Acta Applicandae Mathematicae最新文献

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether.

Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether.

This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid.

We investigate a new class of topological travelling-wave solutions for a macroscopic swarmalator model involving force non-reciprocity. Swarmalators are systems of self-propelled particles endowed with a phase variable. The particles are subject to coupled swarming and synchronization. In previous work, the swarmalator under study was introduced, the macroscopic model was derived and doubly periodic travelling-wave solutions were exhibited. Here, we focus on the macroscopic model and investigate new classes of two-dimensional travelling-wave solutions. These solutions are confined in a strip or in an annulus. In the case of the strip, they are periodic along the strip direction. Both of them have non-trivial topology as their phases increase by a multiple of (2 pi ) from one period (in the case of the strip) or one revolution (in the case of the annulus) to the next. Existence and qualitative behavior of these solutions are investigated.

We study the Lane-Emden system involving the logarithmic Laplacian:

where (p,q>1), (ngeq 2) and (mathcal{L}_{Delta }) denotes the logarithmic Laplacian arising as a formal derivative (partial _{s}|_{s=0}(-Delta )^{s}) of the fractional Laplacian ((-Delta )^{s}) at (s=0). By using a direct method of moving planes for the logarithmic Laplacian, we obtain the symmetry and monotonicity of the positive solutions to the Lane-Emden system. We also establish some key ingredients needed in order to apply the method of moving planes such as the maximum principle for anti-symmetric functions, the narrow region principle, and decay at infinity. Further, we discuss such results for a generalized system of the Lane-Emden type involving the logarithmic Laplacian.

In this paper, we study the controllability of a nonlinear system of coupled second- and fourth-order parabolic equations. This system can be regarded as a simplification of the well-known stabilized Kuramoto-Sivashinsky system. Using only one control applied on the boundary of the second-order equation, we prove that the local-null controllability of the system holds if the square root of the diffusion coefficient of the second-order equation is an irrational number with finite Liouville-Roth constant.

In this paper, we investigate a Cucker-Smale flocking model with varying time delay. We establish exponential asymptotic flocking without requiring smallness assumptions on the time delay size and the monotonicity of the influence function.

We introduce a new norm on (mathcal{C}_{2}times mathcal{C}_{2}), where (mathcal{C}_{2}) is the Hilbert-Schmidt class. We study basic properties of this norm and prove inequalities involving it. As an application of the present study, we deduce a chain of new bounds for the Hilbert-Schmidt numerical radii of (2times 2) operator matrices. Connection with the classical Hilbert-Schmidt numerical radius of a single operator are also provided. Moreover, we refine some related existing bounds, too.

In this paper, we give a direct proof of (t^{-1}) (optimal) linear decay rate for Euler-Coriolis equations in (L^{infty }) space-time. Our proof is based on a proper decomposition of the explicit solution and (L^{infty }) estimate for the kernels, which captures the dispersive mechanism.

We investigate a two-dimensional transmission model consisting of a wave equation and a Kirchhoff plate equation with dynamical boundary controls under geometric conditions. The two equations are coupled through transmission conditions along a steady interface between the domains in which the wave and plate equations evolve, respectively. Our primary concern is the stability analysis of the system, which has not appeared in the literature. For this aim, using a unique continuation theorem, the strong stability of the system is proved without any geometric condition and in the absence of compactness of the resolvent. Then, we show that our system lacks exponential (uniform) stability. However, we establish a polynomial energy decay estimate of type (1/t) for smooth initial data using the frequency domain approach from semigroup theory, which combines a contradiction argument with the multiplier technique. This method leads to certain geometrical conditions concerning the wave’s and the plate’s domains.

We complete previous results about the incompressible limit of both the (n)-dimensional ((ngeq 3)) compressible Patlak-Keller-Segel (PKS) model and its stationary state. As in previous works, in this limit, we derive the weak form of a geometric free boundary problem of Hele-Shaw type, also called congested flow. In particular, we are able to take into account the unsaturated zone, and establish the complementarity relation which describes the limit pressure by a degenerate elliptic equation. Not only our analysis uses a completely different framework than previous approaches, but we also establish two novel uniform estimates in (L^{3}) of the pressure gradient and in (L^{1}) for the time derivative of the pressure. We also prove regularity à la Aronson-Bénilan. Furthermore, for the Hele-Shaw problem, we prove the uniqueness of solutions, meaning that the incompressible limit of the PKS model is unique. In addition, we establish the corresponding incompressible limit of the stationary state for the PKS model with a given mass, where, different from the case of PKS model, we obtain the uniform bound of pressure and the uniformly bounded support of density.

A vector-borne disease model with spatial diffusion with time delays and a general incidence function is studied. We derived conditions under which the system exhibits threshold behavior. The stability of the disease-free equilibrium and the endemic equilibrium are analyzed by using the linearization method and constructing appropriate Lyapunov functionals. It is shown that the given conditions are satisfied by at least two common forms of the incidence function.