Ralph Chill

On the Łojasiewicz–Simon gradient inequality

Abstract We prove a general version of the Łojasiewicz–Simon inequality, and we show how to apply the abstract result to study energy functionals E of the form E(v)= 1 2 a(v,v)+ ∫ Ω F(x,v), defined on a Hilbert space V↪L 2 (Ω) . We show that in some cases it is possible to prove the Łojasiewicz–Simon inequality for such functionals without the assumption of analyticity. The results apply to study the asymptotic behaviour of parabolic and hyperbolic evolution equations.

Convergence to steady states in asymptotically autonomous semilinear evolution equations

Abstract We study the convergence to equilibrium of bounded solutions of the nonautonomous first-order problem u + M u=g(t), t∈ R + and of the second-order problem u + u + M u=g(t), t∈ R + . Applications to diffusion, wave, Cahn–Hilliard and Kirchhoff–Carrier equations are described.

An optimal estimate for the difference of solutions of two abstract evolution equations

AbstractIkehata and Nishihara have established that the difference between any solution u of alinearly damped abstract wave equation and a certain solution v of a related abstract heatequation decays at least like t 1 ðlogtÞ 12þe as time tends to infinity. They conjectured that thedecayisinfactliket 1 : Weproveherethevalidityofthisconjecturebyrelyingonthespectraltheorem for unbounded self-adjoint operators. We also establish the optimality of thisestimate for the wave equation in an exterior domain.r 2003 Elsevier Science (USA). All rights reserved. MSC: primary34D05;secondary35L90Keywords: Abstract heatequation; Abstract waveequation; Diffusionphenomenon 1. IntroductionThe main objective of this work is to establish an estimate of the differencebetween the solution u of the abstract dissipative wave equationu 00 þ u 0 þ Au ¼ 0; uð0Þ¼u 0 ; u 0 ð0Þ¼u 1 ; ð1:1Þand the solution v of the abstract heat equationv 0 þ Av ¼ 0; vð0Þ¼u 0 þ u 1 : ð1:2Þ ARTICLE IN PRESS Corresponding author. E-mail address: haraux@ann.jussieu.fr (A. Haraux).0022-0396/03/

APPLICATIONS OF THE ŁOJASIEWICZ–SIMON, GRADIENT INEQUALITY TO GRADIENT-LIKE EVOLUTION EQUATIONS

-seefrontmatterr 2003 ElsevierScience(USA).Allrights reserved.doi:10.1016/S0022-0396(03)00057-3

Fine scales of decay of operator semigroups

We prove convergence to equilibrium of global and bounded solutions of gradient-like evolution equations. Our abstract results are illustrated by several examples in finite and infinite dimensions.

Generation of Cosine Families on L p (0,1) by Elliptic Operators with Robin Boundary Conditions

Motivated by potential applications to partial differential equations, we develop a theory of fine scales of decay rates for operator semigroups. The theory contains, unifies, and extends several notable results in the literature on decay of operator semigroups and yields a number of new ones. Its core is a new operator-theoretical method of deriving rates of decay combining ingredients from functional calculus, and complex, real and harmonic analysis. It also leads to several results of independent interest.

The Sector of Analyticity of the Ornstein–Uhlenbeck Semigroup on Lp Spaces with Respect to Invariant Measure

Let a ∈ W 1,∞(0,1), a(x) ≥ α > 0, b, c ∈ L ∞ (0,1) and consider the differential operator A given by Au = au″ + bu′ + cu. Let α j , β j (j = 0, 1) be complex numbers satisfying α j , β j ≠ (0,0) for j = 0, 1. We prove that a realization of A with the boundary conditions

Equality of two spectra arising in harmonic analysis and semigroup theory

\alpha _j u\prime \left( j \right) + \beta _j u\left( j \right) = 0,{\text{ }}j = 0,1,