arXiv subjects

Alexander Rezounenko

Publications and source records attributed to Alexander Rezounenko.

8 recordsLinked to original sources

arXiv2026

On the non-uniqueness of continuous solutions to differential equations with a discrete state-dependent delay

We discuss the non-uniqueness of continuous solutions to differential equations with a {\it discrete } state-dependent delay and continuous initial functions. We are interested not only in the fact (conditions) of non-uniqueness, but in additional information on the number of non-unique solutions and discuss an approach to classify them. We provide a few explicit (easy to verify) examples of the non-uniqueness of continuous solutions and propose an approach to their classification. This partial classification may be illustrated by using just three collors and their simple and intuitive combination. We recognize that this initial classification is not exhaustive, and further study is necessary to build a complete picture.

math.CA↗

arXiv2018

Stability of a retrovirus dymanic model

A retrovirus dynamic model is proposed. We pay attention to the case when viral pathogenicity is low and the infected cells are able to reproduce. Using Lyapunov function method we study stability properties of an inner equilibrium of the model. The equilibrium represents a chronic disease steady state.

math.DS↗

arXiv2017

Viral infection model with diffusion and state-dependent delay: stability of classical solutions

A class of reaction-diffusion virus dynamics models with intracellular state-dependent delay and a general non-linear infection rate functional response is investigated. We are interested in classical solutions with Lipschitz in-time initial functions which are adequate to the discontinuous change of parameters due to, for example, drug administration. The Lyapunov functions technique is used to analyse stability of interior infection equilibria which describe the cases of a chronic disease.

math.DS↗

arXiv2016

Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses

A virus dynamics model with intracellular state-dependent delay and nonlinear infection rate of Beddington-DeAngelis functional response is studied. The technique of Lyapunov functionals is used to analyze stability of an interior infection equilibrium which describes the case of both CTL and antibody immune responses activated. We consider first a particular biologically motivated class of discrete state-dependent delays. Next, the general case is investigated.

math.DS↗

arXiv2014

Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay

We deal with a class of parabolic nonlinear evolution equations with state-dependent delay. This class covers several important PDE models arising in biology. We first prove well-posedness in a certain space of functions which are Lipschitz in time. We show that the model considered generates an evolution operator semigroup $S_t$ on a space $CL$ of Lipschitz type functions over delay time interval. The operators $S_t$ are closed for all $t\ge 0$ and continuous for $t$ large enough. Our main result shows that the semigroup $S_t$ possesses compact global and exponential attractors of finite fractal dimension. Our argument is based on the recently developed method of quasi-stability estimates and involves some extension of the theory of global attractors for the case of closed evolutions.

math.AP↗

arXiv2014

Parabolic partial differential equations with discrete state-dependent delay: classical solutions and solution manifold

Classical solutions to PDEs with discrete state-dependent delay are studied. We prove the well-posedness in a set $X_F$ which is an analogous to the solution manifold used for ordinary differential equations with state-dependent delay. We prove that the evolution operators are $C^1$-smooth on the solution manifold.

math.AP↗

arXiv2013

Dynamics of second order in time evolution equations with state-dependent delay

We deal with a class of second order in time nonlinear evolution equations with state-dependent delay. This class covers several important PDE models arising in the theory ofnonlinear plates. Our first result states well-posedness in a certain space of functions which are C1 in time. In contrast with the first order models with discrete state-dependent delay this result does not require any compatibility conditions. The solutions constructed generate a dynamical system in a C1-type space over delay time interval. Our next result shows that this dynamical system possesses compact global and exponential attractors of finite fractal dimension. To obtain this result we adapt the recently developed method of quasi-stability estimates.

math.AP↗

arXiv2010

Non-local PDEs with a state-dependent delay term presented by Stieltjes integral

Parabolic partial differential equations with state-dependent delays (SDDs) are investigated. The delay term presented by Stieltjes integral simultaneously includes discrete and distributed SDDs. The singular Lebesgue-Stieltjes measure is also admissible. The conditions for the corresponding initial value problem to be well-posed are presented. The existence of a compact global attractor is proved.

math.AP↗