Third order homogeneous weakly hyperbolic equations with nonanalytic coefficients

We consider the Cauchy problem for homogeneous linear third order weakly hyperbolic equations with time depending coefficients. We study the relation between the regularity of the coefficients and the Gevrey class in which the Cauchy problem is well-posed.     

A note on well-posedness of bidirectional Whitham equation

We consider the initial-value problem for the bidirectional Whitham equation, a system which combines the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow-water nonlinearity. We prove local well-posedness in classical Sobolev spaces, using a square-root type transformation to symmetrise the system.     

Oscillations of nonlinear hyperbolic equations with functional arguments via Riccati method

This paper is devoted to the study of the oscillatory behavior of solutions of nonlinear hyperbolic equations with functional arguments by using integral averaging method and a generalized Riccati technique. First, we establish oscillation results for nonlinear hyperbolic equations by applying oscillation criteria for Riccati inequality. Secondly, we present interval oscillation results for nonlinear hyperbolic equations. These results improve and extend oscillation results of Cui and Xu [1].     

Numerical verification of solutions for nonlinear hyperbolic equations

We consider a numerical method to verify the solutions for nonlinear hyperbolic problems with guaranteed error bounds in the one-space dimensional case. We present verification procedures and show some numerical examples.     

Invariants of two- and three-dimensional hyperbolic equations

We consider linear hyperbolic equations of the form

     

A note on Jordan type inequalities for hyperbolic functions

In this work the authors present some new lower and upper bounds for the functions sinx/x and x/sinhx, thus improving some inequalities put forward by Klén et al. (2010) in the paper [2].     

Differential invariants for systems of linear hyperbolic equations

In this paper we consider a general class of systems of two linear hyperbolic equations. Motivated by the existence of the Laplace invariants for the single linear hyperbolic equation, we adopt the problem of finding differential invariants for the system. We derive the equivalence group of transformations for this class of systems. The infinitesimal method, which makes use of the equivalence group, is employed for determining the desired differential invariants. We show that there exist four differential invariants and five semi-invariants of first order. Applications of systems that can be transformed by local mappings to simple forms are provided.     

On convexity of the functional for inverse problems of hyperbolic equations

In this paper, a strategy to design a functional for inverse problems of hyperbolic equations is proposed. For an inverse source problem, it is shown that the designed functional is globally strictly convex. For an inverse coefficient problem, we can only prove that it is strictly convex near true solution. This strategy can be generalized to other inverse problems, as long as Lipschitz stability is given.     

On linearization of hyperbolic equations using differential invariants

In this paper we consider the general class of hyperbolic equations uxt=F(x,t,u,ux,ut). We use equivalence transformations to derive differential invariants for this class and for two subclasses. Then we employ these invariants to construct equations that can be linearized via local mappings. Further applications of the differential invariants are given.     

Perturbation index of linear partial differential-algebraic equations with a hyperbolic part

This paper deals with linear partial differential-algebraic equations (PDAEs) which have a hyperbolic part. If the spatial differential operator satisfies a Gårding-type inequality in a suitable function space setting, a perturbation index can be defined. Theoretical and practical examples are considered.     

A note on the tameness of hyperbolic 3-manifolds

Among other related results we prove that a hyperbolic 3-manifold which admits an exhaustion by nested cores is tame.     

Method for solving hyperbolic systems with initial data on non-characteristic manifolds with applications to the shallow water wave equations

We are concerned with hyperbolic systems of order-one linear PDEs originated on non-characteristic manifolds. We put forward a simple but effective method of transforming such initial conditions to standard initial conditions (i.e. when the solution is specified at an initial moment of time). We then show how our method applies in fluid mechanics. More specifically, we present a complete solution to the problem of long waves run-up in inclined bays of arbitrary shape with nonzero initial velocity.     

Quasi-linear weakly hyperbolic equations with Gevrey-Levi conditions

This paper is devoted to the study of the analytic regularity for real solutions of a non-linear weakly hyperbolic equation of the form: $$\sum\limits_{m - (1 - e)r< |a| \leqslant m} {a_a (y,u^{(\beta )} } )\partial _y^a u = g(y,u^{(\beta )} ){\text{, |}}\beta | \leqslant m - (1 - e)r,$$ wherea _α andg are analytic functions of their arguments,r≥2 denotes the largest multiplicity of the characteristic roots of (*) and ?∈(0,(r?1)/r). Assuming that 026 the characteristic roots are of constant multiplicity and that the generalized Levi’s conditions related to the index ? are satisfied, we prove that, ifu is a solution of (*) 026 and belongs to a Gevrey class of index σ<1/?, then the analyticity of Cauchy data propagates according to the geometry of the influence domains of the equation.     

First-order hyperbolic pseudodifferential equations with generalized symbols

We consider the Cauchy problem for a hyperbolic pseudodifferential operator whose symbol is generalized, resembling a representative of a Colombeau generalized function. Such equations arise, for example, after a reduction-decoupling of second-order model systems of differential equations in seismology. We prove existence of a unique generalized solution under log-type growth conditions on the symbol, thereby extending known results for the case of differential operators [J. Math. Anal. Appl. 160 (1991) 93-106, J. Math. Anal. Appl. 142 (1989) 452-467].     

Complete surfaces in the hyperbolic space with a constant principal curvature

In this paper we study complete orientable surfaces with a constant principal curvature R in the 3‐dimensional hyperbolic space H ³. We prove that if R² > 1, such a surface is totally umbilical or umbilically free and it can be described in terms of a complete regular curve in H ³. When R² ≤ 1, we show that this result is not true any more by means of several examples. This contradicts a previous statement by Zhisheng [6]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Energy estimates for regularly hyperbolic operators with log-Lipschitz coefficients

Using the method of Colombini–Lerner, we derive the energy estimates for regularly hyperbolic equations with log-Lipschitz coefficients.     

Local energy decay for a class of hyperbolic equations with constant coefficients near infinity

A uniform local energy decay result is derived to a compactly perturbed hyperbolic equation with spatial vari¬able coefficients. We shall deal with this equation in an N ‐dimensional exterior domain with a star‐shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data and the equation includes anisotropic variable coefficients {a_i(x): i = 1, 2, …, N }, which are not necessarily equal to each other (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

FORCED OSCILLATIONS OF HYPERBOLIC DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

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