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This paper investigates the asymptotic behaviour of solutions of periodic evolution equations. Starting with a general result concerning the quantified asymptotic behaviour of periodic evolution families we go on to consider a special class…
We study the existence of $T$-periodic solutions $(T > 0)$ for the first order differential equations being at resonance at infinity, where the right hand side is the perturbations of a sectorial operator. Our aim is to prove an index…
We are interested in the large-time behavior of periodic entropy solutions in $L^\infty$ to anisotropic degenerate parabolic-hyperbolic equations of second-order. Unlike the pure hyperbolic case, the nonlinear equation is no longer…
In this Note, we review the main existing results, methods, and some key open problems on the controllability of nonlinear hyperbolic and parabolic equations. Especially, we describe our recent universal approach to solve the local…
A simple trick is illustrated, whereby nonlinear evolution equations can be modified so that they feature a lot - or, in some cases, only -- periodic solutions. Several examples (ODEs and PDEs) are exhibited.
We consider linear, time-dependent and skew-adjoint perturbations of periodic transport equations on the one-dimensional torus. We describe the long-time behavior of solutions for all non-degenerate perturbations in resonant regime, proving…
This survey is devoted to the asymptotic behavior of solutions of evolution equations generated by maximal monotone operators in Hilbert spaces. The emphasis is in the comparison of the continuous time trajectories to sequences generated by…
We consider boundary value problems for semilinear hyperbolic systems of the type $$ \partial_tu_j + a_j(x,\la)\partial_xu_j + b_j(x,\la,u) = 0, \; x\in(0,1), \;j=1,\dots,n $$ with smooth coefficient functions $a_j$ and $b_j$ such that…
In this paper, we are devoted to consider the periodic problem for the impulsive evolution equations with delay in Banach space. By using operator semigroups theory and fixed point theorem, we establish some new existence theorems of…
Nonlinear and nonlinear evolution equations of the form $u_t=\L u \pm|\nabla u|^q$, where $\L$ is a pseudodifferential operator representing the infinitesimal generator of a L\'evy stochastic process, have been derived as models for growing…
We present an abstract framework for treating the theory of well-posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on…
For two linear evolution differential equations systems - a normal ordinary differential equations system and a partial differential equations system with Stokes operator in a main part - with rapidly oscillating by time coefficients in a…
We study evolution equations on networks that can be modeled by means of hyperbolic systems. We extend our previous findings in \cite{KraMugNic20} by discussing well-posedness under rather general transmission conditions that might be…
We develop a nonlinear evolution framework for nonlinear parabolic equations with unbounded drift terms formulated in Lorentz spaces. The main contribution lies in the construction of uniformly m-accretive operators based on Lorentz-Sobolev…
Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework…
We consider boundary value problems for quasilinear first-order one-dimensional hyperbolic systems in a strip. The boundary conditions are supposed to be of a smoothing type, in the sense that the $L^2$-generalized solutions to the…
Considering evolutionary equations in the sense of Picard, we identify a certain topology for material laws rendering the solution operator continuous if considered as a mapping from the material laws into the set of bounded linear…
In this work, we study continuity and topological structural stability of attractors for nonautonomous random differential equations obtained by small bounded random perturbations of autonomous semilinear problems. First, we study existence…
We describe a method to model nonlinear dynamical systems using periodic solutions of delay-differential equations. We show that any finite-time trajectory of a nonlinear dynamical system can be loaded approximately into the initial…
The purpose of this paper is to further exemplify an approach to evolutionary problems originally developed in earlier works for a special case and later extended to more general evolutionary problems. We are here concerned with the $(1+1)$…