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

The question of the correct formulation of a Darboux type non-characteristic spatial problem for the wave equation is investigated. The correct solvability of the problem is proved in the Sobolev space for surfaces of the temporal type on which Darboux type boundary conditions are given.     

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 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.     

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 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.     

N-dimensional characteristic problem for the Mangeron equation of non-integer order

In the paper we consider a n-dimensional characteristic problem for a certain partial differential equation of non-integer order.We prove the existence and uniqueness of a solution of the problem in the spaces of integrable and continious functions, respectively. Morever, we give sufficient conditions under which the set of solutions is not empty and relatively compact in the space of integrable functions     

The Riemann problem for the Leray-Burgers equation

For Riemann data consisting of a single decreasing jump, we find that the Leray regularization captures the correct shock solution of the inviscid Burgers equation. However, for Riemann data consisting of a single increasing jump, the Leray regularization captures an unphysical shock. This behavior can be remedied by considering the behavior of the Leray regularization with initial data consisting of an arbitrary mollification of the Riemann data. As we show, for this case, the Leray regularization captures the correct rarefaction solution of the inviscid Burgers equation. Additionally, we prove the existence and uniqueness of solutions of the Leray-regularized equation for a large class of discontinuous initial data. All of our results make extensive use of a reformulation of the Leray-regularized equation in the Lagrangian reference frame. The results indicate that the regularization works by bending the characteristics of the inviscid Burgers equation and thereby preventing their finite-time crossing.     

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.     

The Cauchy problem for the 1-D Dirac–Klein–Gordon equation

The Cauchy problem for the Dirac–Klein–Gordon equation are discussed in one space dimension. Time local and global existence for solutions with rough data, especially the solutions for Klein–Gordon equation in the critical and super critical Sobolev norm of [4] are considered. The solutions with general propagation speeds are dealt with.      

Well-posedness and blow-up phenomena for a higher order shallow water equation

This paper deals with the Cauchy problem for a higher order shallow water equation yt+auxy+buyx=0, where and k=2. The local well-posedness of solutions for the Cauchy problem in Sobolev space Hs(R) with s?7/2 is obtained. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and conditions that lead to the development of singularities in finite time for the solutions are also acquired. Finally, the weak solution for the equation is considered.     

Asymptotic behavior and blow-up of solutions to a nonlinear evolution equation of fourth order

In this paper, the asymptotic behavior of the solution to the initial–boundary value problem for a nonlinear evolution equation of fourth order

equation(1)

_utt−_a1uxx−_a2uxxt−_a3uxxtt=φ_(ux)x$u_{tt}-a_1{u}_{xx}-a_2{u}_{xxt}-a_3{u}_{xxtt}=\phi {\left(u_x\right)}_x$

is studied. The sufficient conditions for blow-up of the solutions to the initial–boundary value problems for Eq. (1) are given.     

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.     

On the Cauchy problem for the generalized Camassa–Holm equation

We establish the local well-posedness for the generalized Camassa–Holm equation. We also prove that the equation has smooth solutions that blow up in finite time.     

Global attractor for a nonlinear wave equation arising in elastic waveguide model

The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in an elastic waveguide model _utt−Δu−Δ_utt+Δ²u−Δ_ut−Δg(u)=f(x)$u_{tt}-\Delta u-\Delta u_{tt}+{\Delta }^{2}u-\Delta u_t-\Delta g\left(u\right)=f\left(x\right)$. It proves that when the space dimension N≤5$N\le 5$, under rather mild conditions the dynamical system associated with the above-mentioned IBVP possesses a global attractor which is connected and has finite fractal and Hausdorff dimension.     

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.     

The semilinear second-order hyperbolic equation with data strongly singular at one point

In this paper we study the initial value problem for the scalar semilinear strictly hyperbolic equation in multidimensional space with data strongly singular at one point. Under the assumption of the initial data being conormal with respect to one point and bounded or regular with a certain low degree, the existence of the solution to this problem is obtained; meanwhile, it is proved that the singularity of the solution will spread on the forward characteristic cone of the hyperbolic operator issuing from this point, and the solution is bounded and conormal with respect to this cone.     

Theorem about existence and uniqueness of continuous solution of nonlocal problem for nonlinear hyperbolic equation

The aim of the paper is to give a theorem about the existence and uniqueness of the continuous solution of a non-linear differential hyperbolic problem with a nonlocal condition in a bounded domain. The Banach theorem about the fixed point is used to prove the existence and uniqueness of the problem considered. The results obtained in this paper can be applied in the theory of elasticity with better effect than the analogous known result with the classical initial condition.     

On boundary-value problem for hyperbolic-type equation of the third order

We formulate some boundary-value problems for a linear third-order equation with hyperbolic operator in the main part and study the unique solvability. Under certain conditions to given functions, using the Riemann method, we obtain an integral representation of solutions. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 591–603, October–December, 2007.