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Semilinear Neumann boundary value problems on a rectangle

Author(s): Junping Shi
Journal: Trans. Amer. Math. Soc. 354 (2002), 3117-3154.
MSC (2000): Primary 35J25, 35B32; Secondary 35J60, 34C11
Posted: April 2, 2002
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Abstract: We consider a semilinear elliptic equation

\begin{displaymath}\Delta u+\lambda f(u)=0, \;\; \mathbf{x}\in \Omega,\;\; \frac{\partial u}{\partial n }=0, \;\; {\mathbf x}\in \partial \Omega, \end{displaymath}

where $\Omega$ is a rectangle $(0,a)\times(0,b)$ in $\mathbf{R}^2$. For balanced and unbalanced $f$, we obtain partial descriptions of global bifurcation diagrams in $(\lambda,u)$ space. In particular, we rigorously prove the existence of secondary bifurcation branches from the semi-trivial solutions, which is called dimension-breaking bifurcation. We also study the asymptotic behavior of the monotone solutions when $\lambda\to\infty$. The results can be applied to the Allen-Cahn equation and some equations arising from mathematical biology.

References:

[ABF]
Alikakos, Nicholas; Bates, Peter W.; Fusco, Giorgio, Slow motion for the Cahn-Hilliard equation in one space dimension. J. Differential Equations, 90, (1991), no. 1, 81-135. MR 92a:35152

[ABrF]
Alikakos, N. D.; Bronsard, L.; Fusco, G., Slow motion in the gradient theory of phase transitions via energy and spectrum. Calc. Var. Partial Differential Equations, 6, (1998), no. 1, 39-66. MR 99d:82025

[AF1]
Alikakos, Nicholas D.; Fusco, Giorgio, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions. I. Spectral estimates. Comm. Partial Differential Equations, 19, (1994), no. 9-10, 1397-1447. MR 95j:35163

[AF2]
Alikakos, Nicholas D.; Fusco, Giorgio, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions: the motion of bubbles. Arch. Rational Mech. Anal., 141, (1998), no. 1, 1-61. MR 99d:35013

[AFK]
Alikakos, Nicholas D.; Fusco, Giorgio; Kowalczyk, Micha\l, Finite-dimensional dynamics and interfaces intersecting the boundary: equilibria and quasi-invariant manifold. Indiana Univ. Math. J., 45, (1996), no. 4, 1119-1155. MR 98b:35077

[AFS]
Alikakos, Nicholas D.; Fusco, Giorgio; Stefanopoulos, Vagelis, Critical spectrum and stability of interfaces for a class of reaction-diffusion equations. J. Differential Equations, 126, (1996), no. 1, 106-167. MR 97d:35100

[BDS]
Bates, Peter W.; Dancer, Norman E.; Shi, Junping, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimensional and instability. Advances of Differential Equations, 4, (1999), no. 1, 1-69. MR 99k:35097

[BFi]
Bates, Peter W.; Fife, Paul C., The dynamics of nucleation for the Cahn-Hilliard equation. SIAM J. Appl. Math., 53, (1993), no. 4, 990-1008. MR 94g:82034

[BFu]
Bates, Peter W.; Fusco, Giorgio, Equilibria with many nuclei for the Cahn-Hilliard equation. J. Differential Equations, 160, (2000), no. 2, 283-356. MR 2001b:35045

[BS]
Bates, Peter W.; Shi, Junping, Existence and instability of spike layer solutions to singular perturbation problems. To appear in J. Funct. Anal., (2002).

