Vol 71, No 1 (2025): Nonlocal and nonlinear problems

The present paper is devoted to the study of the abstract nonlocal boundary value problem with integral type Samarskii–Ionkin conditions for the differential equation of elliptic type \[\hspace{-6em} -u''(t)+Au(t)=f(t)\quad (0\leq t\leq T),\quad u\left( 0\right) =\varphi,\quad u'\left( 0\right) =u'\left( T\right) +\int\limits_{0}^{T}\alpha \left( s\right) u(s)ds+\psi.\quad\] in an arbitrary Banach space \(E\) with the positive operator \(A\). The well-posedness of this problem in various Banach spaces is established. In applications, theorems on the well-posedness of several nonlocal boundary value problems for elliptic equations with integral type Samarskii–Ionkin conditions are proved.

The biharmonic operator plays a central role in a wide array of physical models, such as elasticity theory and the streamfunction formulation of the Navier–Stokes equations. Its spectral theory has been extensively studied. In particular the one-dimensional case (over an interval) serves as the basic model of a high order Sturm-Liouville problem. The need for corresponding numerical simulations has led to numerous works. This review focuses on a discrete biharmonic calculus. The primary object of this calculus is a high-order compact discrete biharmonic operator  (DBO).   The DBO is constructed in terms of the discrete Hermitian derivative.  The surprising strong connection between cubic spline functions (on an interval) and the DBO is recalled.   In particular the kernel of the inverse of the discrete operator is (up to scaling) equal to the  grid evaluation of the  kernel of \( \Big[\Big(\frac{d}{dx}\Big)^4\Big]^{-1}. \)  This fact entails the conclusion that the  eigenvalues of the DBO converge (at an “optimal” \(O(h^4)\) rate) to the continuous ones. Another consequence is the validity of a comparison principle. It is well known that there is no maximum principle for the fourth-order equation.  However, a positivity result is recalled, both  for the continuous and the discrete biharmonic equation, claiming that in both cases the kernels are order preserving.

Necessary and sufficient conditions for the direct representability of one system of ordinary differential equations in the form of Lagrange-Ostrogradsky equations are obtained and the corresponding variational principle (the Hamilton-Ostrogradsky action) is constructed.

In this paper, we study a boundary-value problem for a mathematical model describing the motion of aqueous polymer solutions. Based on the approximation-topological method, we investigate the existence of weak solutions of the problem under study. We consider the case of medium motion both in a bounded domain of two-dimensional or three-dimensional space and in an unbounded domain.

In this paper, we consider a second-order quasilinear elliptic equation with variable nonlinearity exponents and a locally summable right-hand side. The stability property is established and, as a consequence, the existence of a local renormalized solution of the Dirichlet problem in an arbitrary unbounded domain is proved.

Human immunodeficiency virus of type 1 (HIV) attacks the immune system and thereby weakens the defense against other infections and some types of cancer that the immune system of a healthy person can cope with. Despite the use of highly active antiretroviral therapy (HAART), there are no methods yet to completely eliminate HIV from the body of an infected person. However, due to the expansion of access to HIV prevention, diagnosis and treatment with HAART, HIV infection has moved into the category of controllable chronic diseases. Mathematical modeling methods are actively used to study the kinetic mechanisms of HIV pathogenesis and the development of personalized approaches to treatment based on combined immunotherapy. One of the central tasks of HIV infection modeling is to determine the individual parameters of the immune system response during the acute phase of HIV infection by solving inverse problems. To study the kinetics of the pathogenesis of HIV infection, a mathematical model of eight ordinary differential equations formulated by Bank et al. [5] was used. The system of equations of the model describes the change in the number of four subpopulations of CD4+ T cells and two types of CD8+ T cells. A feature of this model is the consideration of latently infected CD4+ T cells, which serve as the main reservoir of the viral population. The viral load on the human body is determined by the combination of populations of infectious and noninfectious viral particles. The inverse problem of parameter identification based on the data of the acute phase of HIV infection was studied. In particular, the identifiability of the parameters was studied and sensitivity analysis from the input data was performed. The inverse problem was reduced to a minimization problem using the evolutionary centers method.

For the Lavrent’ev—Bitsadze equation in a partially perforated model domain with a characteristic size of microinhomogeneities \(\varepsilon,\) we consider the problem with the third-kind boundary condition on the boundary of the cavities (the Fourier condition), which has a small parameter \(\varepsilon^\alpha\) as a multiplier in the coefficients, and the Dirichlet condition on the outer part of the boundary. For this problem, we construct a homogenized problem and prove the convergence of the solutions of the original problem to the solution of the homogenized problem in three cases. The subcritical case with \(\alpha>1\) is characterized by the fact that dissipation at the boundary of the cavities is negligibly small, in the critical case with \(\alpha=1\) a potential appears in the equation due to dissipation, and in the supercritical case with \(\alpha<1\) the dissipation plays the major role, it leads to degeneracy of the solution of the entire problem.