arXiv subjects

Piotr Kalita

Publications and source records attributed to Piotr Kalita.

18 recordsLinked to original sources

arXiv2026

Stability of phase portrait for a gradient ODE with memory

We consider the problem governed by the gradient ODE $x'=\nabla F(x)$ in $\mathbb{R}^d$ on which we assume that it has a finite number of hyperbolic equilibria whose stable and unstable manifolds intersect transversally. This problem is perturbed by the memory term $$x'(t)=\nabla F(x(t))+\varepsilon\int_{-\infty}^t M(t-s)x(s)\, ds$$ where $\varepsilon>0$ is a small constant. The key result is that the structure of connections between the equilibria of the unperturbed problem is exactly preserved for a small $\varepsilon>0$.

math.DS↗

arXiv2026

On $L^{\infty }$-estimates and the structure of the global attractor for weak solutions of reaction-diffusion equations

In this paper, we study the structure of the global attractor for weak and regular solutions of a problem governed by a scalar semilinear reaction-diffusion equation with a non-regular nonlinearity, such that uniquness of solutions can fail to happen. First, using the Moser--Alikakos iterations we obtain the estimates of the weak solutions in the space $L^{\infty }(Ω)$. After that, using these estimates we improve the existing results on the structure of the attractor. Finally, estimates of the Hausdorff and fractal dimension of the attractor are obtained.

math.AP↗

arXiv2025

$C^1$ invariant, stable and inertial manifolds for non-autonomous dynamical systems

We use the version of the Lyapunov--Perron method operating on individual solutions to investigate the existence of invariant manifolds for non-autonomous dynamical systems, focusing in particular on inertial and stable manifolds. We establish a characterization of both types of manifolds in terms of solutions exhibiting a common growth behavior, analogous to the classical characterization involving hyperbolicity. Furthermore, we introduce a unified formulation of the gap condition, from which known sharp versions are derived. Finally, we show that the constructed inertial manifolds have $C^1$ regularity.

math.DS↗

arXiv2024

Pitchfork bifurcation and heteroclinic connections in the Kuramoto--Sivashinsky PDE

We present a method for the complete analysis of the dynamics of dissipative Partial Differential Equations (PDEs) undergoing a pitchfork bifurcation. We apply our technique to the Kuramoto--Sivashinsky PDE on the line to obtain a computer-assisted proof of the creation of two symmetric branches of non-symmetric fixed points and heteroclinic connections between the symmetric fixed point and the new ones. The range of parameters is given explicitly and is large enough to allow for the rigorous continuation of the fixed points and heteroclinic connections created during the bifurcation.

math.DS↗

arXiv2024

Structural stability of invasion graphs for Lotka--Volterra systems

In this paper, we study in detail the structure of the global attractor for the Lotka--Volterra system with a Volterra--Lyapunov stable structural matrix. We consider the invasion graph as recently introduced in [19] and prove that its edges represent all the heteroclinic connections between the equilibria of the system. We also study the stability of this structure with respect to the perturbation of the problem parameters. This allows us to introduce a definition of structural stability in ecology in coherence with the classical mathematical concept where there exists a detailed geometrical structure, robust under perturbation, that governs the transient and asymptotic dynamics.

math.DS↗

arXiv2024

Forwards attractors for non-autonomous Lotka-Volterra cooperative systems: a detailed geometrical description

Non-autonomous differential equations exhibit a highly intricate dynamics, and various concepts have been introduced to describe their qualitative behavior. In general, it is rare to obtain time dependent invariant compact attracting sets when time goes to plus infinity. Moreover, there are only a few papers in the literature that explore the geometric structure of such sets. In this paper we investigate the long time behaviour of cooperative $n$-dimensional non-autonomous Lotka-Volterra systems is population dynamics. We provide sufficient conditions for the existence of a globally stable (forward in time) entire solution in which one species becomes extinct, or where all species except one become extinct. Furthermore, we obtain the precise geometrical structure of the non-autonomous forward attractor in one, two, and three dimensions by establishing heteroclinic connections between the globally stable solution and the semi-stable solutions in cases of species permanence and extinction. We believe that understanding time-dependent forward attractors paves the way for a comprehensive analysis of both transient and long-term behavior in non-autonomous phenomena.