[BX1]
Bates, Peter W.; Xun, Jianping, Metastable patterns for the Cahn-Hilliard equation. I. J. Differential Equations 111 (1994), no. 2, 421-457. MR 95e:35172

[BX2]
Bates, Peter W.; Xun, Jianping, Metastable patterns for the Cahn-Hilliard equation. II. J. Differential Equations 117 (1995), no. 1, 165-216. MR 95m:35188

[BCN]
Berestycki, Henri; Caffarelli, Luis; Nirenberg, Louis, Further qualitative properties for elliptic equations in unbounded domains. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25, (1997), no. 1-2, 69-94. MR 2000e:35053

[BK]
Bronsard, Lia; Kohn, Robert V., On the slowness of phase boundary motion in one space dimension. Comm. Pure Appl. Math., 43, (1990), no. 8, 983-997. MR 91f:35023

[BrF]
Brunovský, Pavol; Fiedler, Bernold, Numbers of zeros on invariant manifolds in reaction-diffusion equations. Nonlinear Anal., 10, (1986), no. 2, 179-193. MR 88c:35080

[CGS]
Carr, Jack; Gurtin, Morton E.; Slemrod, Marshall, Structured phase transitions on a finite interval. Arch. Rational Mech. Anal., 86, (1984), no. 4, 317-351. MR 86i:80001

[CP]
Carr, J.; Pego, R. L., Metastable patterns in solutions of $u_ t=\epsilon^ 2u_ {xx}-f(u)$. Comm. Pure Appl. Math., 42, (1989), no. 5, 523-576. MR 90f:35091

[CaH]
Casten, Richard G.; Holland, Charles J., Instability results for reaction diffusion equations with Neumann boundary conditions. J. Differential Equations, 27, (1978), no. 2, 266-273. MR 80a:35064

[CI]
Chafee, N.; Infante, E. F., A bifurcation problem for a nonlinear partial differential equation of parabolic type. Applicable Anal., 4, (1974/75), 17-37. MR 55:13084

[C]
Cheng, Shiu Yuen, Eigenfunctions and nodal sets. Comment. Math. Helv., 51, (1976), no. 1, 43-55. MR 53:1661

[CR]
Crandall, Michael G.; Rabinowitz, Paul H, Bifurcation from simple eigenvalues. J. Functional Analysis, 8, (1971), 321-340. MR 44:5836

[D1]
Dancer, E. N., On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large. Proc. London Math. Soc. (3), 53, (1986), no. 3, 429-452. MR 88c:35061

[D2]
Dancer, E. N., On the uniqueness of the positive solution of a singularly perturbed problem. Rocky Mountain J. Math., 25, (1995), no. 3, 957-975. MR 96j:35021

[D3]
Dancer, E. N., On positive solutions of some singularly perturbed problems where the nonlinearity changes sign. Topol. Methods Nonlinear Anal., 5, (1995), no. 1, 141-175. MR 96i:35037

[DY1]
Dancer, E. N.; Yan, Shusen, Multipeak solutions for a singularly perturbed Neumann problem. Pacific J. Math., 189, (1999), no. 2, 241-262. MR 2000d:35010

[DY2]
Dancer, E. N.; Yan, Shusen, A singularly perturbed elliptic problem in bounded domains with nontrivial topology. Adv. Differential Equations, 4, (1999), no. 3, 347-368. MR 2000d:35009

[DFP]
Dang, Ha; Fife, Paul C.; Peletier, L. A., Saddle solutions of the bistable diffusion equation. Z. Angew. Math. Phys., 43, (1992), no. 6, 984-998. MR 94b:35041

[FKMW]
Fife, Paul C.; Kielhöfer, Hansjörg; Maier-Paape, Stanislaus; Wanner, Thomas, Perturbation of doubly periodic solution branches with applications to the Cahn-Hilliard equation. Phys. D., 100, (1997), no. 3-4, 257-278. MR 98a:35004

[FH1]
Fusco, G.; Hale, J. K. Stable equilibria in a scalar parabolic equation with variable diffusion. SIAM J. Math. Anal., 16, (1985), no. 6, 1152-1164. MR 87c:35078

[FH2]
Fusco, G.; Hale, J. K., Slow-motion manifolds, dormant instability, and singular perturbations. J. Dynamics Differential Equations, 1, (1989), no. 1, 75-94. MR 90i:35131

[GT]
Gilbarg, David; Trudinger, Neil S., Elliptic partial differential equations of second order. Second edition. Springer-Verlag, Berlin-New York, (1983). MR 86c:35035

[GG]
Ghoussoub, N.; Gui, C., On a conjecture of De Giorgi and some related problems. Math. Ann., 311 (1998), no. 3, 481-491. MR 99j:35049

