Non-autonomous Partial Differential Equations Research Articles

NONLINEAR VIBRATION OF A CANTILEVER WITH A DERJAGUIN–MÜLLER–TOPOROV CONTACT END

In this paper, the principal resonance is investigated for a cantilever with a contact end. The cantilever is modeled as an Euler–Bernoulli beam, and the contact is modeled by the Derjaguin–Müller–Toporov theory. The problem is formulated as a linear nonautonomous partial-differential equation with a nonlinear autonomous boundary condition. The method of multiple scales is applied to determine the steady-state response. The equation of response curves is derived from the solvability condition of eliminating secular terms. The stability of steady-state responses is analyzed by using the Lyapunov-linearized stability theory. Numerical examples are presented to highlight the effects of the excitation amplitude, the damping coefficient, and the coefficients related to the contact.

Balanced growth for solutions of nonautonomous partial differential equations

We show balanced growth for solutions of some nonautonomous partial differential equations which in certain cases also describe the dynamics of structured populations.

FINITE DIMENSIONALITY OF ATTRACTORS FOR NON-AUTONOMOUS DYNAMICAL SYSTEMS GIVEN BY PARTIAL DIFFERENTIAL EQUATIONS

In the last decade, the concept of pullback attractor has become one of the usual tools to describe some qualitative properties of non-autonomous partial differential equations. A pullback attractor is a family of compact sets, invariant for the corresponding process related to the equation, and attracting from the past, and it assumes a natural generalization of the now classical concept of global attractor for autonomous partial differential equations. In this work we give sufficient conditions in order to prove the finite Hausdorff and fractal dimensionality of pullback attractors for non-autonomous infinite dimensional dynamical systems, and we apply our results to a generalized non-autonomous partial differential equation of Navier–Stokes type.

Robustness of asymptotic properties of evolution families under perturbations

In this paper, we present some conditions under which asymptotic properties of an evolution family ${{\mathcal{U}}}:={(U(t,s))_{t\geq s\geq 0}}$ persist under perturbations by a family $\mathcal B:=(B(t),D(B(t))_{t\in{{\mathbb{R_+}}}}$ of linear operators on a Banach space $X$ satisfying suitable conditions. Our results concern asymptotic properties like boundedness, periodicity, and asymptotic almost periodicity (even in the sense of Eberlein). An application of the abstract results to nonautonomous partial differential equations with delay is given.

The dimension of attractors of nonautonomous partial differential equations

AbstractThe concept of nonautonomous (or cocycle) attractors has become a proper tool for the study of the asymptotic behaviour of general nonautonomous partial differential equations. This is a time-dependent family of compact sets, invariant for the associated process and attracting “from –∞”. In general, the concept is rather different to the classical global attractor for autonomous dynamical systems. We prove a general result on the finite fractal dimensionality of each compact set of this family. In this way, we generalise some previous results of Chepyzhov and Vishik. Our results are also applied to differential equations with a nonlinear term having polynomial growth at most.

Bang–bang principle for linear and non-linear Goursat–Darboux problems

In this paper, the bang-bang principle for a control system described by a system of non-linear (in control) non-autonomous partial differential equations of hyperbolic type (the so-called Goursat-Darboux problem) is proved. As a particular case the bang-bang principle for a linear system is obtained. The method used for the proof is based on some properties of the Aumann integral.

Asymptotically finite dimensional pullback behaviour of non-autonomous PDEs

The asymptotic behaviour of general non-autonomous partial differential equations can be described using the concept of pullback attractor. This is, under suitable hypotheses, a time-dependent family of finite-dimensional compact sets. In this work we investigate how this finite-dimensional dynamics on the attractor determines the asymptotic behaviour of non-autonomous PDEs.

A Characteristic Equation for Non-autonomous Partial Functional Differential Equations

We characterize the exponential dichotomy of non-autonomous partial functional differential equations by means of a spectral condition extending known characteristic equations for the autonomous or time-periodic case. From this we deduce robustness results. We further study the almost periodicity of solutions to the inhomogeneous equation. Our approach is based on the spectral theory of evolution semigroups.

The asymptotic behaviour of perturbed evolution families

Given an evolution family $\mathcal U:={(U(t,s))_{t\geq s}}$ on a Banach space $X$, we present some conditions under which asymptotic properties of $\mathcal U$ are stable under small perturbations by a family $$ \mathcal{B}:=(B(t),D(B(t))_{t\in\mathbb{J}},$$ $\mathbb{J} =\mathbb{R}$ or $\mathbb{R}_+$, of linear closed operators on $X$. Our results concern asymptotic properties like periodicity, (asymptotic) almost periodicity (even in the sense of Eberlein), uniform ergodicity and total uniform ergodicity. We present, moreover, an application of the abstract results to non-autonomous partial differential equations with delay.

On a Schwarzian PDE associated with the KdV hierarchy

We present a novel integrable non-autonomous partial differential equation of the Schwarzian type, i.e. invariant under Möbius transformations, that is related to the Korteweg–de Vries hierarchy. In fact, this PDE can be considered as the generating equation for the entire hierarchy of Schwarzian KdV equations. We present its Lax pair, establish its connection with the SKdV hierarchy, its Miura relations to similar generating PDEs for the modified and regular KdV hierarchies and its Lagrangian structure. Finally we demonstrate that its similarity reductions lead to the full Painlevé VI equation, i.e. with four arbitrary parameters.

Model reference adaptive control of a time-varying parabolic system

Related to the error dynamics of an adaptive system, averaging theorems are developed for coupled differential equations which consist of ordinary differential equations and a parabolic partial differential equation. The results are then applied to the convergence analysis of the parameter estimate errors in the model reference adaptive control of a nonautonomous parabolic partial differential equation with lowly time-varying parameters.

Lyapunov stability of columns, pipes and rotating shafts under time-dependent excitation

Structural or mechanical systems governed by non-autonomous partial differential equations are considered. The systems are such that they would be conservative in the absence of dissipation and time variation of the loading parameter. They possess an equilibrium state, and sufficient conditions for its stability are obtained with the use of Lyapunov's direct method. Three problems are treated: a column with a time-varying axial load, a pipe conveying fluid with time-varying velocity, and a rotating shaft with time-varying angular velocity. These excitations appear in coefficients of the equations of motion, and the stability conditions involve the excitations and their time rates of change

Finite-Dimensional Behavior of a Nonautonomous Partial Differential Equation: Forced Oscillations of an Extensible Beam

Finite-Dimensional Behavior of a Nonautonomous Partial Differential Equation: Forced Oscillations of an Extensible Beam

On the resolvent of linear nonautonomous partial functional differential equations

We give sufficient conditions for the existence of the resolvent operator for nonautonomous linear partial differential equations with delay, where the highest order derivatives are undelayed. Furthermore we analyse the connection between the resolvent and the solution operator of the homogeneous equation.