arXiv subjects
Publications and source records attributed to Robert Stańczy.
We consider generalized Kolmogorov system. We prove the existence of heteroclinic trajectory. We apply the results to astrophysical models for self-gravitating particles and predator-prey systems.
We consider fairly general class of dynamical systems under the assumptions guaranteeing the existence of Lyapunov function around some nontrivial stationary point. Moreover, the existence of heteroclinic trajectory is proved motivated by integrated densities approach to some astrophysical models of self-gravitating particles both in relativistic and non--relativistic frameworks. Finally, with the aid of geometric and topological reasoning we find the upper bounds for this trajectory yielding the critical mass--radius theorem for the astrophysical model.
The existence of solutions to Tolman-Openheimer-Volkoff equation with linear equation of state modeling relativistic cloud of interacting particles is proved for mass parameter below certain threshold. For the intermediate values of mass parameters multiplicity result holds. For the small mass parameter the uniqueness is guaranteed. Moreover, there is a threshold value of the mass, which can not be exceeded. It is achieved by considering the related dynamical system in the rescaled mass-density variables which is governed by the global Lyapunov function with sink. Some preliminary extensions to nonlinear equation of state are discussed with numerical density profiles.
The existence of multiple radial solutions to the elliptic equation modeling fermionic cloud of interacting particles is proved for the limiting Planck constant and intermediate values of mass parameters. It is achieved by considering the related nonautonomous dynamical system for which the passage to the limit can be established due to the continuity of the solutions with respect to the parameter going to zero.
In this paper, we consider a nonlinear second order equation modelling rocket motion in the gravitational field obstructed by the drag force. The proofs of the main results are based on topological fixed point approach.
A novel mathematical rigorous method to obtain the vapor-liquid equilibrium curves near the critical point has been proposed covering the cases of well-known and commonly used equations of state for real gases - Van der Waals and Dieterici, which are prototypes to determine the critical exponent value of the coexistence curves system. We explain that the critical exponent value of the vapor-liquid equilibrium curves can be regarded as the analogy to the results based on classical assumptions - mean field theory. The novelty of our method has been exposed in comparison to the standard thermodynamic limit discussed thoroughly and extensively in the literature.
We formulate some global invertibility and implicit function theorems. We extend the result of Idczak, Skowron and Walczak on the invertibility of the operators to the case of the operators with critical points. The proof relies on the Mountain Pass Theorem combined with the Palais-Smale condition guaranteeing the claim by the invertibility of the first or the third derivative. I. e. how to solve $x^3=y$?
The existence of multiple fixed points for the coercive, bilinear, compact forms defined in the cone in the Banach space. Multiple applications to the integral equations derived from BVPs for differential equations are provided.
In this paper we obtain the existence of a radial solution for some elliptic nonlocal problem with constraints. The problem arises from some reaction-diffusion equation modelling among others system of self-gravitating particles when one looks for its stationary solutions.
The global in time existence of solutions of a system describing the interaction of gravitationally attracting particles with a general diffusion term and fixed energy is proved. The presented theory covers the case of the model with diffusion that obeys Fermi-Dirac statistics. Some of the results apply to the dissipative polytropic case as well.