# Taranets R. M.

### Finite speed of propagation for the thin-film equation in the spherical geometry

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 840-851

UDC 517.953

We show that a double degenerate thin-film equation obtained in modeling of a flow of viscous coating on the spherical surface has a finite speed of propagation for nonnegative strong solutions and, hence, there exists an interface or a free boundary separating the regions, where the solution $u>0$ and $u=0.$
Using local entropy estimates, we also obtain the upper bound for the rate of the interface propagation.

### Singular Cauchy problem for the equation of flow of thin viscous films with nonlinear convection

Shishkov A. E., Taranets R. M.

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 250–271

For multidimensional equations of flow of thin capillary films with nonlinear diffusion and convection, we prove the existence of a strong nonnegative generalized solution of the Cauchy problem with initial function in the form of a nonnegative Radon measure with compact support. We determine the exact upper estimate (global in time) for the rate of propagation of the support of this solution. The cases where the degeneracy of the equation corresponds to the conditions of “strong” and “weak” slip are analyzed separately. In particular, in the case of “ weak” slip, we establish the exact estimate of decrease in the $L^2$-norm of the gradient of solution. It is well known that this estimate is not true for the initial functions with noncompact supports.

### Effect of Time Delay of Support Propagation in Equations of Thin Films

Shishkov A. E., Taranets R. M.

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 935-952

We prove the existence of the effect of time delay of propagation of the support of “strong” solutions of the Cauchy problem for an equation of thin films and establish exact conditions on the behavior of an initial function near the free boundary that guarantee the appearance of this effect.