GLOBALLY SMOOTH SOLUTIONS TO CAUCHY PROBLEM OF A QUASILINEAR HYPERBOLIC SYSTEM ARISING IN BIOLOGY

This article considers Cauchy problem u(t) - (uv)(x) = 0, v(t) - u(x) = 0, u(x, 0) = u(0) (x) > 0, v(x, 0) = v(0)(x). A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global C-1-solution on t greater than or equal to 0 is obtained.

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.

GLOBAL EXISTENCE OF WEAKLY DISCONTIN UOUS SOLUTIONS TO THE CAUCHY PROBLEM WITH A KIND OF NON-SMOOTH INITIAL DATA FOR QUASILINEAR HYPERBOLIC SYSTEMS

The authors consider the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems and obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution.

Asymptotic Behavior of Global Classical Solutions to the Cauchy Problem on a Semi-Bounded Initial Axis for Quasilinear Hyperbolic Systems

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.

Global Classical Solutions of Inhomogeneous Quasilinear Hyperbolic Systems

在这份报纸，我们与不同类的条款认为一种 quasilinear 是夸张系统令人满意的消散的状况或匹配的状况。为这种系统的 Cauchy 问题，我们证明如果起始的数据是小的并且满足一些腐烂，状况，和系统是线性地微弱地堕落，那么， Cauchy 问题在 t 上承认一个唯一的全球古典解决方案 0。

ON THE UNIQUENESS OF THE WEAK SOLUTIONS OF A QUASILINEAR HYPERBOLIC SYSTEM WITH A SINGULAR SOURCE TERM

This paper is a continuation of the authors' previous paper [1].In this paper the authors prove,assuming additional conditions on the initial data,some results about the existence and uniqueness of the entropy weak solutions of the Cauchy problem for the singular hyperbolic system at + (au)x + 2aux = 0, x＞0,t≥0. ut + 1/2 (a2 + u2)x = o,

INITIAL-BOUNDARY VALUE PROBLEM AND CAUCHY PROBLEM FOR A QUASILINEAR EVOLUTION EQUATION

In the present paper,the local existence of classical solutions to the periodic boundary problem and the Cauchy problem of a quasilinear evolution equation are studied under the assumptions that do not require the monotonicity of σi(s) (i= 1,…, n). The nonexistence of global solutions to the initial-boundary value problem of the equation is also discussed, a blowup theorem is proved and a concrete example is given.

GLOBAL CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS WITH WEAK LINEAR DEGENERACY

Consider the following Cauchy problem for the first order quasilinear strictly hy- perbolic system ?u ?u + A(u) = 0, ?t ?x t = 0 : u = f(x). We let M = sup |f (x)| < +∞. x∈R The main result of this paper is that under the assumption that the system is weakly linearly degenerated, there exists a positive constant ε independent of M, such that the above Cauchy problem admits a unique global C1 solution u = u(t,x) for all t ∈ R, provided that +∞ |f (x)|dx ≤ ε, ?∞ +∞ ε |f(x)|dx ≤ .

SEMI-GLOBAL C^1 SOLUTION TO THE MIXED INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS

By means of an equivalent invariant form of boundary conditions, the authors get the exis- tence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems with general nonlinear boundary conditions.

GLOBAL CLASSICAL SOLUTIONS TO TYPICAL FREE-BOUNDARY PROBLEMS FOR QUASILINEAR HYPERBOLIC SYSTEMS

In this paper the authors prove the existence and uniqueness of global classical solutions to some kinds of typical boundary-value problems and typical free-boundary problems for quasilinear hyperbolic systems.

Generalized Persistence of Entropy Weak Solutions for System of Hyperbolic Conservation Laws

Let u(t,x)be the solution to the Cauchy problem of a scalar conservation law in one space dimension.It is well known that even for smooth initial data the solution can become discontinuous in finite time and global entropy weak solution can best lie in the space of bounded total variations.It is impossible that the solutions belong to,for example,H^(1) because by Sobolev embedding theorem H^(1) functions are Holder continuous.However,the author notes that from any point(t,x),he can draw a generalized characteristic downward which meets the initial axis at y=α(t,x).If he regards u as a function of(t,y),it indeed belongs to H^(1) as a function of y if the initial data belongs to H^(1).He may call this generalized persistence(of high regularity)of the entropy weak solutions.The main purpose of this paper is to prove some kinds of generalized persistence(of high regularity)for the scalar and 2×2 Temple system of hyperbolic conservation laws in one space dimension.

Global Classical Solutions to Partially Dissipative Quasilinear Hyperbolic Systems with One Weakly Linearly Degenerate Characteristic

For a kind of partially dissipative quasilinear hyperbolic systems without Shizuta-Kawashima condition,in which all the characteristics,except a weakly linearly degenerate one,are involved in the dissipation,the global existence of H 2 classical solution to the Cauchy problem with small initial data is obtained.

Lipschitz Continuous Solutions to the Cauchy Problem for Quasi-linear Hyperbolic Systems

Lipschitz continuous solutions to the Cauchy problem for 1-D first order quasi-linear hyperbolic systems are considered. Based on the methods of approximation and integral equations, the author gives two definitions of Lipschitz solutions to the Cauchy problem and proves the existence and uniqueness of solutions.

On the Two-Dimensional Cauchy Problem of a Hyperbolic System

In this paper, we prove the global existence of generalized solutions to a two-dimensional Cauchy problem of a hyperbolic system by introducing a new definition of generalized solution. Moreover, the solution may involve delta-wave.

LIFE-SPAN OF CLASSICAL SOLUTIONSTO QUASILINEAR HYPERBOLIC SYSTEMSWITH SLOW DECAY INITIAL DATALIFE-SPAN OF CLASSICAL SOLUTIONSTO QUASILINEAR HYPERBOLIC SYSTEMSWITH SLOW DECAY INITIAL DATA

The author considers the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with "slow" decay initial data. By constructing an example, first it is illustrated that the classical solution to this kind of Cauchy problem may blow up in a finite time, even if the system is weakly linearly degenerate. Then some lower bounds of the life-span of classical solutions are given in the case that the system is weakly linearly degenerate. These estimates imply that, when the system is weakly linearly degenerate, the classical solution exists almost globally in time. Finally, it is proved that Theorems 1.1-1.3 in [2] are still valid for this kind of initial data.

GLOBAL CLASSICAL SOLUTIONS WITH SMALL INITIAL TOTAL VARIATION FOR QUASILINEAR HYPERBOLIC SYSTEMS

By means of the continuous Glimm functional, a proof is given on the global existence of classical solutions to Cauchy problem for general first order quasilinear hyperbolic systems with small initial total variation.

Global Classical Solutions to Partially Dissipative Quasilinear Hyperbolic Systems

The author considers the Cauchy problem for quasilinear inhomogeneous hyperbolic systems.Under the assumption that the system is weakly dissipative,Hanouzet and Natalini established the global existence of smooth solutions for small initial data (in Arch.Rational Mech.Anal.,Vol.169,2003,pp.89-117).The aim of this paper is to give a completely different proof of this result with slightly different assumptions.