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Hopf bifurcation for some analytic differential systems in $\R^3$ via averaging theory
Global bifurcation and stable two-phase separation for a phase field model in a disk
1. | Department of Mathematics, Tokyo Institute of Technology, O-Okayama, Meguro-ku, Tokyo 152-8551, Japan |
$E(u)=\int_{D}(\frac{|\nabla u|^2}{2}+\lambda W(u)) dx\ \ $subject to$ \ \ m=\frac{1}{|D|}\int_Dudx $
and study the global structure of critical points. We show that for an arbitrary potential $W\in C^4$ every level set of every nonconstant local minimizer is a $C^1$-curve and it divides $D$ into exactly two simply connected subdomains. Next we consider the case $W(u)=(u^2-1)^2/4$ (Cahn-Hilliard equation). When $\lambda$ varies and $m$ is fixed, we show that this problem has an unbounded continuum of critical points. When $m$ varies and $\lambda$ is fixed, we show that this problem has a bounded continuum meeting at two different points on the trivial branch. Moreover, we show that in each case a bifurcating critical point is stable (a local minimizer) near the bifurcation point in a certain parameter range. The main technique is the nodal curve which relates the shape with the Morse index. We do not use a small parameter or the $\Gamma$-convergence technique.
References:
[1] |
J. Carr, M. Gurtin and M. Slemrod, Structured phase transitions on a finite interval, Arch. Rational Mech. Anal., 86 (1984), 317-351.
doi: 10.1007/BF00280031. |
[2] |
R. Casten and C. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Differential Equations, 27 (1978), 266-273.
doi: 10.1016/0022-0396(78)90033-5. |
[3] |
P. Freitas and C. Rocha, Lyapunov functionals and stability for FitzHugh-Nagumo systems, J. Differential Equations, 169 (2001), 208-227.
doi: 10.1006/jdeq.2000.3901. |
[4] |
M. Gurtin and H. Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids, Quart. Appl. Math., 46 (1988), 301-317. |
[5] |
M. Grinfeld and A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 351-370. |
[6] |
B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and M. P. Owen, Nodal sets for groundstates of Schrödinger operators with zero magnetic field in non-simply connected domains, Comm. Math. Phys., 202 (1999), 629-649.
doi: 10.1007/s002200050599. |
[7] |
P. Hartman and A. Wintner, On the local behavior of solutions of non-parabolic partial differential equations, Amer. J. Math., 75 (1953), 449-476.
doi: 10.2307/2372496. |
[8] |
H. Kielhöfer, Pattern formation of the stationary Cahn-Hilliard model, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1219-1243. |
[9] |
H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004, viii+346 pp. |
[10] |
S. Kosugi, Y. Morita and S. Yotsutani, Stationary solutions to the one-dimensional Cahn-Hilliard equation: proof by the complete elliptic integrals, Discrete Contin. Dyn. Syst., 19 (2007), 609-629.
doi: 10.3934/dcds.2007.19.609. |
[11] |
S. Luckhaus and L. Modica, The Gibbs-Thompson relation within the gradient theory of phase transitions, Arch. Rational Mech. Anal., 107 (1989), 71-83.
doi: 10.1007/BF00251427. |
[12] |
H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15 (1979), 401-454.
doi: 10.2977/prims/1195188180. |
[13] | |
[14] |
B. McCartin, Eigenstructure of the equilateral triangle. II. The Neumann problem, Math. Probl. Eng., 8 (2002), 517-539.
doi: 10.1080/1024123021000053664. |
[15] |
Y. Miyamoto, An instability criterion for activator-inhibitor systems in a two-dimensional ball, J. Differential Equations, 229 (2006), 494-508.
doi: 10.1016/j.jde.2006.03.015. |
[16] |
Y. Miyamoto, An instability criterion for activator-inhibitor systems in a two-dimensional ball II, J. Differential Equations, 239 (2007), 61-71.
doi: oi:10.1016/j.jde.2007.05.006. |
[17] |
Y. Miyamoto, On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains, Quart. Appl. Math., 65 (2007), 357-374. |
[18] |
Y. Miyamoto, Global branches of non-radially symmetric solutions to a semilinear Neumann problem in a disk, J. Funct. Anal., 256 (2009), 747-776.
doi: 10.1016/j.jfa.2008.11.023. |
[19] |
Y. Miyamoto, The "hot spots" conjecture for a certain class of planar convex domains, J. Math. Phys., 50 (2009), 103530, 7pp. |
[20] |
L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[21] |
S. Maier-Paape and U. Miller, Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square, Discrete Contin. Dyn. Syst., 15 (2006), 1137-1153.
