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 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 $t \to \infty$.

[1]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[2]  Lawrence E. Payne,et al.  Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time , 1974 .

[3]  J. Hale.  Asymptotic Behavior of Dissipative Systems , 1988 .

[4]  M. A. Krasnoselʹskii,et al.  Geometrical Methods of Nonlinear Analysis , 1984 .

[5]  Amy Novick-Cohen,et al.  Blow up and growth in the directional solidification of dilute binary alloys , 1992 .

[6]  Robert D. Russell,et al.  Moving Mesh Methods for Problems with Blow-Up , 1996, SIAM J. Sci. Comput..

[7]  A. Lunardi.  Analytic Semigroups and Optimal Regularity in Parabolic Problems , 2003 .

[8]  Arieh Iserles,et al.  On the Global Error of Discretization Methods for Highly-Oscillatory Ordinary Differential Equations , 2002 .

[9]  Victor A. Galaktionov,et al.  The problem of blow-up in nonlinear parabolic equations , 2002 .

[10]  Andrea L. Bertozzi,et al.  Singularities in a modified Kuramoto-Sivashinsky equation describing interface motion for phase transition , 1995 .

[11]  Andrea L. Bertozzi,et al.  Finite-time blow-up of solutions of some long-wave unstable thin film equations , 2000 .

[12]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[13]  Charles M. Elliott,et al.  On the Cahn-Hilliard equation , 1986 .

[14]  Arieh Iseries,et al.  Think globally, act locally: solving highly-oscillatory ordinary differential equations , 2002 .

[15]  Mary C. Pugh,et al.  Long-wave instabilities and saturation in thin film equations , 1998 .

[16]  J. F. Williams,et al.  Self-Similar Blow-Up in Higher-Order Semilinear Parabolic Equations , 2004, SIAM J. Appl. Math..

[17]  Howard A. Levine,et al.  Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put=−Au+ℱ(u) , 1973 .

[18]  M. Kreĭn,et al.  Introduction to the theory of linear nonselfadjoint operators , 1969 .

[19]  A. P. Mikhailov,et al.  Blow-Up in Quasilinear Parabolic Equations , 1995 .

[20]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[21]  M. Naderi.  Think globally... , 2004, HIV prevention plus!.

[22]  Weizhang Huang,et al.  A moving collocation method for solving time dependent partial differential equations , 1996 .

[23]  Thomas P. Witelski,et al.  Blowup and dissipation in a critical-case unstable thin film equation , 2004, European Journal of Applied Mathematics.

[24]  J. F. Williams,et al.  Blow-up in a fourth-order semilinear parabolic equation from explosion-convection theory , 2003, European Journal of Applied Mathematics.

[25]  Victor A. Galaktionov,et al.  Global solutions of higher-order semilinear parabolic equations in the supercritical range , 2002, Advances in Differential Equations.

[26]  J. F. Williams,et al.  On very singular similarity solutions of a higher-order semilinear parabolic equationResearch suppor , 2004 .

[27]  A. Iserles.  On the numerical quadrature of highly‐oscillating integrals I: Fourier transforms , 2004 .

[28]  Robert D. Russell,et al.  Adaptive mesh movement — the MMPDE approach and its applications , 2001 .

[29]  Fred B. Weissler,et al.  Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations , 1986 .

[30]  J. C Kurtz.  Weighted Sobolev spaces with applications to singular nonlinear boundary value problems , 1983 .

[31]  J. R. Ockendon,et al.  SIMILARITY, SELF‐SIMILARITY AND INTERMEDIATE ASYMPTOTICS , 1980 .

[32]  L. Segel,et al.  Nonlinear aspects of the Cahn-Hilliard equation , 1984 .