# Shishkov A. E.

### 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.

### Phragmen-Lindelof Principle for Some Quasilinear Evolution Equations of the Second Order

Shishkov A. E., Sleptsova I. P.

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 239–249

We consider the equation $u_{tt} + A (u_t) + B(u) = 0$, where $A$ and $B$ are quasilinear operators with respect to the variable x of the second order and the fourth order, respectively. In a cylindrical domain unbounded with respect to the space variables, we obtain estimates that characterize the minimum growth of any nonzero solution of the mixed problem at infinity.

### 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.

### Propagation of Perturbations in Quasilinear Multidimensional Parabolic Equations with Convective Term

Sapronov D. A., Shishkov A. E.

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 953-969

We establish estimates for the initial evolution of the supports of solutions of a broad class of quasilinear parabolic equations of arbitrary order that have the structure of the equation of strong nonlinear convective diffusion.

### Solvability of boundary-value problems for quasilinear elliptic and parabolic equations in unbounded domains in classes of functions growing at infinity

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 277–289

For divergent elliptic equations with the natural energetic space*W* _{p} ^{m} (Ω),*m*≥1,*p*>2, we prove that the Dirichlet problem is solvable in a broad class of domains with noncompact boundaries if the growth of the right-hand side of the equation is determined by the corresponding theorem of Phragmén-Lindelöf type. For the corresponding parabolic equation, we prove that the Cauchy problem is solvable for the limiting growth of the initial function % MathType!MTEF!2!1!+- $$u_0 (x) \in L_{2.loc} (R^n ): \int\limits_{|x|< \tau } {u_0^2 dx \leqslant c\tau ^{n + 2mp/(p - 2)} \forall \tau< \infty } $$

### Uniqueness of solutions of mixed problems and a Cauchy problem for parabolic equations of high order with unbounded coefficients

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 149–155

New uniqueness classes of generalized solutions (generalized Tacklind classes) of initial-boundary-value problems are found for linear and quasilinear divergent parabolic equations of high order with coefficients increasing at infinity.

### Behavior of generalized solutions of mixed problems for quasilinear parabolic equations of high order in unbounded domains

Ukr. Mat. Zh. - 1987. - 39, № 5. - pp. 624–631

### Existence of generalized solutions, increasing at infinity, of boundary-value problems for linear and quasilinear parabolic equations

Ukr. Mat. Zh. - 1985. - 37, № 4. - pp. 473–481