Higher Order Hyperbolic Equations Research Articles

Homogenization of the Higher-Order Hyperbolic Equations with Periodic Coefficients

Homogenization of the Higher-Order Hyperbolic Equations with Periodic Coefficients

Dirichlet–Neumann Problem for Systems of Hyperbolic Equations with Constant Coefficients

In a domain obtained as the Cartesian product of a segment by a circle of unit radius, we investigate a boundary-value problem with Dirichlet–Neumann conditions with respect to the time variable for a system of high-order hyperbolic equations with constant coefficients. We establish the conditions of unique solvability of the problem in the Sobolev spaces and construct its solution in the form of a vector series in a system of orthogonal functions. To establish lower estimates of small denominators encountered in the construction of solutions of the problem, we use the metric approach.

Dirichlet–Neumann Problem in a Strip for Hyperbolic Equations with Constant Coefficients

We study the conditions of unique solvability of the problem with Dirichlet–Neumann conditions with respect to the time variable in a strip and the conditions of periodicity or almost periodicity with respect to a spatial coordinate for linear hyperbolic equations of higher order with constant coefficients. The solutions of the considered problems in the form of series in systems of orthogonal functions are constructed. To obtain the lower bound for small denominators appearing in the construction of solutions of the problems, we use the metric approach.

A damping term for higher-order hyperbolic equations

We construct a damping term for general higher-order strictly hyperbolic homogeneous equations with constant coefficients. We derive long-time decay estimates for the solution to the Cauchy problem, and we show that no better dissipative effect can be obtained with a different damping term.

On some three-dimensional variants of Goursat and Darboux problems for higher-order hyperbolic equations with dominating principal parts

Some three-dimensional variants of Goursat and Darboux problems for higher-order hyperbolic equations with dominating principal part are set and investigated. Conditions to the problems' data which in some cases ensure the well-posedness of the problem in question and in other cases, despite the solvability of the problem, imply the presence of an infinite set of linearly independent solutions of corresponding homogeneous problem, are found. We consider both generalized and classical solutions.

Regularity of the Solution of the First Initial-Boundary Value Problem for Hyperbolic Equations in Domains with Cuspidal Points on Boundary

The goal of this paper is to establish the regularity of the solution of the first initial-boundary value problem for general higher-order hyperbolic equations in cylinders with the bases containing cuspidal points.

On Doubly Periodic Solutions of Nonlinear Hyperbolic Equations of Higher Order

Abstract Unimprovable conditions of the existence and uniqueness of doubly periodic solutions are established for nonlinear hyperbolic equations of higher order with two independent variables.

Some inequalities of Glaeser-Bronšteĭn type

The classical Glaeser estimate is a special case of the Lemma of Bronšteĭn which states the Lipschitz continuity of the roots \lambda_j(x) of a hyperbolic polynomial P(x,X) with coefficients a_j(x) depending on a real parameter. Here we prove a pointwise estimate for the successive derivatives of the a_j(x) 's in term of certain nonnegative functions which are symmetric polynomials of the roots \{\lambda_j(x)\} (hence also of the coefficients \{a_j(x)\}) . These inequalities are very helpful in the study of the Cauchy problem for homogeneous weakly hyperbolic equations of higher order.

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

AbstractWe study the wellposedness in the Gevrey classes Gs 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)

Well-posedness of the cauchy problem in gevrey classes for some weakly hyperbolic equations of higher order

This article is devoted to the study of the Cauchy problem in Gevrey classes for some higher order weakly hyperbolic equations with time-dependent coefficients and without lower order terms.

Qualitative properties of solutions for linear and nonlinear hyperbolic PDE's

We present a number of results concerning large-time qualitative behavior of solutions for high-order hyperbolic equations and first-order hyperbolic systems. We discuss the properties of exponential stability and exponential dichotomy, construction of stable, unstable, and center manifolds, Grobman--Hartman type theorems on linearization of the phase portrait, and existence and uniqueness of time-bounded and almost periodic (AP) solutions.

