Blow-up and global asymptotics of the limit unstable Cahn--Hilliard equation

We study the asymptotic behavior of classes of global and blow-up solutions of a semilinear parabolic equation of the "limit" Cahn--Hilliard type \[u_t = -\Delta(\Delta u + |u|^{p-1}u)\quad \mbox{in} \,\,\, \ren \times \re_+, \quad p>1, \] with bounded integrable initial data. We show t...

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Veröffentlicht in:SIAM journal on mathematical analysis 2006-01, Vol.38 (1), p.64-102
Hauptverfasser: Evans, J. D., Galaktionov, V. A., Williams, J. F.
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Sprache:eng
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Zusammenfassung:We study the asymptotic behavior of classes of global and blow-up solutions of a semilinear parabolic equation of the "limit" Cahn--Hilliard type \[u_t = -\Delta(\Delta u + |u|^{p-1}u)\quad \mbox{in} \,\,\, \ren \times \re_+, \quad p>1, \] with bounded integrable initial data. We show that in some $\{p,N\}$-parameter ranges it admits a {\em countable} set of blow-up similarity patterns. The most interesting set of blow-up solutions is constructed at the first critical exponent $p=p_0=1+\frac 2N$, where the first simplest profile is shown to be stable. Unlike the blow-up case, we show that, for $p=p_0$, the set of global decaying source-type similarity solutions is {\em continuous} and determine the stable mass-branch. We prove that there exists a countable spectrum of critical exponents $\{p=p_l=1+\frac 2{N+l}, \,\, l =0,1,2,\ldots\}$ creating bifurcation branches, which play a key role in general description of solutions globally decaying as $t \to \infty$.
ISSN:0036-1410
1095-7154
DOI:10.1137/S0036141004440289