On the solvability of a spatial problem of Darboux type for the wave equation

The question of the correct formulation of one spatial problem of Darboux type for the wave equation has been investigated. The correct formulation of that problem in the Sobolev space has been proved for surfaces having a quite definite orientation on which are given the boundary value conditions of the problem of Darboux type.     

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 spatial problem of Darboux type for a second-order hyperbolic equation

The theorem of unique solvability of a spatial problem of Darboux type in Sobolev space is proved for a second-order hyperbolic equation.     

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 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.     

Spatial problem of Darboux type for one model equation of third order

For a hyperbolic type model equation of third order a Darboux type problem is investigated in a dihedral angle. It is shown that there exists a real number ₀ such that for > ₀ the problem under consideration is uniquelly solvable in the Frechet space. In the case where the coefficients are constants, Bochner's method is developed in multidimensional domains, and used to prove the uniquely solvability of the problem both in Frechet and in Banach spaces.     

On the Cauchy problem for the wave equation on time-dependent domains

We introduce a notion of solution to the wave equation on a suitable class of time-dependent domains and compare it with a previous definition. We prove an existence result for the solution of the Cauchy problem and present some additional conditions which imply uniqueness.     

A blow up result for a fractionally damped wave equation

In this paper we prove a blow up result for solutions of the wave equation with damping of fractional order and in presence of a polynomial source. This result improves a previous result in [5]. There we showed that the classical energy is unbounded provided that the initial data are large enough.     

On the global solvability of Kirchhoff equation for non-analytic initial data

We give sufficient conditions for the global solvability of Kirchhoff equation in terms of the spectral resolutions of the initial data . We assume no smallness conditions and only “Sobolev-type” regularity.     

半线性波动方程的高维古沙问题

In this paper we study the Goursat problem for semilinear wave equations with zero boundary condition in which the boundary is the characteristic cone for wave operator. Our result states that the solution is Lipschitz and is smooth awayfrom the characteristic cone.     

On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms

In this paper we consider the decay and blow-up properties of a viscoelastic wave equation with boundary damping and source terms. We first extend the decay result (for the case of linear damping) obtained by Lu et al. (On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution, Nonlinear Analysis: Real World Applications 12 (1) (2011), 295-303) to the nonlinear damping case under weaker assumption on the relaxation function g(t). Then, we give an exponential decay result without the relation between g^′(t) and g(t) for the linear damping case, provided that ‖g‖_L¹(0,∞) is small enough. Finally, we establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases, the other is for certain solutions with arbitrarily positive initial energy for the linear damping case.     

On the Cauchy problem for a generalized Camassa–Holm equation

The local well-posedness of a generalized Camassa–Holm equation is established by means of Kato's theory for quasilinear evolution equations and two types of results for the blow-up of solutions with smooth initial data are given.     

Homogenization and behaviour of optimal controls for the wave equation in domains with oscillating boundary

In this paper the asymptotic behaviour of a second-order linear evolution problem is studied in a domain, a part of wich has an oscillating boundary. An homogeneous Neumann condition is given on the whole boundary of the domain. Moreover the behaviour of associated optimal control problem is analyzed.      

Global solvability for the Kirchhoff equations in exterior domains of dimension three

We consider the initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole space of dimension three, and show that these problems admit time-global solutions, provided the norms of the initial data in the usual Sobolev spaces of appropriate order are sufficiently small. We obtain uniform estimates of the L¹(R) norms with respect to time variable at each point in the domain, of solutions of initial (boundary) value problem for the linear wave equations. We then show that the estimates above yield the unique global solvability for the Kirchhoff equations.     

On the wave equation with a large rough potential

We prove an optimal dispersive L^∞ decay estimate for a three-dimensional wave equation perturbed with a large nonsmooth potential belonging to a particular Kato class. The proof is based on a spectral representation of the solution and suitable resolvent estimates for the perturbed operator.     

Asymptotically self-similar solutions of the damped wave equation

We consider the damped hyperbolic equation

(1)     

An identification problem for a time periodic nonlinear wave equation in non cylindrical domains

The existence of a time-periodic solution of a free boundary nonlinear wave equation in non cylindrical domains is established. The problem arises in the study of the identification of the coefficient of the wave equation and of the boundary of the region from the observed values of the solution in a fixed subregion.     

Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions

This article investigates optimal decay rates for solutions to a semilinear hyperbolic equation with localized interior damping and a source term. Both dissipation and the source are fully nonlinear   and the growth rate of the source map may include critical exponents (for Sobolev’s embedding H¹→L²$H^1\to L^2$). Besides continuity and monotonicity, no growth or regularity assumptions are imposed on the damping. We analyze the system in the presence of Neumann-type boundary conditions including the mixed cases: Dirichlet–Neumann–Robin.     

Finite-dimensional attractors for the quasi-linear strongly-damped wave equation

We present a new method of investigating the so-called quasi-linear strongly-damped wave equations