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.     

Formation of Singularities for Quasilinear Hyperbolic Systems with Characteristics with Constant Multiplicity

In this paper we consider the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Without restriction on characteristics with constant multiplicity(> 1), under the assumptions that there is a genuinely nonlinear simple characteristic and the initial data possess certain decaying properties, the blow-up result is obtained for the C¹ solution to the Cauchy problem.     

Life-span of Classical Solutions to Nonlinear Wave Equations in Two-space-dimensions II

In two-space-dimensional case we get the sharp lower bound of the life-span of classical solutions to the Cauchy problem with small initial data for fully nonlinear wave equations of the form ◻u = F (u, Du, D_zDu) in which F(\hat{λ}) = O(|\hat{λ}|^{1+α}) with α = 2 in a neighbourhood of \hat{λ} = 0. The cases α = 1 and α ≥ 3 have been considered respectively in [1] and [2].     

Gradient Catastrophe in the Classical Solutions of Nonlinear Hyperbolic Systems

Classical solutions of hyperbolic systems, generally, collapse in finite time, even for small and smooth initial data. Here, we consider a type of these systems and prove a blow up result.     

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.     

Formation of Singularities of Solutions for Cauchy Problem of Quasilinear Hyperbolic Systems with Dissipative Terms

ln this paper, for a class of 2 × 2 quasilinear hyperbolic systems, we get existence theorems of the global smooth solutions of its Cauchy problem, under a certain hypotheses. In addition, Tor two concrete quasilinear hyperbolic systems, we study the formation of the singularities of the C¹-solution to its Cauchy problem.     

Analytic Propagation for Nonlinear Weakly Hyperbolic Systems

The propagation of analyticity for sufficiently smooth solutions to either strictly hyperbolic, or smoothly symmetrizable nonlinear systems, dates back to Lax [14 Lax , P.D. ( 1953 ). Nonlinear hyperbolic equations . Comm. Pure Appl. Math. 6 : 231 – 258 . [Google Scholar]] and Alinhac and Métivier [2 Alinhac , S. , Métivier G. ( 1984 ). Propagation de l'analyticité des solutions de systèmes hyperboliques nonlinéaires [Propagation of analyticity for solutions of nonlinear hyperbolic systems]. Invent. Math. 75, 189–204 . [Google Scholar]]. Here we consider the general case of a system with real, possibly multiple, characteristics, and we ask which regularity should be a priori required of a given solution in order that it enjoys the propagation of analyticity. By using the technique of the quasi-symmetrizer of a hyperbolic matrix, we prove, in the one-dimensional case, the propagation of analyticity for those solutions which are Gevrey functions of order s for some s < m/(m ? 1), m being the maximum multiplicity of the characteristics.     

一类高阶非线性双曲型方程Cauchy问题解的爆破

本文研究一类高阶非线性双曲型方程utt-uxx+μuxxx-αuxxtt+βuxxxxtt=f(ux)x的Cauchy问题,证明问题解的存在性与唯一性,并给出解在有限时刻爆破的充分条件.     

一维磁流体力学方程组经典解的生命区间

考虑具耗散项的一维磁流体力学方程组Cauchy问题．对于非耗散情形证明了如果初始能量和磁场强度弱于声波的能量,则Cauchy问题的光滑解在有限时间内破裂;对于耗散情形,如果初始能量、磁场强度和耗散强度弱于声波的能量,则Cauchy问题的光滑解在有限时间内破裂,而且给出了生命区间估计．     

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

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

On Quasilinear Generalized Canonical Hyperbolic Systems of First-Order Partial Differential Equations

By using the method of characteristics, we prove theorems on continuous solvability and on properties of solutions of the mixed Cauchy boundary-value problem for the generalized canonical hyperbolic system of quasilinear partial differential equations of the first order in a general connected domain in (m+ 1)-dimensions.     

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.     

Breakdown of Classical Solutions for Quasilinear Hyperbolic Systems of Diagonal Form

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Cauchy Problem for General Rst Order Inhomogeneous Quasilinear Hyperbolic Systems

In this paper, we consider Cauchy problem for general first order inho- mogeneous quasilinear strictly hyperbolic systems. Under the matching condition, we first give an estimate on inhomogeneous terms. By this estimate, we obtain the asymptotic behaviour for the life-span of C¹ solutions with “slowly” decaying and small initial data and prove that the formation of singularity is due to the envelope of characteristics of the same family.     

Blow-up Mechanism of Classical Solutions to Quasilinear Hyperbolic Systems in the Critical Case

This paper deals with the blow-up phenomenon, particularly, the geometric blow-up mechanism, of classical solutions to the Cauchy problem for quasilinear hyperbolic systems in the critical case. We prove that it is still the envelope of the same family of characteristics which yields the blowup of classical solutions to the Cauchy problem in the critical case.     

三步完全可化约拟线性双曲组型方程组的整体经典解

本文对三步完全可化约拟线性双曲型方程组的整体经典解进行讨论，得到了整体经典解的存在性．这一结果说明非线性特征向量也可能引发奇性．     

两类方程Cauchy问题经典解的等价性

对于给出的一类二阶线性双曲型方程,通过未知变量替换,将其化为一阶对称双曲型方程组.可以证明这个一阶对称双曲型方程组与原来的二阶线性双曲型方程的Cauchy问题的经典解在某种意义下是等价的.     

一阶非齐次拟线性双曲组的柯西问题整体经典解的存在性

本文考察了弱线性退化的一阶非齐次拟线性严格双曲组具有小初值的柯西问题．在非齐次项满足匹配条件的假设下,给出了精细的波的分解公式,利用这些公式,证明了整体C1解的存在唯一性和稳定性．     

一类高阶非线性双曲型方程初边值问题解的爆破

本文研究一类高阶非线性双曲型方程的初边值问题，证明问题局部广义解的存在性与唯一性，同时给出解爆破的充分条件。     

On the Decay of Solutions for Some Nonlinear Dissipative Hyperbolic Equations

In this paper we show the decay of solutions to the initial-boundary value problem for somenonlinear hyperbolic equation with a nonlinear dissipative term,by using a difference inequality.