On the fast solution of evolution equations with a rapidly decaying source term

Alain Haraux

doi:10.3934/mcrf.2011.1.1

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2011, Volume 1, Issue 1: 1-20. Doi: 10.3934/mcrf.2011.1.1

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On the fast solution of evolution equations with a rapidly decaying source term

Alain Haraux^1,

1.
UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris

Received: September 2010

Revised: December 2010

Published: March 2011

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Abstract

If $L$ is the generator of a uniformly bounded group of operators $T(t)$ on a Banach space $X$, the abstract evolution equation $ u' + Lu(t) = h(t) $ has a (weak) solution tending to $0$ as $t\rightarrow +\infty $ if, and only if $\int_0^{+\infty}T(s) h(s) ds $ is semi-convergent, and then this solution is unique. For the semi-linear equation $ u' + Lu(t) + f(u) = h(t) $, if $f$ such that $f(0) = 0$ is Lipschitz continuous on bounded subsets of $X$ and has a Lipschitz constant bounded by $ Cr^\alpha $ in the ball $B(0, r)$ for $r\leq r_0$, for any $h$ satisfiying

$||h(t)|| \leq c(1+t)^{-(1+ \lambda )} $

with $\lambda >\frac{1}{\alpha}$ and $c$ small enough there exists a unique solution tending to $0$ at least like $(1+t)^{- \lambda}.$ When the system is dissipative, this special solution makes it sometimes possible to estimate from below the rate of decay to $0$ of the other solutions.

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Mathematics Subject Classification: 2000 Mathematics Subject Classification: 34C11, 34D05, 34D30, 34G20, 35B40.

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