Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations
We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation u t − div a ( x , ∇ u ) + f ( x , u ) = 0 on a bounded domain, subject to Dirichlet boundary and to initial conditions. The data are supposed to satisfy suitable regularity and growth co...
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| Published in | Journal of Differential Equations Vol. 228; no. 2; pp. 611 - 632 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
15.09.2006
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0022-0396 1090-2732 |
| DOI | 10.1016/j.jde.2006.02.009 |
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| Abstract | We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation
u
t
−
div
a
(
x
,
∇
u
)
+
f
(
x
,
u
)
=
0
on a bounded domain, subject to Dirichlet boundary and to initial conditions. The data are supposed to satisfy suitable regularity and growth conditions. Our approach to the convergence result and decay estimate is based on the Łojasiewicz–Simon gradient inequality which in the case of the semilinear heat equation is known to give optimal decay estimates. The abstract results and their applications are discussed also in the framework of Orlicz–Sobolev spaces. |
|---|---|
| AbstractList | We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation
u
t
−
div
a
(
x
,
∇
u
)
+
f
(
x
,
u
)
=
0
on a bounded domain, subject to Dirichlet boundary and to initial conditions. The data are supposed to satisfy suitable regularity and growth conditions. Our approach to the convergence result and decay estimate is based on the Łojasiewicz–Simon gradient inequality which in the case of the semilinear heat equation is known to give optimal decay estimates. The abstract results and their applications are discussed also in the framework of Orlicz–Sobolev spaces. |
| Author | Fiorenza, Alberto Chill, Ralph |
| Author_xml | – sequence: 1 givenname: Ralph surname: Chill fullname: Chill, Ralph email: chill@univ-metz.fr, chill@mathematik.uni-ulm.de organization: Abteilung Angewandte Analysis, Universität Ulm, 89069 Ulm, Germany – sequence: 2 givenname: Alberto surname: Fiorenza fullname: Fiorenza, Alberto email: fiorenza@unina.it organization: Dipartimento di Costruzioni e Metodi Matematici in Architettura, Universitá di Napoli, Via Monteoliveto, 3, I-80134 Napoli, Italy |
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| Cites_doi | 10.1080/00036819108840034 10.1007/s00028-004-0163-x 10.1016/S0294-1449(00)00121-9 10.1215/kjm/1250522572 10.1016/0022-0396(74)90012-6 10.1016/S0022-1236(02)00102-7 10.1023/A:1026467729263 10.1007/s00028-003-1112-8 10.1016/0022-0396(86)90121-X 10.1016/j.jde.2004.08.011 10.7146/math.scand.a-12162 10.1006/jdeq.1997.3362 10.1006/jdeq.1996.0020 10.1007/s005260050133 10.2307/2006981 10.1016/0022-1236(71)90018-8 10.1006/jfan.1997.3174 10.1007/s00028-004-0162-y 10.1112/S0024610705006630 10.3792/pja/1195518909 10.1051/cocv:2003015 |
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| Keywords | secondary Decay rate Łojasiewicz–Simon inequality Orlicz–Sobolev space Convergence of solutions Quasilinear parabolic problems primary |
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| Snippet | We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation
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| SubjectTerms | Convergence of solutions Decay rate Orlicz–Sobolev space Quasilinear parabolic problems Łojasiewicz–Simon inequality |
| Title | Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations |
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