TITLE

Global–in–time Uniform Convergence for Linear Hyperbolic–Parabolic Singular Perturbations

AUTHOR(S)
Ghisi, Marina; Gobbino, Massimo
PUB. DATE
July 2006
SOURCE
Acta Mathematica Sinica;Jul2006, Vol. 22 Issue 4, p1161
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
We consider the Cauchy problem $$ \varepsilon {u}\ifmmode{''}\else$''$\fi_{\varepsilon } + \delta {u}\ifmmode{'}\else$'$\fi_{\varepsilon } + Au_{\varepsilon } = 0,\;u_{\varepsilon } (0) = u_{0} ,\;{u}\ifmmode{'}\else$'$\fi_{\varepsilon } (0) = u_{1} , $$ where ∈ > 0, δ > 0, H is a Hilbert space, and A is a self–adjoint linear non–negative operator on H with dense domain D( A). We study the convergence of { u ∈} to the solution of the limit problem δ u' + Au = 0, u(0) = u 0. For initial data ( u 0, u 1) ∈ D( A 1/2) × H, we prove global–in–time convergence with respect to strong topologies. Moreover, we estimate the convergence rate in the case where ( u 0, u 1) ∈ D( A 3/2) × D( A 1/2), and we show that this regularity requirement is sharp for our estimates. We give also an upper bound for $$ {\left| {{u}\ifmmode{'}\else$'$\fi_{\varepsilon } (t)} \right|} $$ which does not depend on ∈.
ACCESSION #
21690502

 

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