[G]
Gui, Changfeng, Multipeak solutions for a semilinear Neumann problem. Duke Math. J., 84, (1996), no. 3, 739-769. MR 97i:35052

[GW1]
Gui, Changfeng; Wei, Juncheng, Multiple interior peak solutions for some singularly perturbed Neumann problems. J. Differential Equations, 158, (1999), no. 1, 1-27. MR 2000g:35035

[GW2]
Gui, Changfeng; Wei, Juncheng, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems. Canad. J. Math., 52, (2000), no. 3, 522-538. MR 2001b:35023

[GWW]
Gui, Changfeng; Wei, Juncheng; Winter, Matthias, Multiple boundary peak solutions for some singularly perturbed Neumann problems. Ann. Inst. H. Poincaré Anal. Non Linéaire, 17, (2000), no. 1, 47-82. MR 2001a:35018

[HK]
Healey, Timothy J.; Kielhöfer, H, Symmetry and nodal properties in the global bifurcation analysis of quasi-linear elliptic equations. Arch. Rational Mech. Anal., 113, (1990), no. 4, 299-311. MR 92d:35042

[H]
Henry, Daniel, Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, (1981). MR 83j:35084

[Ki]
Kielhöfer, Hansjörg, Pattern formation of the stationary Cahn-Hilliard model. Proc. Roy. Soc. Edinburgh Sect. A, 127, (1997), no. 6, 1219-1243. MR 99b:35055

[KS]
Kohn, Robert V.; Sternberg, Peter, Local minimisers and singular perturbations. Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), no. 1-2, 69-84. MR 90c:49021

[Kor]
Korman, Philip, Exact multiplicity of solutions for a class of semilinear Neumann problems. Preprint, (2000).

[Kow]
Kowalczyk, Micha\l, Multiple spike layers in the shadow Gierer-Meinhardt system: existence of equilibria and the quasi-invariant manifold. Duke Math. J., 98, (1999), no. 1, 59-111. MR 2000d:35073

[L]
Li, Yanyan, On a singular perturbed equation with Neumann boundary condition. Comm. Partial Differential Equations, 23, (1998), no. 3-4, 487-545. MR 2000a:35013

[LNT]
Lin, C.-S.; Ni, W.-M.; Takagi, I., Large amplitude stationary solutions to a chemotaxis system. J. Differential Equations, 72, (1988), no. 1, 1-27. MR 89e:35075

[M1]
Matano, Hiroshi, Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ. Res. Inst. Math. Sci., 15, (1979), no. 2, 401-454. MR 80m:35046

[M2]
Matano, Hiroshi, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 29, (1982), no. 2, 401-441. MR 84m:35060

[MM]
Maier-Paape, Stanislaus; Miller, Ulrich, The set of equilibria for the Allen-Cahn equation on the square. Preprint, (2000).

[Me]
Mei, Zhen, Solution branches of a semilinear elliptic problem at corank-$2$ bifurcation points with Neumann boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A, 123, (1993), no. 3, 479-495. MR 94j:35025

[MS]
Mizoguchi, Noriko; Suzuki, Takashi, Equations of gas combustion: $S$-shaped bifurcation and mushrooms. J. Differential Equations, 134, (1997), no. 2, 183-215. MR 97m:35090

[N]
Ni, Wei-Ming, Diffusion, cross-diffusion, and their spike-layer steady states. Notices Amer. Math. Soc., 45, (1998), no. 1, 9-18. MR 99a:35132

[NT1]
Ni, Wei-Ming; Takagi, Izumi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type. Trans. Amer. Math. Soc., 297, (1986), no. 1, 351-368. MR 87k:35091

[NT2]
Ni, Wei-Ming; Takagi, Izumi, On the shape of least-energy solutions to a semilinear Neumann problem. Comm. Pure Appl. Math., 44, (1991), no. 7, 819-851. MR 92i:35052

[NT3]
Ni, Wei-Ming; Takagi, Izumi, Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J., 70, (1993), no. 2, 247-281. MR 94h:35072

