On the Gevrey well posedness of the Cauchy problem for weakly hyperbolic equations of higher order

We shall consider the Cauchy problem for weakly hyperbolic equations of higher order with coefficients depending only on time. The regularities of the distinct characteristic roots and the multiple characteristic roots independently influence Gevrey well posedness of the Cauchy problem.     

On a Darboux type multidimensional problem for second-order hyperbolic systems

The correct formulation of a Darboux type multidimensional problem for second-order hyperbolic systems is investigated. The correct formulation of such a problem in the Sobolev space is proved for temporal type surfaces on which the boundary conditions of a Darboux type problem are given.     

On the solvability of one multidimensional version of the first Darboux problem for some nonlinear wave equations

In the present paper, for wave equations with power nonlinearity we investigate the problem of the existence or nonexistence of global solutions of a multidimensional version of the first Darboux problem in the conic domain.     

On a Darboux problem for a third-order hyperbolic equation with multiple characteristics

A Darboux type problem for a model hyperbolic equation of the third order with multiple characteristics is considered in the case of two independent variables. The Banach space, 0, is introduced where the problem under consideration is investigated. The real number ₀ is found such that for > ₀ the problem is solved uniquely and for < ₀ it is normally solvable in Hausdorff's sense. In the class of uniqueness an estimate of the solution of the problem is obtained which ensures stability of the solution.     

Cauchy–Dirichlet problem for second-order hyperbolic equations in cylinders with non-smooth base

This paper is concerned with the smoothness of generalized solutions of the Cauchy–Dirichlet problem for the second-order hyperbolic equation in domains with a conical point.     

Cauchy problem for higher order p-evolution equations

We consider p  -evolution equations of order m?2$m?2$ in (t,x)$(t,x)$ with real characteristics. We give sufficient conditions for the well-posedness of the Cauchy problem in Sobolev spaces, in terms of decay estimates of the coefficients as the space variable x→∞$x\to \infty$.     

On the wellposedness of the Cauchy problem for weakly hyperbolic equations of higher order

We study the wellposedness in the Gevrey classes G^s and in C^∞ of the Cauchy problem for weakly hyperbolic equations of higher order. In this paper we shall give a new approach to the case that the characteristic roots oscillate rapidly and vanish at an infinite number of points. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Cauchy problem for second order strictly hyperbolic equations with non–regular coefficients

In this paper we shall consider some necessary and sufficient conditions for well–posedness of second order hyperbolic equations with non–regular coefficients with respect to time. We will derive some optimal regularities for well–posedness from the intensity of singularity to the coefficients by WKB representation of the solution and some counter examples which are constructed by using ideas of Floquet theory.     

On degenerate quasilinear parabolic equations of higher order

In the paper existence results for degenerate quasilinear parabolic initial boundary value problems of higher order are proved. The weak solution is sought in a suitable weighted Sobolev space using the generalized degree theory. Supported by the funds of State Educational Commission of China for returned scholars from abroad.     

Existence and nonexistence of global solutions for a class of nonlinear wave equations of higher order

In this paper, we investigate the existence and uniqueness of the solution to the Cauchy problem for a class of nonlinear wave equations of higher order and prove the existence and nonexistence of global solutions to this problem by a potential well method.     

On some multidimensional versions of a characteristic problem for second-order degenerating hyperbolic equations

Some multidimensional versions of a characteristic problem for second-order degenerating hyperbolic equations are considered. Using the technique of functional spaces with a negative norm, the correctness of these problems in the Sobolev weighted spaces are proved.     

An existence result for dissipative nonhomogeneous hyperbolic equations via a minimization approach

We discuss a purely variational approach to the study of a wide class of second order nonhomogeneous dissipative hyperbolic PDEs. Precisely, we focus on the wave-like equations that present also a nonzero source term and a first-order-in-time linear term. The paper carries on the research program initiated in [14], and developed in [15], [21], on the De Giorgi approach to hyperbolic equations.     

On the Riemann problem for 2D compressible Euler equations in three pieces

The Riemann problem for two-dimensional isentropic Euler equations is considered. The initial data are three constants in three fan domains forming different angles. Under the assumption that only a rarefaction wave, shock wave or contact discontinuity connects two neighboring constant initial states, it is proved that the cases involving three shock or rarefaction waves are impossible. For the cases involving one rarefaction (shock) wave and two shock (rarefaction) waves, only the combinations when the three elementary waves have the same sign are possible (impossible).     

Global existence and causality for a hyperbolic transmission problem with a repulsive nonlinearity

It is well known that the solution of the classical linear wave equation with an initial condition with compact support and vanishing initial velocity also has a compact support included in a set depending on time: the support of the solution at time t$t$ is causally related to that of the initial condition. Reed and Simon have shown that for a real-valued Klein–Gordon equation with (nonlinear) right-hand side −λu³$-\lambda u^3$ (λ>0$\lambda >0$), causality still holds. We show the same property for a one-dimensional Klein–Gordon problem but with transmission and with a more general repulsive nonlinear right-hand side F(u)$F\left(u\right)$. We also prove the global existence of a solution using the repulsiveness of F$F$. In the particular case F(u)=−λu³$F\left(u\right)=-\lambda u^3$, the problem is a relativistic model for a quantum particle with repulsive self-interaction and tunnel effect at a semi-infinite potential step.     

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

It is proven that the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws admits a unique global piecewise C¹$C^1$ solution u=u(t,x)$u=u(t,x)$ containing only n$n$ shock waves with small amplitude on t?0$t?0$ and this solution possesses a global structure similar to that of the similarity solution u=U(x/t)$u=U(x/t)$ of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data.