Cauchy problem for a parabolic higher-order equation with pulse action

We prove the existence and establish some estimates of a solution of the Cauchy problem for a parabolic pulse-action equation of higher order in t. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 17–24, January–March, 2008.     

Inverse problem for the strongly degenerate heat equation in a domain with free boundary

In a domain with free boundary, we establish conditions for the existence and uniqueness of a solution of the inverse problem of finding the time-dependent coefficient of heat conductivity. We study the case of strong degeneration where the unknown coefficient tends to zero as t → +0 as a power function t ^β , where β ≥ 1. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 28–43, January, 2009.     

Existence and stability of a summable generalized solution to a mathematical model of a catalytic fluidized bed process

For a system of first-order partial differential equations describing a catalytic process in a fluidized bed, we consider a mixed problem in the half-strip 0 ≤ x ≤ h, t ≥ 0. We prove the existence and uniqueness of a bounded summable generalized solution and study its stability. We prove the stabilization as t → ∞ of the values of some physically meaningful functionals of the solution.     

On certain nonlinear pseudoparabolic variational inequalities without initial conditions

We consider a nonlinear pseudoparabolic variational inequality in a tube domain semibounded in variablet. Under certain conditions imposed on coefficients of the inequality, we prove the theorems of existence and uniqueness of a solution without any restriction on its behavior ast→−∞. Lvov University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 328–337, March, 1999.     

Integral representation of a solution of the Cauchy problem for a degenerating hyperbolic equation with retarded argument

By using the method of integral equations, we prove the existence and uniqueness of a regular solution of the Cauchy problem for a degenerating hyperbolic equation with retarded argument. Orel Pedagogic Institute, Russia. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1332–1336, October, 1997.     

Existence, uniqueness, and dependence on a parameter of solutions of differential-functional equations with ordinary and partial derivatives

For a system of quasilinear hyperbolic equations with a system of differential equations with lag, we prove theorems on the existence and uniqueness of a solution of the Cauchy problem and its continuous dependence on the initial conditions. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 715–719, May, 1997.     

Inverse problem for a weakly degenerate parabolic equation in a domain with free boundary

In a domain with free boundary, we consider an inverse problem of determining the time-dependent leading coefficient of a parabolic equation, which tends to zero as t → 0 like a certain given function. Conditions for the existence and uniqueness of a classical solution in the case of weak degeneration are established.     

Stability of solutions of an equation arising in hydromechanics

We study the strong stability of the equation describing small oscillations of an elastic pipe conveying an ideal fluid. We prove the existence and uniqueness theorem for the generalized solution and find sufficient conditions for strong stability in the case of a pulsing fluid. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 15–24, January, 2000.     

On a smooth solution of a nonlinear periodic boundary-value problem

We establish conditions for the existence of a smooth solution of a quasilinear hyperbolic equationu _tt - u_xx = ƒ(x, t, u, u, u _x),u (0,t) = u (π,t) = 0,u (x, t+ T) = u (x, t), (x, t) ∈ [0, π] ×R, and prove a theorem on the existence and uniqueness of a solution. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1574–1576, November, 1999.     

On a class of stochastic partial differential equations related to turbulent transport

Summary. We consider the Cauchy problem for the mass density ρ of particles which diffuse in an incompressible fluid. The dynamical behaviour of ρ is modeled by a linear, uniformly parabolic differential equation containing a stochastic vector field. This vector field is interpreted as the velocity field of the fluid in a state of turbulence. Combining a contraction method with techniques from white noise analysis we prove an existence and uniqueness result for the solution ρ∈C ^1,2([0,T]×ℝ ^d ,(S)^*), which is a generalized random field. For a subclass of Cauchy problems we show that ρ actually is a classical random field, i.e. ρ(t,x) is an L ²-random variable for all time and space parameters (t,x)∈[0,T]×ℝ ^d . Received: 27 March 1995 / In revised form: 15 May 1997