doi: 10.3934/dcds.2006.15.1137. |
[22] |
Y. Nishiura, Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit, Dynamics Reported, 3 (1994), 25-103. |
[23] |
W. M. Ni, P. Poláčik and E. Yanagida, Monotonicity of stable solutions in shadow systems, Trans. Amer. Math. Soc., 353 (2001), 5057-5069.
doi: 10.1090/S0002-9947-01-02880-X. |
[24] |
P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260.
doi: 10.1007/BF00253122. |
[25] |
T. Suzuki and S. Tasaki, Stationary Fix-Caginalp equation with non-local term, Nonlinear Anal., 71 (2009), 1329-1349.
doi: 10.1016/j.na.2008.12.007. |
[26] |
P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400.
doi: 10.1007/s002050050081. |
[27] |
show all references
References:
[1] |
J. Carr, M. Gurtin and M. Slemrod, Structured phase transitions on a finite interval, Arch. Rational Mech. Anal., 86 (1984), 317-351.
doi: 10.1007/BF00280031. |
[2] |
R. Casten and C. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Differential Equations, 27 (1978), 266-273.
doi: 10.1016/0022-0396(78)90033-5. |
[3] |
P. Freitas and C. Rocha, Lyapunov functionals and stability for FitzHugh-Nagumo systems, J. Differential Equations, 169 (2001), 208-227.
doi: 10.1006/jdeq.2000.3901. |
[4] |
M. Gurtin and H. Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids, Quart. Appl. Math., 46 (1988), 301-317. |
[5] |
M. Grinfeld and A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 351-370. |
[6] |
B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and M. P. Owen, Nodal sets for groundstates of Schrödinger operators with zero magnetic field in non-simply connected domains, Comm. Math. Phys., 202 (1999), 629-649.
doi: 10.1007/s002200050599. |
[7] |
P. Hartman and A. Wintner, On the local behavior of solutions of non-parabolic partial differential equations, Amer. J. Math., 75 (1953), 449-476.
doi: 10.2307/2372496. |
[8] |
H. Kielhöfer, Pattern formation of the stationary Cahn-Hilliard model, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1219-1243. |
[9] |
H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004, viii+346 pp. |
[10] |
S. Kosugi, Y. Morita and S. Yotsutani, Stationary solutions to the one-dimensional Cahn-Hilliard equation: proof by the complete elliptic integrals, Discrete Contin. Dyn. Syst., 19 (2007), 609-629.
doi: 10.3934/dcds.2007.19.609. |
[11] |
S. Luckhaus and L. Modica, The Gibbs-Thompson relation within the gradient theory of phase transitions, Arch. Rational Mech. Anal., 107 (1989), 71-83.
doi: 10.1007/BF00251427. |
[12] |
H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15 (1979), 401-454.
doi: 10.2977/prims/1195188180. |
[13] | |
[14] |
B. McCartin, Eigenstructure of the equilateral triangle. II. The Neumann problem, Math. Probl. Eng., 8 (2002), 517-539.
doi: 10.1080/1024123021000053664. |
[15] |
Y. Miyamoto, An instability criterion for activator-inhibitor systems in a two-dimensional ball, J. Differential Equations, 229 (2006), 494-508.
doi: 10.1016/j.jde.2006.03.015. |
[16] |
Y. Miyamoto, An instability criterion for activator-inhibitor systems in a two-dimensional ball II, J. Differential Equations, 239 (2007), 61-71.
doi: oi:10.1016/j.jde.2007.05.006. |
[17] |
Y. Miyamoto, On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains, Quart. Appl. Math., 65 (2007), 357-374. |
[18] |
Y. Miyamoto, Global branches of non-radially symmetric solutions to a semilinear Neumann problem in a disk, J. Funct. Anal., 256 (2009), 747-776.
doi: 10.1016/j.jfa.2008.11.023. |
[19] |
Y. Miyamoto, The "hot spots" conjecture for a certain class of planar convex domains, J. Math. Phys., 50 (2009), 103530, 7pp. |
[20] |
L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[21] |
S. Maier-Paape and U. Miller, Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square, Discrete Contin. Dyn. Syst., 15 (2006), 1137-1153.
doi: 10.3934/dcds.2006.15.1137. |
[22] |
Y. Nishiura, Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit, Dynamics Reported, 3 (1994), 25-103. |
[23] |
W. M. Ni, P. Poláčik and E. Yanagida, Monotonicity of stable solutions in shadow systems, Trans. Amer. Math. Soc., 353 (2001), 5057-5069.
doi: 10.1090/S0002-9947-01-02880-X. |
[24] |
P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260.
doi: 10.1007/BF00253122. |
[25] |
T. Suzuki and S. Tasaki, Stationary Fix-Caginalp equation with non-local term, Nonlinear Anal., 71 (2009), 1329-1349.
doi: 10.1016/j.na.2008.12.007. |
[26] |
P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400.
doi: 10.1007/s002050050081. |
[27] |
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