On a nonlocal problem for a parabolic equation

We study a nonlocal boundary-value problem for a parabolic equation in a two-dimensional domain, establish ana priori estimate in the energy norm, prove the existence and uniqueness of a generalized solution from the classW ₂ ^1,0 (Q _T ), and construct a difference scheme for the second-order approximation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 6, pp. 790–800, June, 1995.     

On a mixed problem with integral conditions for one-dimensional parabolic equation

We consider an initial-boundary value problem for a one-dimensional parabolic equation with nonlocal boundary conditions. These nonlocal conditions are given in terms of integrals. Based on solution of the Dirichlet problem for the parabolic equation, we constructively establish the well-posedness for the nonlocal problem.     

On a linear inverse problem for a second-order parabolic equation

The question is studied of existence and uniqueness of a solution to the inverse problem of finding the right-hand side of a special kind for a second-order parabolic equation.     

On a certain nonlocal problem for a hyperbolic equation

The author proves the existence and uniqueness of a generalized solution of a nonlocal problem with an integral condition for a hyperbolic equation with n spatial variables. This work is a continuation of the studies started in [3–5], where the solvability problem of nonlocal problems with an integral condition was studied for hyperbolic equations on the plane. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.     

Solution of a boundary control problem for a nonlinear parabolic equation

An optimal boundary control problem for a nonlinear parabolic equation of second order is considered. An existence theorem is proved and necessary optimality conditions are obtained in the form of point and integral maximum principles.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 30–37, 1986.     

Inverse problem for a parabolic equation

Mathematical Notes -     

On a coefficient inverse problem for a parabolic equation in a domain with free boundary

The present paper deals with the inverse problem of determination of the coefficient of the first derivative of the unknown function with respect to a spatial variable for a one-dimensional parabolic equation in the domain whose boundary is determined by two unknown functions. The conditions of local existence and uniqueness of a solution to the inverse problem are established.     

Inverse problem for a parabolic equation with strong power degeneration

We consider the inverse problem of determining the time-dependent coefficient of the leading derivative in a full parabolic equation under the assumption that this coefficient is equal to zero at the initial moment of time. We establish conditions for the existence and uniqueness of a classical solution of the problem under consideration. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1487–1500, November, 2006.     

Cauchy problem for a degenerate parabolic equation with non-divergence form

In this article, it is shown that there exists a unique viscosity solution of the Cauchy problem for a degenerate parabolic equation with non-divergence form.     

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.     

On a singularly perturbed mixed problem for a linear parabolic equation with nonlinear boundary conditions

A mixed problem for a linear parabolic equation with a small parameter multiplying the time derivative and with nonlinear boundary conditions is solved. Such boundary conditions arise in some heat and mass transfer problems, for example, in cooling thin spherical gas-filled shells or in the case of a gas filling such shells with gas-permeable walls.     

An inverse problem for a semilinear parabolic equation

Summary In this paper we are concerned with the study of the stability of an unknown non-linear term in a parabolic equation in dependence on over specified Cauchy-Dirichlet data prescribed on the parabolic boundary of the open set under consideration. Since, in general, the dependence of the nonlinear term upon the data is not stable with respect to L -metrics, we show how a Hölder continuity may be restored under mild restrictions for the set of admissible solutions.Lavoro eseguito nell'amMto del G.N.A.F.A. del C.N.R.     

An inverse problem for a quasilinear parabolic equation

Sunto Si considera un problema parabolico sovraderminato in una sola variabile spaziale per l'operatore quasilineare definito da · a₃ o u. Tale operatore contiene un termine incognito a(k {0, 1, 2, 3})del quale si studia la dipendenza dalle condizioni iniziali e alla frontiera. Si determinano due classi, rispettivamente di dati e di soluzioni ammissibili, ed una coppia di metriche rispetto alle quali l'applicazione dati(u, a_k)è lipschitziana. Si mostra infine che tale applicazione si mantiene hölderiana quando le metriche ammesse per i dati sono soltanto del tipo L.

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Lavoro eseguito con contributo del Ministero della Pubblica Istruzione e nell'ambito del Gruppo G.N.A.F.A. del C.N.R.     

Nonlocal Neumann problem for a degenerate parabolic equation

In the spaces of classical functions with power weight, we prove the existence and uniqueness of a solution of the nonlocal Neumann problem for nonuniformly parabolic equations without restrictions on the power order of coefficient degeneration. We find an estimate of the solution of this problem in the spaces considered. Chernovtsy State University, Chernovtsy. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1232–1243, September, 1999.