Point‐wise decay estimate for the global classical solutions to quasilinear hyperbolic systems

In this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19 :1263–1317; Nonlinear Anal. 1997; 28 :1299–1322; Chin. Ann. Math. 2004; 25B :37–56). We give a new, very simple proof of this result and also give a sharp point‐wise decay estimate of the solution. Then, we consider the mixed initial‐boundary‐value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12 (1):59–78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point‐wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Global classical solutions to partially dissipative quasilinear hyperbolic systems with weaker restrictions on wave interactions

A kind of partially dissipative quasilinear hyperbolic systems in which a part of equations possesses the strict dissipation and the others are weakly linearly degenerate are discussed. Under some weaker hypotheses on the interactions between two parts of equations, it is proved that for any given initial data with small W^1,1 and C¹ norms, the corresponding Cauchy problem admits a unique C¹ global classical solution. Copyright © 2012 John Wiley & Sons, Ltd.     

BREAKDOWN OF CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS

This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with weaker decaying initial data, and obtains a blow-up result for C1 solution to Cauchy problem.     

Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems with BV data

We investigate the existence of a global classical solution to the Goursat problem for linearly degenerate quasilinear hyperbolic systems. As the result in [A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409–421] suggests that one may achieve global smoothness even if the C¹ norm of the initial data is large, we prove that, if the C¹ norm of the boundary data is bounded but possibly large, and the BV norm of the boundary data is sufficiently small, then the solution remains C¹ globally in time. Applications include the equation of time‐like extremal surfaces in Minkowski space R^{1 + (1 + n)} and the one‐dimensional Chaplygin gas equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Global existence of classical solutions to the Cauchy problem on a semi-bounded initial axis for a nonhomogeneous quasilinear hyperbolic system

It is proven that if the leftmost eigenvalue is weakly linearly degenerate, then the Cauchy problem for a class of nonhomogeneous quasilinear hyperbolic systems with small and decaying initial data given on a semi-bounded axis admits a unique global C¹ solution on the domain , where x=xn(t) is the fastest forward characteristic emanating from the origin. As an application of our result, we prove the existence of global classical, C¹ solutions of the flow equations of a model class of fluids with viscosity induced by fading memory with small smooth initial data given on a semi-bounded axis.     

Global classical solutions to a kind of quasilinear hyperbolic systems in several space variables

For a kind of quasilinear hyperbolic systems in several space variables whose coefficient matrices commute each other, by means of normalized coordinates, formulas of wave decomposition and pointwise decay estimates, the global existence of classical solution to the Cauchy problem for small and decaying initial data is obtained, under hypotheses of weak linear degeneracy and weakly strict hyperbolicity.     

Global existence of weak discontinuous solutions to the Cauchy problem with small BV initial data for quasilinear hyperbolic systems

In this paper, we study the Cauchy problem for quasilinear hyperbolic system with a kind of non‐smooth initial data. Under the assumption that the initial data possess a suitably small bounded variation norm, a necessary and sufficient condition is obtained to guarantee the existence and uniqueness of global weak discontinuous solution. Copyright © 2014 John Wiley & Sons, Ltd.     

Global existence of weakly discontinuous solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems

In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x） ｜ t ≥ 0, x ≥0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.     

对角型拟线性双曲组整体经典解的渐近行为

This paper deals with the asymptotic behavior of global classical solutions to quasilinear hyperbolic systems of diagonal form with weakly linearly degenerate characteristic fields. On the basis of global existence and uniqueness of C^1 solution, we prove that the solution to the Cauchy problem approaches a combination of C^1 traveling wave solutions when t tends to the infinity.     

Asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries

In this paper, we investigate the asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries. Under some suitable assumptions, we prove that the solution approaches a combination of Lipschitz continuous and piecewise C¹ traveling wave solution. As an application, we apply the result to the equation for time-like extremal surfaces in the Minkowski space-time R1+(1+n).     

Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws II

This work is a continuation of our previous work. In the present paper, we study the existence and uniqueness of global piecewise C¹ solutions with shock waves to the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping in the presence of a boundary. It is shown that the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping with nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0} admits a unique global piecewise C¹ solution u = u (t, x) containing only shock waves with small amplitude and this solution possesses a global structure similar to that of a self‐similar solution u = U (x /t) of the corresponding homogeneous Riemann problem, if each characteristic field with positive velocity is genuinely nonlinear and the corresponding homogeneous Riemann problem has only shock waves but no rarefaction waves and contact discontinuities. This result is also applied to shock reflection for the flow equations of a model class of fluids with viscosity induced by fading memory. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and stability of traveling wave solutions to one‐sided mixed initial‐boundary value problem for first‐order quasilinear hyperbolic systems

When one characteristic of the system is linearly degenerate, under suitable boundary conditions, we get the existence of traveling wave solutions located on the corresponding characteristic trajectory to the one‐sided mixed initial‐boundary value problem. When the system is linearly degenerate, by introducing the semi‐global normalized coordinates, we derive the related formulas of wave decomposition to prove the stability of traveling wave solutions corresponding to all leftward and the rightmost characteristic trajectories. Finally, for the traveling wave solutions corresponding to other rightward characteristic trajectories, some examples show their possible instability. Copyright © 2014 John Wiley & Sons, Ltd.     

Blow-up of periodic solutions to linearly degenerate quasilinear hyperbolic systems

An example is given to show that for the Cauchy problem of n×n$n\times n$ homogeneous linearly degenerate quasilinear hyperbolic systems with periodic initial data, when n>2$n>2$, C¹$C^1$ solutions may blow up in a finite time, no matter how small and smooth the initial data are.     

Simple waves and characteristic decompositions of quasilinear hyperbolic systems in two independent variables

This paper is concerned with the simple waves for the general quasilinear strictly hyperbolic systems in two independent variables. By using the method of characteristic decomposition, we first establish a more general sufficient condition for the existence of characteristic decompositions. These decompositions allow us to extend a well‐known result on reducible equations by Courant and Friedrichs to the non‐reducible equations. Then we construct a simple wave solution, in which the characteristics may be a set of non‐straight curves, around a given curve under the obtained sufficient condition. Copyright © 2014 John Wiley & Sons, Ltd.     

拟线性双曲组在半有界初始轴上的柯西问题整体经典解的渐近性态

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.     

A note on the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems

This paper is concerned with the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields. On the basis of the existence result for the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions, provided that the C¹ norm and the BV norm of the initial data are bounded but possibly large. In contrast to former results obtained by Liu and Zhou [J. Liu, Y. Zhou, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Methods Appl. Sci. 30 (2007) 479-500], ours do not require their assumption that the system is rich in the sense of Serre. Applications include that to the one-dimensional Born-Infeld system arising in string theory and high energy physics.     

Global classical solutions for a class of quasilinear hyperbolic systems

In this paper we investigate the mechanism of singularity formation to the Cauchy problem of quasilinear hyperbolic system.Moreover, we present some examples to show some significant problems.