MR2108676
Ralph CHILL and Alain HARAUX
Ralph CHILL and Alain HARAUX
An Optimal Estimate for the Time Singular Limit of an Abstract Wave Equation
Funkcialaj Ekvacioj. Serio Internacia
47
2004
277--290
http://fe.math.kobe-u.ac.jp/FE/FullPapers/47-2/47_277.pdf
http://www.ams.org/mathscinet-getitem?mr=MR2108676
R. Ikehata recently proved some integral estimate for the difference between the solution of an abstract heat equation and the solution of an abstract wave equation which results from the heat equation by a time singular perturbation. The estimate is obtained if the initial values are chosen appropriately. We prove a pointwise estimate which improves the above result for large times into several directions, and we also establish the optimality of this estimate for the wave equation in an exterior domain. Our proofs rely on the spectral theorem for unbounded self-adjoint operators.
Heat equation, Wave equation, Time singular perturbation.
34C99.
47-277
2004
An Optimal Estimate for the Time Singular Limit of an Abstract Wave Equation
Ralph CHILL and Alain HARAUX
Ralph CHILL and Alain HARAUX
1
Bellout, H.; Friedman, A.
Blow-up estimates for a nonlinear hyperbolic heat equation
SIAM J. Math. Anal.
20
1989
354-366
MR0982663
2
Chill, R.; Haraux, A.
An optimal estimate for the difference of solutions of two abstract evolution equations
J. Differential Equations
193
2003
385-395
MR1998640
3
Engel, K. J.
On singular perturbations of second order Cauchy problems
Pacific J. Math.
512
1992
79-91
MR1139973
4
Esham, B. F.; Weinacht, R. J.
Hyperbolic-parabolic singular perturbations for scalar nonlinearities
Appl. Anal.
29
1988
19-44
MR0960576
5
Esham, B. F.; Weinacht, R. J.
Hyperbolic-parabolic singular perturbations for quasilinear equations
SIAM J. Math. Anal.
20
1989
1344-1365
MR1019305
6
Fattorini, H. O.
Second Order Linear Differential Equations in Banach Spaces
North-Holland, Amsterdam
1985
MR0797071
7
Ikehata, R.
$L^2$-convergence results for linear dissipative wave equations in unbounded domains
Asymptotic Anal.
36
2003
63-74
MR2018018
8
Ikehata, R.; Matsuyama, T.
$L^2$-behaviour of solutions to the linear heat and wave equations in exterior domains
Sci. Math. Japon.
55
2002
33-42
MR1885774
9
Kisynski, J.
Sur les équations hyperboliques avec petit paramètre
Colloq. Math.
10
1963
331-343
MR0156104
10
Najman, B.
Time singular limit of semilinear wave equations with damping
J. Math. Anal. Appl.
174
1993
95-117
MR1212920
11
Najman, B.
Hyperbolic singular perturbation of a quasilinear parabolic equation
Evolution equations, control theory, and biomathematics (Han sur Lesse, 1991), Dekker, New York
1994
405-416
MR1254917
12
Nur, H. S.
Singular perturbations of differential equations in abstract spaces
Pacific J. Math.
36
1971
775-780
MR0282036
13
Reed, M.; Simon, B.
Methods of Modern Mathematical Physics, I: Functional Analysis. Second edition
Academic Press, New York
1980
MR0751959
14
Smoller, J. A.
Singular perturbations and a theorem of Kisynski
J. Math. Anal. Appl.
12
1965
105-114
MR0181814