arXiv subjects

Alain Haraux

Publications and source records attributed to Alain Haraux.

27 records · Page 2Linked to original sources

arXiv2016

A Liapunov function approach to the stabilization of second order coupled systems

In 2002, Fatiha Alabau, Piermarco Cannarsa and Vilmos Komornik investigated the extent of asymptotic stability of the null solution for weakly coupled partially damped equations of the second order in time. The main point is that the damping operator acts only on the first component and, whenever it is bounded, the coupling is not strong enough to produce an exponential decay in the energy space associated to the conservative part of the system. As a consequence, for initial data in the energy space, the rate of decay is not exponential. Due to the nature of the result it seems at first sight impossible to obtain the asymptotic stability result by the classical Liapunov method. Surprisingly enough, this turns out to be possible and we exhibit, under some compatibility conditions on the operators, an explicit class of Liapunov functions which allows to do 3 different things: 1) When the problem is reduced to a stable finite dimensional space, we recover the exponential decay by a single differential inequality and we estimate the logarithmic decrement of the solutions with worst (slowest) decay. The estimate is optimal at least for some values of the parameters.

math.AP↗

arXiv2015

Some simple problems for the next generations

A list of open problems on global behavior in time of some evolution systems, mainly governed by P.D.E, is given together with some background information explaining the context in which these problems appeared. The common characteristic of these problems is that they appeared a long time ago in the personnal research of the author and received almost no answer till then at the exception of very partial results which are listed to help the readers' understanding of the difficulties involved.

math.AP↗

arXiv2015

The remarkable effectiveness of time-dependent damping terms for second order evolution equations

We consider a second order linear evolution equation with a dissipative term multiplied by a time-dependent coefficient. Our aim is to design the coefficient in such a way that all solutions decay in time as fast as possible. We discover that constant coefficients do not achieve the goal, as well as time-dependent coefficients that are too big. On the contrary, pulsating coefficients which alternate big and small values in a suitable way prove to be more effective. Our theory applies to ordinary differential equations, systems of ordinary differential equations, and partial differential equations of hyperbolic type.

math.AP↗

arXiv2015

The convergence problem for dissipative autonomous systems: classical methods and recent advances

The initial motivation of this text was to provide an up to date translation of the monograph [45] written in french by the first author, taking account of more recent developments of infinite dimensional dynamics based on the Łojasiewicz gradient inequality. In order to keep the present work within modest size bounds and to make it available to the readers without too much delay, we decided to make a first volume entirely dedicated to the so-called convergence problem for autonomous systems of dissipative type. We hope that this volume will help the interested reader to make the connection between the rather simple background developed in the french monograph and the rather technical specialized literature on the convergence problem which grew up rather fast in the recent years.

math.DS↗

arXiv2014

Finding the exact decay rate of all solutions to some second order evolution equations with dissipation

We consider an abstract second order evolution equation with damping. The "elastic" term is represented by a self-adjoint nonnegative operator A with discrete spectrum, and the nonlinear term has order greater than one at the origin. We investigate the asymptotic behavior of solutions. We prove the coexistence of slow solutions and fast solutions. Slow solutions live close to the kernel of A, and decay as negative powers of t as solutions of the first order equation obtained by neglecting the operator A and the second order time-derivatives in the original equation. Fast solutions live close to the range of A and decay exponentially as solutions of the linear homogeneous equation obtained by neglecting the nonlinear terms in the original equation. The abstract results apply to semilinear dissipative hyperbolic equations.

math.AP↗

arXiv2014

Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation

We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the "elastic" operator. In the homogeneous case, we investigate the phase spaces in which the initial value problem gives rise to a semigroup, and the further regularity of solutions. In the non-homogeneous case, we study how the regularity of solutions depends on the regularity of forcing terms, and we characterize the spaces where a bounded forcing term yields a bounded solution. What we discover is a variety of different regimes, with completely different behaviors, depending on the exponent in the friction term. We also provide counterexamples in order to show the optimality of our results.

math.AP↗

arXiv2014

A description of all possible decay rates for solutions of some semilinear parabolic equations

We consider an abstract first order evolution equation in a Hilbert space in which the linear part is represented by a self-adjoint nonnegative operator A with discrete spectrum, and the nonlinear term has order greater than one at the origin. We investigate the asymptotic behavior of solutions. We prove that two different regimes coexist. Close to the kernel of A the dynamic is governed by the nonlinear term, and solutions (when they decay to 0) decay as negative powers of t. Close to the range of A, the nonlinear term is negligible, and solutions behave as solutions of the linearized problem. This means that they decay exponentially to 0, with a rate and an asymptotic profile given by a simple mode, namely a one-frequency solution of the linearized equation. The abstract results apply to semilinear parabolic equations.

math.AP↗

arXiv2013

Global behavior of the Solutions to a Class of Nonlinear, Singular Second Order ODE

In this paper the initial value problem and global properties of solutions are studied for the scalar second order ODE: $ (|u'|^{l}u')' + c|u'|^αu' + d|u|^βu=0$, where $α,β,l,c, d$ are positive constants. In particular, existence, uniqueness and regularity as well as optimal decay rates of solutions to 0 are obtained depending on the various parameters, and the oscillatory or non-oscillatory behavior is elucidated.

math.CA↗

arXiv2013

Optimal decay estimates for the general solution to a class of semi-linear dissipative hyperbolic equations

We consider a class of semi-linear dissipative hyperbolic equations in which the operator associated to the linear part has a nontrivial kernel. Under appropriate assumptions on the nonlinear term, we prove that all solutions decay to 0, as t -> +infinity, at least as fast as a suitable negative power of t. Moreover, we prove that this decay rate is optimal in the sense that there exists a nonempty open set of initial data for which the corresponding solutions decay exactly as that negative power of t. Our results are stated and proved in an abstract Hilbert space setting, and then applied to partial differential equations.

math.AP↗