# Shkhanukov-Lafishev M. Kh.

### Finite-Time Stabilization in Problems with Free Boundary for Nonlinear Equations in Media with Fractal Geometry

Berezovsky A. A., Mitropolskiy Yu. A., Shkhanukov-Lafishev M. Kh.

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 997–1001

By using the method of a priori estimates, we establish differential inequalities for energetic norms in $W^l_{2,r}$ of solutions of problems with a free bound in media with the fractal geometry for one-dimensional evolutionary equation. On the basis of these inequalities, we obtain estimates for the stabilization time $T$.

### Stabilization for a finite time in problems with free boundary for some classes of nonlinear second-order equations

Berezansky Yu. M., Mitropolskiy Yu. A., Shkhanukov-Lafishev M. Kh.

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 214–223

We obtain estimates for the time of stabilization of solutions of problems with free boundary for one-dimensional quasilinear parabolic equations.

### Nonlinear nonlocal problems for a parabolic equation in a two-dimensional domain

Berezovsky A. A., Mitropolskiy Yu. A., Shkhanukov-Lafishev M. Kh.

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 244–254

We establish the convergence of the Rothe method for a parabolic equation with nonlocal boundary conditions and obtain an *a priori* estimate for the constructed difference scheme in the grid norm on a ball. We prove that the suggested iterative process for the solution of the posed problem converges in the small.

### Space-time localization in problems with free boundaries for a nonlinear second-order equation

Berezovsky A. A., Mitropolskiy Yu. A., Shkhanukov-Lafishev M. Kh.

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 202-211

For thermal and diffusion processes in active media described by nonlinear evolution equations, we study the phenomena of space localization and stabilization for finite time.

### On a nonlocal problem for a parabolic equation

Berezansky Yu. M., Mitropolskiy Yu. A., Shkhanukov-Lafishev M. Kh.

Ukr. Mat. Zh. - 1995. - 47, № 6. - pp. 790–800

We study a nonlocal boundary-value problem for a parabolic equation in a two-dimensional domain, establish an*a priori* estimate in the energy norm, prove the existence and uniqueness of a generalized solution from the class*W* _{2} ^{1,0} (*Q* _{ T }), and construct a difference scheme for the second-order approximation.

### Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems

Berezovsky A. A., Kerefov A. A., Shkhanukov-Lafishev M. Kh.

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1289–1398

Boundary-value problems for the heat conduction equation are considered in the case where the boundary conditions contain a fractional derivative. Problems of this type arise when the heat processes are simulated by a nonstationary heat flow by using the one-dimensional thermal model of a two-layer system (coating — base). It is proved that the problem under consideration is correct. A one-parameter family of difference schemes is constructed; it is shown that these schemes are stable and convergent in the uniform metric.