The Riemann Function for Higher-Order Hyperbolic Equations and Systems with Dominated Lower-Order Terms

The Riemann Function for Higher-Order Hyperbolic Equations and Systems with Dominated Lower-Order Terms

Blow-up and critical exponents for nonlinear hyperbolic equations

We prove nonexistence results for the Cauchy problem for the abstract hyperbolic equation in a Banach space X, u tt=f′(u), t>0; u(0)=u 0, u t(0)=u 1, where f : X→ R is a C 1-function. Several applications to the second- and higher-order hyperbolic equations with local and nonlocal nonlinearities are presented. We also describe an approach to Kato's and John's critical exponents for the semilinear equations u t= Δu+b(x,t)|u| p, p>1 , which are responsible for phenomena of stability, unstability, blow-up and asymptotic behaviour.

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.

A Boundary-Value Problem for Weakly Nonlinear Hyperbolic Equations with Data on the Entire Boundary of a Domain

In a domain that is the Cartesian product of a segment and a p-dimensional torus, we investigate a boundary-value problem for weakly nonlinear hyperbolic equations of higher order. For almost all (with respect to Lebesgue measure) parameters of the domain, we establish conditions for the existence of a unique solution of the problem.

Bounded and Almost Periodic Solutions of Linear High-Order Hyperbolic Equations

The present paper is devoted to investigating high-order strictly hyperbolic operators with nearly constant coefficients. The invertibility of these operators in spaces of functions uniformly bounded or almost periodic in time is established, provided the symbol of the operator in question has no roots in an open strip containing the real line. Under the additional condition that the strip coincides with a half-plane, exponentially decaying solutions of the nonhomogeneous equation with right-hand side of exponential decay are constructed. In the case of equations with constant coef- ficients the necessity of the derived conditions is proved. The results of this paper were announced without proofs in (SV).

On the Correct Formulation of a Multidimensional Problem for Strictly Hyperbolic Equations of Higher Order

Abstract A theorem of the unique solvability of the first boundary value problem in the Sobolev weighted spaces is proved for higher-order strictly hyperbolic systems in the conic domain with special orientation.

On the correct formulation of a multidimensional problem for strictly hyperbolic equations of higher order

A theorem of the unique solvability of the first boundary value problem in the Sobolev weighted spaces is proved for higherorder strictly hyperbolic systems in the conic domain with special orientation.

Reflection and interaction of progressing waves for semilinear wave equations

Recently, many works have been devoted to the discussion of the propagation and interaction of singularities for solutions of nonlinear hyperbolic equations. Among them the study of the interaction of nonlinear progressing waves is noticeable. First in [ 11, J. M. Bony discussed the behavior of’the interaction of two progressing waves. He showed that for semilinear hyperbolic equations of second order, when two progressing waves intersect transversally, no anomalous singularities are caused by the interaction, and for semilinear hyperbolic equations of higher order anomalous progressing waves appear issuing from the intersection. However, the interaction is much more complicated when three progressing waves intersect transversally with each other at one point. J. Rauch and M. Reed constructed an example showing that in the triple-intersection case, besides the linear-predicted singularities, a new progressing wave caused by interaction issuing from the intersection point propagates along the characteristic cone. Recently, R. Melrose and N. Ritter [3], J. M. Bony 141, and M. Beals [5] discussed the general triple-intersection of progressing waves for semilinear wave equations with two space variables by different methods. They pointed out that in the generalized case the singularities have the same structure as that in RauchhReed’s example. On the other hand, M. Beals and G. Metivier considered reflection of progressing waves for boundary value problems of semilinear hyperbolic equations. On the basis of these works, we study in this paper the interaction of two progressing waves with reflection on the boundary for boundary value problems of semilinear wave equations in two space variables. Our result shows that, in this case, beside the linear-predicted singularities there appears an anomalous reflected wave, which issues from the intersection

Equipartition of energy for higher-order hyperbolic equations

Let A(0), A(1),...,A(2(N)-1) be commuting skew-adjoint operators on a Hilbert space [unk]. Then the equation Pi(j=0) (2(N)-1) (d/dt - A(j))v(t) = 0 (t real) admits equipartition of energy [in the sense that the jth partial energy E(j)(t) of any solution at time t satisfies lim(t-->+/-infinity)E(j)(t) = 2(-N).(total energy) for each of the 2(N) values of j] if and only if the closure B(jk) of A(j) - A(k) satisfies weak-operator-limit exp(tB(jk)) = 0 as t --> +/-infinity whenever j not equal k.