2015 / December Volume 10 No.4
Asymptotic Profile of Solutions to a Hyperbolic Cahn-Hilliard Equation
| Published Date |
2015 / December
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|---|---|
| Title | Asymptotic Profile of Solutions to a Hyperbolic Cahn-Hilliard Equation |
| Author | |
| Keyword | |
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| Pagination | 479-539 |
| Abstract | We study the initial value problem for a hyperbolic Cahn-Hilliard equation
in n-dimensional space. The dissipative structure of our linearized equation
is of the regularity-loss type. We overcome the difficulty caused by the
regularity-loss property by introducing a set of suitable time-weighted spaces
and prove the global existence and optimal decay of solutions under smallness
and enough regularity assumptions on the initial data. Moreover, we
investigate the asymptotic behavior of our nonlinear solutions as t → $\infty$.
When n $\geq$ 3, they are asymptotic to the linear diffusion wave expressed by
the fundamental solution of the equation $v_{t}$ +$\Delta ^{2}v$ = 0. On the other hand,
when n = 1 or n = 2, they are asymptotic to the nonlinear diffusion wave
which can be expressed in terms of the self-similar solution of the equation
$v_{t}$ + $\Delta ^{2}v$ = $\Delta$ $v^{1+\frac{2}{n}}$.
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| AMS Subject Classification |
35L30, 35L76, 35B40, 35C06.
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| Received |
2015-06-08
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| Accepted |
2015-06-08
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