math.DS↗

arXiv2023

Computer-assisted validation of the existence of periodic orbit in the Brusselator system

We investigate the Brusselator system with diffusion and Dirichlet boundary conditions on one dimensional space interval. Our proof demonstrates that, for certain parameter values, a periodic orbit exists. This proof is computer-assisted and rooted in the rigorous integration of partial differential equations. Additionally, we present the evidence of the occurrence of period-doubling bifurcation.

math.DS↗

arXiv2022

Autonomous and non-autonomous unbounded attractors in evolutionary problems

If the semigroup is slowly non-dissipative, i.e., its solutions can diverge to infinity as time tends to infinity, one still can study its dynamics via the approach by the unbounded attractors - the counterpart of the classical notion of global attractors. We continue the development of this theory started by Chepyzhov and Goritskii [CG92]. We provide the abstract results on the unbouded attractor existence, and we study the properties of these attractors, as well as of unbounded $ω$-limit sets in slowly non-dissipative setting. We also develop the pullback non-autonomous counterpart of the unbounded attractor theory. The abstract theory that we develop is illustrated by the analysis of the autonomous problem governed by the equation $u_t = Au + f(u)$. In particular, using the inertial manifold approach, we provide the criteria under which the unbounded attractor coincides with the graph of the Lipschitz function, or becomes close to the graph of the Lipschitz function for large argument.

math.DS↗

arXiv2021

Convergence of non-autonomous attractors for subquintic weakly damped wave equation

We study the non-autonomous weakly damped wave equation with subquintic growth condition on the nonlinearity. Our main focus is the class of Shatah--Struwe solutions, which satisfy the Strichartz estimates and are coincide with the class of solutions obtained by the Galerkin method. For this class we show the existence and smoothness of pullback, uniform, and cocycle attractors and the relations between them. We also prove that these non-autonomous attractors converge upper-semicontinuously to the global attractor for the limit autonomous problem if the time-dependent nonlinearity tends to time independent function in an appropriate way.

math.AP↗

arXiv2020

Rigorous FEM for 1D Burgers equation

We propose a method to integrate dissipative PDEs rigorously forward in time with the use of Finite Element Method (FEM). The technique is based on the Galerkin projection on the FEM space and estimates on the residual terms. The proposed approach is illustrated on a periodically forced one-dimensional Burgers equation with Dirichlet conditions. For two particular choices of the forcing we prove the existence of the periodic globally attracting trajectory and give precise bounds on its shape.

math.AP↗

arXiv2020

On renormalized solutions to elliptic inclusions with nonstandard growth

We study the elliptic inclusion given in the following divergence form \begin{align*} & -\mathrm{div}\, A(x,\nabla u) \ni f\quad \mathrm{in}\quad Ω, & u=0\quad \mathrm{on}\quad \partial Ω. \end{align*} As we assume that $f\in L^1(Ω)$, the solutions to the above problem are understood in the renormalized sense. We also assume nonstandard, possibly nonpolynomial, heterogeneous and anisotropic growth and coercivity conditions on the maximally monotone multifunction $A$ which necessitates the use of the nonseparable and nonreflexive Musielak--Orlicz spaces. We prove the existence and uniqueness of the renormalized solution as well as, under additional assumptions on the problem data, its relation to the weak solution. The key difficulty, the lack of a Carathéodory selection of the maximally monotone multifunction is overcome with the use of the Minty transform.

math.AP↗

arXiv2019

Micropolar meets Newtonian in three dimensions. The Rayleigh--B{é}nard problem for large Prandtl numbers

We consider the Rayleigh--Bénard problem for the three--dimensional Boussinesq system for the micropolar fluid. We introduce the notion of the multivalued eventual semiflow and prove the existence of the two-space global attractor $\mathcal{A}^K$ corresponding to weak solutions, for every micropolar parameter $K\geq 0$ denoting the deviation of the considered system from the classical Rayleigh--Bénard problem for the Newtonian fluid. We prove that for every $K$ the attractor $\mathcal{A}^K$ is the smallest compact, attracting, and invariant set. Moreover, the semiflow restricted to this attractor is single-valued and governed by strong solutions. Further, we prove that the global attractors $\mathcal{A}^K$ converge to $\mathcal{A}^0$ upper semicontinuously in Kuratowski sense as $K\to 0$, and that the projection of $\mathcal{A}^0$ on the restricted phase space corresponding to the classical Rayleigh--Bénard problem is the global attractor for the latter problem, having the invariance property. These results are established under the assumption that the Prandtl number is relatively large with respect to the Rayleigh number.

