# Tedeev A. F.

### On the behavior of solutions of the Cauchy problem for a degenerate parabolic equation with source in the case where the initial function slowly vanishes

Martynenko A. V., Shramenko V. N., Tedeev A. F.

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1500-1515

We study the Cauchy problem for a degenerate parabolic equation with source and inhomogeneous density of the form $$u_t = \text{div}(\rho(x)u^{m-1}|Du|^{\lambda-1}Du) + u ^p $$ in the case where initial function decreases slowly to zero as $|x| \rightarrow \infty$. We establish conditions for the existence and nonexistence of a global-in-time solution, which substantially depend on the behavior of the initial data as $|x| \rightarrow \infty$. In the case of global solvability, we obtain an exact estimate of a solution for large times.

### Bilateral estimates for the support of a solution of the Cauchy problem for an anisotropic quasilinear degenerate equation

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1477–1486

We establish exact-order bilateral estimates for the size of the support of a solution of the Cauchy problem for a doubly nonlinear parabolic equation with anisotropic degeneration in the case where the initial data are finite and have finite mass.

### Initial-boundary-value problems for quasilinear degenerate hyperbolic equations with damping. Neumann problem

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 272–282

We study the behavior of the total mass of the solution of Neumann problem for a broad class of degenerate parabolic equations with damping in spaces with noncompact boundary. New critical indices for the investigated problem are determined.

### Theorems on the existence and nonexistence of solutions of the Cauchy problem for degenerate parabolic equations with nonlocal source

Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1443–1464

We consider the Cauchy problem for a doubly nonlinear degenerate parabolic equation with nonlocal source under the assumption that the initial function is integrable. We establish the existence and nonexistence of time-global solutions of the problem.

### The International Conference “Nonlinear Partial Differential Equations”

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 1007–1008

### Two-sided estimates of a solution of the Neumann problem as $t \rightarrow \infty$ for a second-order Quasilinear Parabolic Equation

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 989-998

We establish exact upper and lower bounds as $t \rightarrow \infty$ for the norm ‖*u*(·, *t*)‖_{ L } _{∞(Ω)} of a solution of the Neumann problem for a second-order quasilinear parabolic equation in the region *D*=Ω×{>0}, where Ω is a region with noncompact boundary.

### Method for symmetrization and estimation of solutions of the Neumann problem for the equation of a porous medium in domains with noncompact boundary for infinitely increasing time

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 147–157

We consider the initial boundary-value Neumann problem for the equation of a porous medium in a domain with noncompact boundary. By using a symmetrization method, we obtain exact*L* _{p}-estimates, 1≤*p*≤∞, for solutions as t→∞.

### Qualitative properties of solutions of the Neumann problem for a higher-order quasilinear parabolic equation

Ukr. Mat. Zh. - 1993. - 45, № 11. - pp. 1571–1579

The property of localization of perturbations is proved for a solution of an initial boundary-value Neumann problem in a region $D = \Omega x,\quad t > 0$, where $\Omega$ is a region in $R^n$ with a noncompact boundary.

### Symmetrization and initial boundary-value problems for certain classes of nonlinear second order parabolic equations

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 884–892

Initial boundary-value problems are considered for equations of porous medium and $p$-Laplace types. By using the Schwarz symmetrization method, the $L_p$-estimates for the solutions of the original problems are obtained in terms of the analogous estimates for the corresponding symmetric solutions.

### Multiplicative inequalities in domains with noncompact boundary

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 260–268

Exact embedding theorems of the multiplicative type are established for functions of Sobolev spaces defined in a domain Ω ⊂*R* ^{n},*n*⩾2, whose boundary is not compact. The main condition on the domain is of the isoperimetric type.

### Conferences on nonlinear problems of mathematical physics and problems with free boundaries

Bazalii B. V., Skrypnik I. V., Tedeev A. F.

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 295-297