infinity. The proof is based on the decay estimates for the derivatives of the solution. By virtue of the stability results for the heat equation, our result gives the sufficient and necessary condition on the stability of constant solutions that represent stationary lines of the curvature flow in the whole plane.     

Interaction of Plane Stress Waves in a Three-Layer Structure. 1. Degenerate Solutions in Terms of Characteristics for a Homogeneous Structure

Exact expressions in terms of characteristics for calculating the normal-stress waves propagating across the layers of different materials are deduced. A one-dimensional boundary-value problem is considered for a three-layer structure of sandwich type. The faces of the layered structure are free from loads or one of them is rigidly fixed (variant 1), or one face is rigidly fixed and the other is subjected to an impact of a mass M with a speedV₀ (variant 2). For the boundary conditions of variant 1, relationships are obtained which allow one to reduce the analytical continuation of a solution in time to a periodic procedure if solely the initial disturbances of the strain field in the layers are given. It is shown that, in this case, the Cauchy problem with the initial strain field is reduced to graphoanalytically constructing the superposition patterns of the forward and backward waves. The fundamental features of the construction are demonstrated for a uniform bar with a piecewise constant distribution of strains along its length. To solve the problem of impact loading in variant 2, analytical results for a uniform plate are used, which allows us to account for the direction of mass forces in collision. In the latter case, the possibility of mass recoil is revealed in the first and second time cycles. The analytical constructions presented are focused on an exact calculation of stresses upon response of a layered plate to initial disturbances within its layers, as well as to an external dynamic action. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 5, pp. 585–606, September–October, 2005.     

Global existence for three‐dimensional incompressible isotropic elastodynamics via the incompressible limit

The existence of global‐in‐time classical solutions to the Cauchy problem for incompressible nonlinear isotropic elastodynamics for small initial displacements is proved. Solutions are constructed via approximation by slightly compressible materials. The energy for the approximate solutions remains uniformly bounded on a time scale that goes to infinity as the material approaches incompressibility. A necessary component to the long‐time existence of the approximating solution is a null or linear degeneracy condition, inherent in the isotropic case, which limits the quadratic interaction of the shear waves. The proof combines energy and decay estimates based on commuting vector fields and a compactness argument. © 2004 Wiley Periodicals, Inc.     

相变演化Suliciu模型中波的相互作用

描述相变演化的Suliciu模型，其基本波可由行波分析得到．对于任何给定分两段常值的初始状态，相应的Riemann解是某些基本波的组合．对分三段常值的初始状态，解由上述Piemann解构成，其中相邻两状态间以基本波连接．当基本波发生碰撞时，新的Riemann问题形成．通过研究这些Riemann。问题，可以在适当的参数空间中对基本波之间复杂的相互作用加以分类．     

Initial trace for a doubly nonlinear parabolic equation

This paper studies the Cauchy problem for a doubly nonlinear parabolic equation. The main result shows that if there is a nonnegative solution of the Cauchy problem, then the initial trace of the solution is uniquely given as a nonnegative Borel measure satisfying an exponential growth condition. This extends the known result for the heat equation to the nonlinear case.     

Traveling waves in a one-dimensional heterogeneous medium

We consider solutions of a scalar reaction–diffusion equation of the ignition type with a random, stationary and ergodic reaction rate. We show that solutions of the Cauchy problem spread with a deterministic rate in the long time limit. We also establish existence of generalized random traveling waves and of transition fronts in general heterogeneous media.