math-ph↗

arXiv2019

A frictional contact problem with wear diffusion

This paper constructs and analyzes a model for the dynamic frictional contact between a viscoelastic body and a moving foundation. The contact involves wear of the contacting surface and the diffusion of the wear debris. The relationships between the stresses and displacements on the contact boundary are modeled by the normal compliance law and a version of the Coulomb law of dry friction. The rate of wear of the contact surface is described by the differential form of the Archard law. The effects of the diffusion of the wear particles that cannot leave the contact surface on the surface are taken into account. The novelty of this work is that the contact surface is a manifold and, consequently, the diffusion of the debris takes place on a curved surface. The interest in the model is related to the wear of mechanical joints and orthopedic biomechanics where the wear debris are trapped, they diffuse and often cause the degradation of the properties of joint prosthesis and various implants. The model is in the form of a differential inclusion for the mechanical contact and the diffusion equation for the wear debris on the contacting surface. The existence of a weak solution is proved by using a truncation argument and the Kakutani--Ky Fan--Glicksberg fixed point theorem.

math-ph↗

arXiv2019

On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions

We study the non-autonomously forced Burgers equation $$ u_t(x,t) + u(x,t)u_x(x,t) - u_{xx}(x,t) = f(x,t) $$ on the space interval $(0,1)$ with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique $H^1$ bounded trajectory of this equation defined for all $t\in \mathbb{R}$. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential.

math.AP↗

arXiv2015

Smooth attractors for weak solutions of the SQG equation with critical dissipation

We consider the evolution of weak vanishing viscosity solutions to the critically dissipative surface quasi-geostrophic equation. Due to the possible non-uniqueness of solutions, we rephrase the problem as a set-valued dynamical system and prove the existence of a global attractor of optimal Sobolev regularity. To achieve this, we derive a new Sobolev estimate involving Hölder norms, which complement the existing estimates based on commutator analysis.

math.AP↗

arXiv2014

Minimality properties of set-valued processes and their pullback attractors

We discuss the existence of pullback attractors for multivalued dynamical systems on metric spaces. Such attractors are shown to exist without any assumptions in terms of continuity of the solution maps, based only on minimality properties with respect to the notion of pullback attraction. When invariance is required, a very weak closed graph condition on the solving operators is assumed. The presentation is complemented with examples and counterexamples to test the sharpness of the hypotheses involved, including a reaction-diffusion equation, a discontinuous ordinary differential equation and an irregular form of the heat equation.

math.AP↗

arXiv2014

Semidiscrete variable time-step θ-scheme for nonmonotone evolution inclusion

In this paper we show the convergence of a semidiscrete time stepping θ-scheme on a time grid of variable length to the solution of parabolic operator differential inclusion in the framework of evolution triple. The multifunction is assumed to be strong-weak upper-semicontinuous and to have nonempty, closed and convex values, while the quasilinear operator present in the problem is required to be pseudomonotone, coercive and satisfy the appropriate growth condition. The convergence of piecewise constant and piecewise linear interpolants constructed on the solutions of time discrete problems is shown. Under an additional assumption on the sequence of time grids and regularity of quasilinear operator strong convergence results are obtained.

math.AP↗

arXiv2012

Convergence of Rothe scheme for hemivariational inequalities of parabolic type

This article presents the convergence analysis of a sequence of piecewise constant and piecewise linear functions obtained by the Rothe method to the solution of the first order evolution partial differential inclusion $u'(t)+Au(t)+ι^*\partial J(ιu(t))\ni f(t)$, where the multivalued term is given by the Clarke subdifferential of a locally Lipschitz functional. The method provides the proof of existence of solutions alternative to the ones known in literature and together with any method for underlying elliptic problem, can serve as the effective tool to approximate the solution numerically. Presented approach puts into the unified framework known results for multivalued nonmonotone source term and boundary conditions, and generalizes them to the case where the multivalued term is defined on the arbitrary reflexive Banach space as long as appropriate conditions are satisfied. In addition the results on improved convergence as well as the numerical examples are presented.

math.AP↗