LIFE-SPAN OF CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS WITH SLOW DECAY INITIAL DATA LIFE-SPAN OF CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS WITH SLOW DECAY INITIAL DATA

1. IntroductionConsider the following quasilinear systeman on~ A(u)~ = 0, (1.1)ot oxwhere u ~ (ul,'' t u.)" is the unknown vector function of (t, x) and A(u) ~ (ail(u)) is ann x n matrix with suitably smooth elements ail(u) (i, j = 1,... ) n).Suppose that the system (1.1) is strictly hyperbolic in a neighbourhood of u = 0, namely,for any given u in this domain, A(u) has n distinct real eigenvalues Al(u), AZ(u),'' j A.(u)such thatAl(u) < AZ(u) <'' < A.(u). (1.2)For i = 1,'',nl let h(u…     

LIFE-SPAN OF CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS WITH SLOWDECAY 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 casethat 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.     

LIFE-SPAN OF CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS WITH SLOWDECAY 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 TO QUASILINEAR HYPERBOLIC SYSTEMS WITH WEAK LINEAR DEGENERACY

§1. Introduction and Main Results Consider the following ?rst order quasilinear strictly hyperbolic system ?u ?u A(u) = 0, (1.1) ?t ?xwhere u = (u1, ···,un)T is the unknown vector function of (t,x) and A(u) is an n×n matrixwith suitably smooth elements aij(u) (i,j = 1, ···,n). By the de?nition …     

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 datafor quasilinear hyperbolic systems and obtain a necessary and sufficient condition toguarantee the existence and uniqueness of global weakly discontinuous solution.     

BREAKDOWN OF CLASSICAL SOLUTION TO A KIND OF QUASILINEAR NON-STRICTLY HYPERBOLIC SYSTEMS

In this article, the author considers the Cauchy problem for quasilinear non-strict ly hyperbolic systems and obtain a blow-up result for the C1 solution to the Cauchy problem with weaker decaying initial data.     

FORMATION OF SINGULARITIES FOR A KIND OF QUASILINEAR NON－STRICTLY HYPERBOLIC SYSTEM

The author gets a blow-up result of C1 solution to the Cauchy problem for a first order quasilinear non-strictly hyperbolic system in one space dimension.     

拟线性双曲组的经典解的奇性形成

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.     

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 authorsprove,assuming additional conditions on the initial data,some results about the existence anduniqueness of the entropy weak solutions of the Cauchy problem for the singular hyperbolicsystem a_t+(au)_x_2au/x=0,u_t+1/2(a~2+u~2)_x=0,x>0,t≥0.     

EXACT CONTROLLABILITY FOR FIRST ORDER QUASILINEAR HYPERBOLIC SYSTEMS WITH ZERO EIGENVALUES＊＊＊

For a class of mixed initial-boundary value problem for general quasilinear hyperbolic sys-tems with zero eigenvalues, the authors establish the local exact controllability with boundary controls acting on one end or on two ends and internai controls acting on a part of equations in the system.     

BREAKDOWN OF CLASSICAL SOLUTION TO A KIND OF QUASILINEAR NON-STRICTLY HYPERBOLIC SYSTEMS

In this article, the author considers the Cauchy problem for quasilinear non-strictly hyperbolic systems and obtain a blow-up result for the C¹ solution to the Cauchy problem with weaker decaying initial data.     

EXISTENCE OF GLOBAL SMOOTH SOLUTIONS FOR TWO IMPORTANT NONSTRICTLY QUASILINEAR HYPERBOLIC SYSTEMS

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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 dissi...     

A NECESSARY AND SUFFICIENT CONDITION FOR GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO CAUCHY PROBLEM OF QUASILINEAR HYPERBOLIC SYSTEMS IN DIAGONAL FORM

This article considers Cauchy problem for quasilinear hyperbolic systems in diagonal form.A necessary and sufficient condition in guaranteeing that Cauchy problem admits a unique global classical solution on t 0 is obtained,and a sharp estimate of the life span for the classical solution is given.     

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.     

LOCAL EXACT BOUNDARY CONTROLLABILITY FOR A CLASS OF QUASILINEAR HYPERBOLIC SYSTEMS

InroductionLet us consider the following irst order quaslllnear hyperbolic     

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 existence and uniqueness of semi-global C^1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems with general nonlinear boundary conditions.     

CONSTRUCTION OF SOLUTIONS TO M-D RIEMANN PROBLEMS FOR A 2*2 QUASILINEAR HYPERBOLIC SYSTEM

The author studies M-D Riemann problems for a quasilinear nonstrictly byperbolic system.The initial date are taken as three different constants in three sections divided by three rays starting from the origin.From each direction of these rays two waves coming from infinity are allowed.All possible local singularity structures are carefully studied and classified.Then based on such analysis,existence and global singularity structure of the solution are obtained under some assumptions.     

THE CAUCHY PROBLEMS FOR HIGH DIMENSIONAL QUASILINEAR HYPERBOLIC SYSTEMS

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