Asymptotic behavior for an almost periodic, strongly dissipative wave equation
This paper deals with asymptotic behavior for (weak) solutions of the equation u tt − Δu + β(u t) ∋ ƒ(t, x) , on R + × Ω; u( t, x) = 0, on R + × ∂Ω. If ƒ∈L∞(R +,L 2(Ω)) and β is coercive, we prove that the solutions are bounded in the energy space, under weaker assumptions than those used by G. Prou...
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| Published in | Journal of Differential Equations Vol. 38; no. 3; pp. 422 - 440 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
01.01.1980
|
| Online Access | Get full text |
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| Summary: | This paper deals with asymptotic behavior for (weak) solutions of the equation
u
tt − Δu + β(u
t) ∋ ƒ(t, x)
, on
R
+ × Ω;
u(
t,
x) = 0, on
R
+ × ∂Ω. If
ƒ∈L∞(R
+,L
2(Ω))
and β is coercive, we prove that the solutions are bounded in the energy space, under weaker assumptions than those used by
G. Prouse in a previous work. If in addition
ƒ
t∈S
2(R
+,L
2(Ω))
and ƒ is srongly almost-periodic, we prove for
strongly monotone β that all solutions are asymptotically almost-periodic in the energy space. The assumptions made on β are much less restrictive than those made by G. Prouse: mainly, we allow β to be multivalued, and in the one-dimensional case β need not be defined everywhere. |
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| ISSN: | 0022-0396 1090-2732 |
| DOI: | 10.1016/0022-0396(80)90017-0 |