This article considers a nonlocal problem in a cylindrical domain for the third-order mixed-composite type equation of the form
uttt −µ(x1) ∂
∂x1
�u −a(x, t)�u = f(x, t),
where x1µ(x1) > 0 for x1 ̸= 0, µ(0) = 0, x = (x1, x2, . . . , xn) ∈Rn.
Using the Galerkin method, it is proved that this nonlocal problem, under certain conditions on the coefficients and the right side of the equation, has a unique solution in Sobolev spaces. The proof is based on the Galerkin method with the choice of a special basis and a priori estimates. New theorems are also proved regarding the existence and uniqueness of the solution of the nonlocal problem, which allow expanding the range of solvable problems in the theory of boundary value problems for nonclassical equations of mathematical physics.

The work is devoted to the study of solvability of boundary value problems with nonlocal conditions of integral form for the differential equations
uxt −auxx + c(x, t)u = f(x, t),
in which x ∈�= (0, 1), t ∈(0, T), 0 < T < +∞, a ∈R, and c(x, t) and f(x, t) are
known functions. 
The peculiarity of these equations is that any of variables t and x can be considered a temporary variable, and in accordance with this, for these equations, for- mulations of boundary value problems with different carriers of boundary conditions can be proposed. For the problems under study, the work proves existence and uniqueness theorems for regular solutions; namely, solutions that have all derivatives generalized according to S. L. Sobolev and included in the equation.

We study the global behavior of the trajectories of the polynomial system˙
x = x −x2y + pxy2 + y3, ˙
y = y + py3, p ∈R.
Our study is related to the paper arXiv:2106/07516v2 [math.DS].

A nonlinear mathematical model describing equilibrium of a two-dimensional elastic body with two thin rigid inclusions is investigated. It is assumed that two rigid inclusions have one common connection point. Moreover, a connection between two inclusions at a given point is characterized by a positive damage parameter. Rectilinear inclusions are located at a given angle to each other in an initial state. Nonlinear Signorini conditions are imposed, which describe the contact with the obstacle, as well as a homogeneous Dirichlet condition is set on corresponding parts of the outer boundary of the body. An optimal control problem for the parameter that specifies the angle between inclusions is formulated. The quality functional is given by an arbitrary continuous functional defined on the Sobolev space. The solvability of the optimal control problem is proved. A continuous dependence of solutions on varying angle parameter between the inclusions is established.

We investigate the solvability of inverse problems of determination, along with the solution u(x, t) to the pseudoparabolic equation, of the unknown source function. Similar problems are considered when studying wave processes, filtration in porous media, and heat transfer processes. A theorem for the existence of a regular solution is proved. Inverse problems for pseudoparabolic equations with an unknown external influence depending on x and a final redefinition have not been considered previously.

We consider two-parameter families of planar vector fields with central symmetry. Assume that for zero values of the parameters, the field has a hyperbolic saddle at the origin O and two symmetric loops of the separatrices of this saddle. The saddle value – the trace of the matrix of the linear part of the field at the point O – is assumed to be zero. We describe the bifurcation diagram of a generic family – a partition of a neighborhood of the origin on the parameter plane into topological equivalence classes of dynamical systems defined by these vector fields in a fixed neighborhood U of the polycycle formed by loops of separatrices. In particular, for each element of the partition, the number and type of the field belonging to U are indicated.

The intention of this paper is to introduce and study certain new analytic spaces in the disk and to show that certain Blaschke type products belong to new large Nevanlinna type classes in the unit disk. We also provide parametric representation of such classes. These results extend and complement some previously known assertions of this type obtained earlier by other authors. Our arguments are based on certain new embeddings which relate the well-known Sp α area Nevanlinna spaces in the unit disk with our new large area Nevanlinna type spaces in the unit disk.

A numerical method based on the CABARET scheme for modeling unsteady flow over arbitrary topography in the shallow water approximation is developed. The method allows simulating a wide range of flow conditions, including transcritical. To model transcritical transitions, a hybrid approach is used based on solving the local Riemann problem, as is done in Godunov-type schemes. The presented numerical method has a well-balance condition–the fulfillment of the condition of hydrostatic equilibrium or the condition of a fluid at rest on an uneven bottom topography. A robust technique is used to simulate the movement of wet/dry fronts caused by flooding or recession. A number of physical processes are taken into account, such as bed friction and rain. Numerical results are compared with analytical solutions and data from the dam-break experiment.