On the fast solution of evolution equations with a rapidly decaying source term
Alain Haraux  UPMC Univ Paris 06, UMR 7598, Laboratoire JacquesLouis Lions, F75005, Paris, France (email) 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 semiconvergent, and then this solution is unique. For the semilinear 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.
Keywords: Fast solution, evolution equations, rapidly decaying source.
Received: September 2010; Revised: December 2010; Available Online: March 2011. 
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