Limit behaviour of solutions to boundary value problem with equivalued surface

In this paper, we discuss the limit behaviour of solutions to boundary value problem with equivalued surface with m inner holes and give a different proof from that of Li Ta-tsien et al. (1998).     

Limit Behaviour of Solutions to a Class of Equivalued Surface Boundary Value Problems for Parabolic Equations

In this paper, we discuss the limit behaviour of solutions for a class of equivalued surface boundary value problems for parabolic equations. When the equivalued surface boundary \overline{\Gamma}^\varepsilon_1 shrinks to a fixed point on boundary \Gamma_1, only homogeneous Neumann boundary conditions or Neumann boundary conditions with Dirac function appear on \Gamma_1.     

ASYMPTOTIC BEHAVIOR FOR A CLASS OF ELLIPTIC EQUIVALUED SURFACE BOUNDARY VALUE PROBLEM WITH DISCONTINUOUS INTERFACE CONDITIONS

Spontaneous potential well-logging is one of the important techniques in petroleum exploitation. A spontaneous potential satisfies an elliptic equivalued surface boundary value problem with diseontinuous interface conditlons. In practice, the measuring electrode is so small that we can simplify the corresponding equivalued surface to a point. In this paper, we give a positive answer to this approximation process: when the equivalued surface shrinks to a point, the solution of the original equivalued surface boundary value problem converges to the solution of the corresponding limit boundary value problem.     

Limit behaviour of solutions to equivalued surface boundary value problem for p‐Laplacian equations: II

In this paper (which is a continuation of Part‐I), we discuss the limit behaviour of solutions to boundary value problem with equivalued surface for p‐Laplacian equations in the case of 1<p?2?1/N when the equivalued surface boundary shrinks to a point in certain way. Copyright © 2001 John Wiley & Sons, Ltd.     

Insensitizing controls for the parabolic equation with equivalued surface boundary conditions

This paper is devoted to the study of the existence of insensitizing controls for the parabolic equation with equivalued surface boundary conditions. The insensitizing problem consists in finding a control function such that some energy functional of the equation is locally insensitive to a perturbation of the initial data. As usual, this problem can be reduced to a partially null controllability problem for a cascade system of two parabolic equations with equivalued surface boundary conditions. Compared the problems with usual boundary conditions, in the present case we need to derive a new global Carleman estimate, for which, in particular one needs to construct a new weight function to match the equivalued surface boundary conditions.     

Existence and uniqueness of bounded weak solution for non‐linear parabolic boundary value problem with equivalued surface

In this paper, we discuss the existence and uniqueness of bounded weak solution for non‐linear parabolic boundary value problem with equivalued surface and correct the mistake in Zhang Xu (Math. Meth. Appl. Sci. 1999; 22 : 259). The approach is based on L^∞ estimate of solution. Copyright © 2004 John Wiley & Sons, Ltd.     

Estimates of regularized solutions of convolution-type equations in the spaces of functions with derivatives of fractional orders

The paper deals with convolution type integral equations of the first kind. For exact solutions it is typical that the solutions are discontinuous or unbounded. The aim of the paper is to construct approximate solutions which may also be discontinuous or unbounded. A priori information about the solution and estimates for the approximate solutions, as well as for the differences between the approximate solution and the exact one, are given in terms of Nikol'skii-Besov spaces with differentiation of fractional orders. Bibliography: 8 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 18, pp. 170–194, 1995. This work was supported by the Russian Fundamental Research Foundation (Project No 93-01-00255).     

Non-Newton Filtration Equation with Nonconstant Medium Void and Critical Sobolev Exponent

In this paper we consider the existence and asymptotic estimates of global solutions and finite time blowup of local solutions of quasilinear parabolic equation with critical Sobolev exponent and with lower energy initial value; we also describe the asymptotic behavior of global solutions with high energy initial value.     

Limit behaviour of optimal control problems governed by elliptic boundary value problems with equivalued surface

In this paper, we discuss the limit behaviour of optimal control problems governed by elliptic boundary value problems with equivalued surface when the equivalued surface boundary shrinks to a fixed point on the outer boundary of a bounded domain.     

Junctions of singularly degenerating domains with different limit dimensions. II

This paper is aimed at finding asymptotic formulas for solutions to the mixed boundary problem for the Poisson equation in a domain obtained by joining singularly degenerating domains. In this paper, which is the second part of the work (the first part was published in No. 18), the main attention is given to three-dimensional problems in which a thin plate or a periodic family of thin rods is joined to a massive body (the distances between the rods are comparable with the diameters of their cross-sections). The distinctive feature of such problems is that an integral equation arises as one of the limit problems. Bibliography: 48 titles. Translated from Trudy Seminara imeni l. G. Petrovskogo, No. 20, pp. 155–195, 1997.