Volume 68, Nº 4 (2022)

We study the existence of solutions to the problem

\[\label{eng_A1} \begin{array}{rl} -\Delta u+u^p-M|\nabla u|^q=0 & \text{in }\;\Omega,\\ u=\mu & \text{on }\;\partial\Omega \end{array}\]

in a bounded domain \(\Omega\), where \(p>1\), \(1, \(M>0\), \(\mu\) is a nonnegative Radon measure in \(\partial\Omega\), and the associated problem with a boundary isolated singularity at \(a\in\partial\Omega,\)

\[\label{eng_A2} \begin{array}{rl} -\Delta u+u^p-M|\nabla u|^q=0 & \text{in }\;\Omega,\\ u=0 & \text{on }\;\partial\Omega\setminus\{a\}. \end{array}\]

The difficulty lies in the opposition between the two nonlinear terms which are not on the same nature. Existence of solutions to [eng_A1] is obtained under a capacitary condition \[\mu(K)\leq c\min\left\{cap^{\partial\Omega}_{\frac{2}{p},p'},cap^{\partial\Omega}_{\frac{2-q}{q},q'}\right\}\quad\text{for all compacts }K\subset\partial\Omega.\] Problem [eng_A2] depends on several critical exponents on \(p\) and \(q\) as well as the position of \(q\) with respect to \(\dfrac{2p}{p+1}\).

The work is a brief review of the results obtained in nonautonomous dynamics based on the concept of uniform equivalence of nonautonomous systems. This approach to the study of nonautonomous systems was proposed in [10] and further developed in the works of the second author, and recently - jointly by both authors. Such an approach seems to be fruitful and promising, since it allows one to develop a nonautonomous analogue of the theory of dynamical systems for the indicated classes of systems and give a classi cation of some natural classes of nonautonomous systems using combinatorial type invariants. We show this for classes of nonautonomous gradient-like vector elds on closed manifolds of dimensions one, two, and three. In the latter case, a new equivalence invariant appears, the wild embedding type for stable and unstable manifolds [14,17], as shown in a recent paper by the authors [5].

In this paper, a model of deformations of Stieltjes strings system located along a geometric star graph with a nonlinear condition at the node is studied. This kind of condition arises due to the presence of a limiter for the movement of strings in the node under the in uence of an external load. In the present paper, the necessary and sufficient conditions for the extremum of the energy functional are established; existence and uniqueness theorems for the solution are proved; the critical loads at which the strings come into contact with the limiter are analyzed; the dependence of the solution on the limiter length is studied.

In this paper, we study the homogenization of a parabolic equation given in a domain perforated by ''tiny'' balls. On the boundary of these perforations, a unilateral dynamic boundary constraints are specified. We address the so-called ''critical'' case that is characterized by a relation between the coefficient in the boundary condition, the period of the structure and the size of the holes. In this case, the homogenized equation contains a nonlocal ''strange'' term. This term is obtained as a solution of the variational problem involving ordinary differential operator.

We describe the semiclassical asymptotic behavior of the solution of the Cauchy problem for the Schrödinger equation with a delta potential localized on a surface of codimension 1. The Schrödinger operator with a delta potential is defined using the theory of extensions and is given by the boundary conditions on this surface. The initial data are selected as a narrow peak, which is a Gaussian packet localized in a small neighborhood of the point. To construct the asymptotics, we use the Maslov complex germ method. We describe the re ection of the complex germ from the carrier of the delta potential.