arXiv subjects

Alexander V. Rezounenko

Publications and source records attributed to Alexander V. Rezounenko.

6 recordsLinked to original sources

arXiv2010

Non-local PDEs with discrete state-dependent delays: well-posedness in a metric space

Partial differential equations with discrete (concentrated) state-dependent delays are studied. The existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of continuously differentiable (with respect to an appropriate norm) functions is used to construct a dynamical system. This subset is an analogue of \textit{the solution manifold} proposed for ordinary equations in [H.-O. Walther, The solution manifold and $C\sp 1$-smoothness for differential equations with state-dependent delay, J. Differential Equations, {195}(1), (2003) 46--65]. The existence of a compact global attractor is proven.

math.AP↗

arXiv2009

Non-linear partial differential equations with discrete state-dependent delays in a metric space

We investigate a class of non-linear partial differential equations with discrete state-dependent delays. The existence and uniqueness of strong solutions for initial functions from a Banach space are proved. To get the well-posed initial value problem we restrict our study to a smaller metric space, construct the dynamical system and prove the existence of a compact global attractor.

math.AP↗

arXiv2008

Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions

Partial differential equations with discrete (concentrated) state-dependent delays in the space of continuous functions are investigated. In general, the corresponding initial value problem is not well posed, so we find an additional assumption on the state-dependent delay function to guarantee the well posedness. For the constructed dynamical system we study the long-time asymptotic behavior and prove the existence of a compact global attractor.

math.AP↗

arXiv2007

Partial Inertial Manifolds for infinite-dimensional dynamical systems: Example for P.D.E.s with a state-dependent delay

We propose a new notion of Partial Inertial Manifold to study the long-time asymptotic behavior of dissipative differential equations. As shown on an example, such manifolds may exist in the cases when the classical Inertial manifold does not exist (or not known to exist).

math.DS↗

arXiv2006

On a class of PDEs with nonlinear distributed in space and time state-dependent delay term

A new class of nonlinear partial differential equations with distributed in space and time state-dependent delay is investigated. We find appropriate assumptions on the kernel function which represents the state-dependent delay and discuss advantages of this class. Local and long-time asymptotic properties, including the existence of global attractor, are studied.

math.DS↗

arXiv2005

Two models of partial differential equations with discrete and distributed state-dependent delays

This work is the first attempt to treat partial differential equations with discrete (concentrated) state-dependent delay. The main idea is to approximate the discrete delay term by a sequence of distributed delay terms (all with state-dependent delays). We study local existence and long-time asymptotic behavior of solutions and prove that the model with distributed delay has a global attractor while the one with discrete delay possesses the trajectory attractor.

math.DS↗