[OS1]
Ouyang Tiancheng; Shi, Junping, Exact multiplicity of positive solutions for a class of semilinear problem. Jour. Differential Equations, 146, no. 1, 121-156, (1998). MR 99f:35061

[OS2]
Ouyang Tiancheng; Shi, Junping, Exact multiplicity of positive solutions for a class of semilinear problems: II. Jour. Differential Equations, 158, no. 1, 94-151, (1999). MR 2001b:35117

[PT]
Padilla, Pablo; Tonegawa, Yoshihiro, On the convergence of stable phase transitions. Comm. Pure Appl. Math., 51, (1998), no. 6, 551-579. MR 99b:58054

[R]
Rabinowitz, Paul H., Some global results for nonlinear eigenvalue problems. J. Func. Anal., 7, (1971), 487-513. MR 46:745

[Sc]
Schaaf, Renate, Global solution branches of two-point boundary value problems. Lecture Notes in Mathematics, 1458. Springer-Verlag, Berlin, (1990). MR 92a:34003

[S1]
Shi, Junping, Persistence and bifurcation of degenerate solutions. Jour. Func. Anal., 169, no. 2, 494-531, (1999). MR 2001h:47115

[S2]
Shi, Junping, Exact multiplicity of solutions to superlinear and sublinear problems. To appear in Nonlinear Anal., (2001).

[S3]
Shi, Junping, Blow-up points of solution curves for a semilinear problem. Topological Meth. of Nonlinear Anal., 15, no. 2, 251-266, (2000). MR 2002b:35060

[S4]
Shi, Junping, Saddle solutions of the balanced bistable diffusion equation. To appear in Comm. Pure Appl. Math., (2002).

[S5]
Shi, Junping, Entire solutions and solutions to singularly perturbed semilinear problems with unbalanced nonlinearities. In preparation.

[SS]
Shi, Junping; Shivaji, Ratnasingham, Global bifurcation for concave semipositon problems. To appear in Recent Advances in Evolution Equations, (2002).

[SWj]
Shi, Junping; Wang, Junping, Morse indices and exact multiplicity of solutions to semilinear elliptic problems. Proc. of Amer. Math. Soc., 127, (1999), no. 12, 3685-3695. MR 2000d:35082

[SmW]
Smoller, J.; Wasserman, A., Global bifurcation of steady-state solutions. J. Differential Equations, 39, (1981), no. 2, 269-290; errata, ibid. 77, (1989), 199-202. MR 82d:58056; MR 90f:58136

[T]
Taniguchi, Masaharu, Bifurcation from flat-layered solutions to reaction diffusion systems in two space dimensions. J. Math. Sci. Univ. Tokyo, 1, (1994), no. 2, 339-367. MR 96b:35109

[TN]
Taniguchi, Masaharu; Nishiura, Yasumasa, Instability of planar interfaces in reaction-diffusion systems. SIAM J. Math. Anal. 25, (1994), no. 1, 99-134. MR 94b:35029

[W]
Wei, Juncheng, On the boundary spike layer solutions to a singularly perturbed Neumann problem. J. Differential Equations, 134, (1997), no. 1, 104-133. MR 98e:35076

[WW1]
Wei, Juncheng; Winter, Matthias, Stationary solutions for the Cahn-Hilliard equation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 15, (1998), no. 4, 459-492. MR 2000b:35093

[WW2]
Wei, Juncheng; Winter, Matthias, Multi-peak solutions for a wide class of singular perturbation problems. J. London Math. Soc., 59, (1999), no. 2, 585-606. MR 2000h:35012

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Additional Information:

Junping Shi
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187, and Department of Mathematics, Harbin Normal University, Harbin, Heilongjiang, P. R. China 150080
Email: shij@math.wm.edu

DOI: 10.1090/S0002-9947-02-03007-6
PII: S 0002-9947(02)03007-6
Keywords: Semilinear elliptic equations, secondary bifurcations, global bifurcation diagrams, asymptotic behavior of solutions
Received by editor(s): April 17, 2001
Posted: April 2, 2002
Copyright of article: Copyright 2002, American Mathematical Society

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