For a nonlinear third-order functional differential equation, we consider a boundary value problem with an integral boundary condition at one of the ends of the segment under study. Using special topological tools, we establish sufficient conditions for the existence and uniqueness of a positive solution to the problem under consideration and construct a corresponding example.

The paper considers a nonlinear equation of motion of a pendulum the suspension point of which performs high-frequency harmonic oscillations. The problem of robust stability for the upper equilibrium position is studied. Conditions for perturbations of the equation coefficients are established, under which the corresponding stationary solution is exponentially stable. Estimates for attraction sets are given and estimates for stabilization rates of solutions at infinity are obtained.

The article is devoted to the study of behavior of solutions to one initial value problem for a delay equation at infinity. Estimates for the solutions are established which give exponential decay of the solutions as t → ∞ under some conditions.

We study the problems similar to the known Samarskii–Ionkin’s problem for the composite type differential equation with degenerating operator of heat conduction in higher part. In this work we prove existence and uniqueness of regular solutions for such problems.

The solvability of boundary value problems for fourth-order parabolic equations with changing time direction in Hölder spaces is established, related to the application of the theory of singular integral equations, as well as systems of these equations. It is shown that the Hölder classes of solutions to the Gevrey-type problem in the case of weighted gluing functions depend both on the non-integer Hölder exponent and on the weight coefficients of the gluing conditions. Singular integral operators in Hölder spaces with piecewise continuous matrix coefficients are considered. In contrast to the classical case, these operators, in addition to the singular Cauchy operator, may contain non-compact integral operators with a kernel homogeneous of degree −1 with respect to the distances to the end points of the integration contour.

We consider the third boundary value problem for a p-Laplace equation with a low-order term that does not satisfy the Bernstein–Nagumo condition. The solvability of the problem in the class of radially symmetric solutions is investigated. A class of gradient nonlinearities is defined, for which the existence of a weak Sobolev radially symmetric solution with a Hölder continuous derivative with exponent 1 p−1 is proven. It is shown that nonlinearity in the gradient can be arbitrary, provided that the low order term containing the gradient is Lipschitz continuous in the spatial variable and strictly monotone in the variable u. The solution to the original problem is approximated by classical solutions to the corresponding regularized problem. The a priori estimates obtained for the regularized problem do not depend on the regularization parameter, which allows us to obtain a solution to the original problem of the specified smoothness by passing to the limit.

We study the problem of radial fingering in immiscible liquid/liquid flow in a Hele-Shaw cell under injection or suction of liquid of less viscosity. The addition of new effects of viscosity and surface tension at the liquid/liquid interface brings our theory into a much better agreement with experiments than other theories. In this paper the classical solvability of the original Hele-Shaw problem (a nonlinear problem with a free boundary for elliptic equations) is established by means of its parabolic regularization with a small parameter ε (> 0) in the time-derivative term and the nonhomogeneous term (the parabolic regularized Hele-Shaw problem) and by vanishing along some subsequence of {ε > 0}. The similar result for a one-phase problem has been already studied in “H. Tani, Classical solvability of the radial viscous fingering problem in a Hele-Shaw cell with surface tension, Sib. J. Pure Appl. Math., 16, 79–92 (2016);” “A. Tani and H. Tani, On the uniqueness of the classical solution of the radial viscous fingering problem in a Hele-Shaw cell with surface tension, J. Appl. Mech. Tech. Phys., 65, No. 5, 178–191 (2024).”

The non-homogeneous Dirichlet problem for degenerate quasilinear parabolic equations is considered. We prove the existence of a solution u such that ut belongs to L∞. The L∞ estimate of ut is obtained by introducing a new time variable.