Initial-boundary Value Problem For Equations Research Articles

On initial-boundary value problem for nonlinear integro-differential equations with variable exponents of nonlinearity

Some nonlinear parabolic integro-differential equations with variable exponents of the nonlinearity are considered. The initial-boundary value problem for these equations is investigated and the existence theorem for the problem is proved.

On a class of initial-boundary value problems for equations of Korteweg-de Vries type

We consider a special class of initial-boundary value problems on the positive halfline x > 0 for the Korteweg-de Vries equation and its generalizations. For this class, we prove theorems on the nonexistence of global solutions for t > 0.

Remarks on equations of Benjamin–Bona–Mahony type

We investigate an initial-boundary value problem for equations of Benjamin–Bona–Mahony (BBM) type in two different physical situations. In the first, the mixed problem is considered on a cylinder domain Q of R n × R t . In the second one, the mixed problem is studied inside of an increasing noncylindrical domain Q ˆ of R n × R t . In both situations we show the existence of a unique nonlocal solution. In cylindrical case it is proved the existence of weak and strong solutions, regularity of strong solutions, and in noncylindrical case weak solutions. One of the goals of this paper is to show that the noncylindrical problem is well-posed by using the penalty method idealized by Lions [J.L. Lions, Une remarque sur les problèmes d'évolution non linéaires dans des domaines non cylindriques, Rev. Roumaine Math. Pures Appl. 9 (1964) 11–18].

On classical solvability of the first initial-boundary value problem for equations generated by curvatures

On classical solvability of the first initial-boundary value problem for equations generated by curvatures

Smooth global solutions of initial boundary-value problems for the equations of Oldroyd fluids and of their ε-approximations

The paper considers the solvability of the firsst and second initial boundary-value problems (with sticking or sliding conditions, respectively) for equations describing the motion of Oldroyd fluids and of their e-approximations in classes of functions of higher smoothness. Bibliography: 23 titles.

Initial boundary-value problems for equations of slightly compressible Jeffreys-Oldroyd fluids

The solvability of initial boundary value problems with adhesion and slippage boundary conditions for the equations of slightly compressible Jeffreys-Oldroyd fluids and penalized equations of Jeffreys-Oldroyd fluids in domains with smooth boundary is studied. Bibliography: 31 titles.

The solvability of the initial boundary-value problem for equations of motion of a viscous compressible barotropic liquid in the spacesW 2 l+1,l/2+1 (Q T )

A new method of estimating the solutions of the Navier-Stokes equations for a viscous compressible barotropic fluid in a bounded domain {Omega} {contained_in} R{sup 3} suggested, which makes it possible to investigate the problem for the whole scale of anisotropic spaces W{sub 2}{sup l+2, l/2+1} (Q{sub T}), Q{sub T} = {Omega} x (0,T), for arbitrary l > 1/2.

Asymptotic solution of initial boundary-value problems for hyperbolic systems

A theory is developed for the derivation of formal asymptotic solutions for initial boundary-value problems for equations of the form ^ ° S + l / ^ + 'LBu+Cu = f(‘>x ' A)' where A0, Av, B, and C are m xm matrix functions of t and X = (xv ..., xn), u X; A) is an ^-component column vector, and A is a large positive parameter. Our procedure is to consider a formal asymptotic solution of the form 1 00 u(t, X; A) ~ e £ (iA)-i z jt, X). Substitution of this formal solution into the equation yields, for the function s(t, X), a first order partial differential equation which can be solved by the method of characteristics. If the coefficient matrices satisfy certain conditions then we obtain, for the functions z X), linear systems of ordinary differential equations called transport equations along space-time curves called rays . They may be solved explicitly under suitable conditions. A proof is presented of the asymptotic nature of the formal solution when the coefficient matrices and initial data for u are appropriately chosen. The problem of reflexion and refraction at an interface is considered.