Breakdown of Classical Solutions for Quasilinear Hyperbolic Systems of Diagonal Form

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A Class of Cauchy Problem for Quasilinear Hyperbolic Systems with the Weakly Dissipative Terms

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Asymptotic Behavior of Global Classical Solutions of Quasilinear Non-strictly Hyperbolic Systems with Weakly Linear Degeneracy

In this paper, we study the asymptotic behavior of global classical solutions of the Cauchy problem for general quasilinear hyperbolic systems with constant multiple and weakly linearly degenerate characteristic fields. Based on the existence of global classical solution proved by Zhou Yi et al., we show that, when t tends to infinity, the solution approaches a combination of C1 travelling wave solutions, provided that the total variation and the L1 norm of initial data are sufficiently small.     

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.     

The Problem of Periodic Solutions for a Class of Dynamical Systems（Ⅰ

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A Class of the Quasilinear Hyperbolic Equations with the Inhomogenous Terms

In this paper,we discuss a class of the quasillinear hyperbolic equations with the inhomogeneous terms:ur σ(v)x 2α(t)u=0,vr-ux=0 Under the certain of hypothesis,we prove the globally existence theorems of the smooth solutions for is cauchy problem.     

Hugoniót–Maslov Chains for Singular Vortical Solutions to Quasilinear Hyperbolic Systems and Typhoon Trajectory

According to Maslov, many 2D quasilinear systems of PDE possess only three algebras of singular solutions with properties of structural self-similarity and stability. They are the algebras of shock waves, narrow solitons, and square-root point singularities (solitary vortices). Their propagation is described by infinite chains of ODE (the Hugoniót–Maslov chains). We consider the Hugoniót-Maslov chain for the square-root point singularities of the shallow water equations. We discuss different related mathematical questions (in particular, unexpected integrability effects) as well as their possible application to the problem of typhoon dynamics.     

Structure of Coexistence States for a Class of Quasilinear Elliptic Systems

The structure of positive solutions of the p-Laplacian systems is discussed via bifurcation theory and monotone techniques.     

Stability of Solutions for a Class of Quasilinear Degenerate Parabolic Equations

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Global Smooth Solutions for Quasilinear Hyperbolic Equations with "Really Large" Initial Date

Consider the Cauehy problem of the first order quasilinear hyperbolie equations z_t λ(z,w)z_x=0,w_t (z,w)w_x=0, (E) z(0,x)=2_0(x),w(0,x)=w_0(x). (I)We suppose F(z,w)=μ(z,w)-λ(z,w)>O,in D,(1) λ_z(z,w)>O,μ_w(z,w)>O.in D.(2)It is well known that if Q∈∈D (V)where Q≡{(z,w)|z_*≤z≤z~*,w_*≤w≤w~*},z_*=inf z_0(x),~*=8up z_0(x),w_*=infw_0(x),w~*=supw_0(x),then the necessary and 8ufficient COildition for the existence ofglobal smooth solution is[1—3] z_0~＇≥O,w_0~(x)≥0 (M)asz_0(x),w_0(x)∈C~1(R).However,under the COIldition(V),the initial data under     

Strong （Weak） Exact Controllability and Strong （Weak） Exact Observability for Quasilinear Hyperbolic Systems

In this paper,the authors define the strong （weak） exact boundary controllability and the strong （weak） exact boundary observability for first order quasilinear hyperbolic systems,and study their properties and the relationship between them.     

Exact Controllability for Nonautonomous First Order Quasilinear Hyperbolic Systems

By means of the theory on the semi-global C1 solution to the mixed initial-boundary value problem (IBVP) for first order quasilinear hyperbolic systems, we establish the exact controllability for general nonautonomous first order quasilinear hyperbolic systems with general nonlinear boundary conditions.     

Sign-Changing Solutions for a Class of Nonlinear Elliptic Problems

In the present paper,the author studies the existence of sign-changing solutions for nonlinear elliptic equations,which have jumping nonlinearities,and may or may not be resonant with respect to Fu(?)ik spectrum,via linking methods under Cerami condition.     

Existence and Uniqueness of Weak Solutions for Quasilinear Parabolic Systems

Consider the following IBVP for quasilinear parabolic systemsWhereΩis a bounded domain in R~n with a sufficiently smooth boundary(?)Ω,u=     

Oscillation Theorem of Systems of Quasilinear Impulsive Delay Hyperbolic Equations

In this paper, oscillatory properties for solutions of the systems of certain quasilinear impulsive delay hyperbolic equations with nonlinear diffusion coefficient are investigated. A sufficient criterion for oscillations of such systems is obtained.     

Strong （Weak） Exact Controllability and Strong （Weak） Exact Observability for Quasilinear Hyperbolic Systems

Numerical approximations of Cahn-Hilliard phase-field model for the two-phase incompressible flows are considered in this paper.Several efficient and energy stable time discretization schemes for the coupled nonlinear Cahn-Hilliard phase-field system for both the matched density case and the variable density case are constructed,and are shown to satisfy discrete energy laws which are analogous to the continuous energy laws.     

Existence of Positive Solutions to Fourth Order Quasilinear Boundary Value Problems

In this paper, we generalize the fixed point theorem of cone expansion and compression of norm type to the theorem of functional type. As an application, the existence of positive solutions for some fourth-order beam equation boundary value problems is obtained. The emphasis is put on that the nonlinear term is dependent on all lower order derivatives.     

Existence of Solutions to Generalized Vector Quasi-Equilibrium Problems with Discontinuous Mappings

Let X, Y be two finite-dimensional topological vector spaces, Z a Hausdorff topological vector space, K C X and D C Z be two nonempty sets, C be a pointed, closed, and convex cone in Y with int C ≠θ Let S ： K → 2^K and T ： K → 2^D be two multivalued mappings, and φ ： K × D × K → Y be a trifunction. In this paper, we consider the generalized vector quasi-equilibrium problem, which is formulated by finding X∈ K and y∈ T（x） such that x∈ E S（x） and φ（x,y, u） （∈/） -int C for all u ∈ S（x）. We establish an existence result in which T is not supposed to have any continuity property. Our results extend and improve the corresponding results of Cubiotti, Yao and Guo.     

Solutions of Robin Problems for Overdetermined Inhomogeneous Cauchy–Riemann Systems on the Unit Polydisc

The inhomogeneous Robin condition with general coefficient for the overdetermined inhomogeneous Cauchy–Riemann system of equations on the polydisc is studied using Fourier analysis. It is shown that this problem for the case of nonholomorphic general coefficient, is actually a problem with essential singularity in the domain, but still well-posed under certain compatibility conditions. Under proper assumptions, the unique solution is given explicitly.     

Cauchy Problem for Quasilinear Hyperbolic Systems with Higher Order Dissipative Terms

Abstract In this paper, the author studies the global existence, singularities and life span of smoothsolutions of the Cauchy probleth for a class of quasilinear hypetbolic systems with higher order dissipativeterms and gives their applications to nonlinear wave equations with higher order dissipative terms.