Semilinear Neumann boundary value problems on a rectangle

Shi, Junping

doi:10.1090/S0002-9947-02-03007-6

Semilinear Neumann boundary value problems on a rectangle

Author: Junping Shi
Journal: Trans. Amer. Math. Soc. 354 (2002), 3117-3154
MSC (2000): Primary 35J25, 35B32; Secondary 35J60, 34C11
DOI: https://doi.org/10.1090/S0002-9947-02-03007-6
Published electronically: April 2, 2002
MathSciNet review: 1897394
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Abstract: We consider a semilinear elliptic equation \begin{equation*} \Delta u+\lambda f(u)=0, \;\; \mathbf {x}\in \Omega ,\;\; \frac {\partial u}{\partial n }=0, \;\; \mathbf {x}\in \partial \Omega , \end{equation*} 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.

Nicholas Alikakos, Peter W. Bates, and Giorgio Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differential Equations 90 (1991), no. 1, 81–135. MR 1094451, DOI https://doi.org/10.1016/0022-0396%2891%2990163-4
N. D. Alikakos, L. Bronsard, and G. Fusco, 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 1488493, DOI https://doi.org/10.1007/s005260050081
Nicholas D. Alikakos and Giorgio Fusco, 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 1294466, DOI https://doi.org/10.1080/03605309408821059
Nicholas D. Alikakos and Giorgio Fusco, 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 1613496, DOI https://doi.org/10.1007/s002050050072
Nicholas D. Alikakos, Giorgio Fusco, and Michał Kowalczyk, Finite-dimensional dynamics and interfaces intersecting the boundary: equilibria and quasi-invariant manifold, Indiana Univ. Math. J. 45 (1996), no. 4, 1119–1155. MR 1444480, DOI https://doi.org/10.1512/iumj.1996.45.1123
Nicholas D. Alikakos, Giorgio Fusco, and Vagelis Stefanopoulos, Critical spectrum and stability of interfaces for a class of reaction-diffusion equations, J. Differential Equations 126 (1996), no. 1, 106–167. MR 1382059, DOI https://doi.org/10.1006/jdeq.1996.0046
Peter W. Bates, E. Norman Dancer, and Junping Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Differential Equations 4 (1999), no. 1, 1–69. MR 1667283
Peter W. Bates and Paul C. Fife, The dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math. 53 (1993), no. 4, 990–1008. MR 1232163, DOI https://doi.org/10.1137/0153049
Jian Wei Hu, A strongly discrete maximum principle and a domain decomposition method for nonselfadjoint elliptic problems, Math. Numer. Sin. 21 (1999), no. 3, 283–292 (Chinese, with English summary). MR 1762986

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

Peter W. Bates and Jian Ping Xun, Metastable patterns for the Cahn-Hilliard equation. I, J. Differential Equations 111 (1994), no. 2, 421–457. MR 1284421, DOI https://doi.org/10.1006/jdeq.1994.1089

Peter W. Bates and Jian Ping Xun, Metastable patterns for the Cahn-Hilliard equation. II. Layer dynamics and slow invariant manifold, J. Differential Equations 117 (1995), no. 1, 165–216. MR 1320187, DOI https://doi.org/10.1006/jdeq.1995.1052

Henri Berestycki, Luis Caffarelli, and Louis Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 69–94 (1998). Dedicated to Ennio De Giorgi. MR 1655510

Lia Bronsard and Robert V. Kohn, On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math. 43 (1990), no. 8, 983–997. MR 1075075, DOI https://doi.org/10.1002/cpa.3160430804

Pavol Brunovský and Bernold Fiedler, Numbers of zeros on invariant manifolds in reaction-diffusion equations, Nonlinear Anal. 10 (1986), no. 2, 179–193. MR 825216, DOI https://doi.org/10.1016/0362-546X%2886%2990045-3

Jack Carr, Morton E. Gurtin, and Marshall Slemrod, Structured phase transitions on a finite interval, Arch. Rational Mech. Anal. 86 (1984), no. 4, 317–351. MR 759767, DOI https://doi.org/10.1007/BF00280031

J. Carr and R. L. Pego, Metastable patterns in solutions of $u_t=\epsilon ^2u_{xx}-f(u)$, Comm. Pure Appl. Math. 42 (1989), no. 5, 523–576. MR 997567, DOI https://doi.org/10.1002/cpa.3160420502

Richard G. Casten and Charles J. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Differential Equations 27 (1978), no. 2, 266–273. MR 480282, DOI https://doi.org/10.1016/0022-0396%2878%2990033-5

N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal. 4 (1974/75), 17–37. MR 440205, DOI https://doi.org/10.1080/00036817408839081

Shiu Yuen Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), no. 1, 43–55. MR 397805, DOI https://doi.org/10.1007/BF02568142

Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340. MR 0288640, DOI https://doi.org/10.1016/0022-1236%2871%2990015-2

E. N. Dancer, 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 868453, DOI https://doi.org/10.1112/plms/s3-53.3.429

E. N. Dancer, On the uniqueness of the positive solution of a singularly perturbed problem, Rocky Mountain J. Math. 25 (1995), no. 3, 957–975. MR 1357103, DOI https://doi.org/10.1216/rmjm/1181072198

E. N. Dancer, On positive solutions of some singularly perturbed problems where the nonlinearity changes sign, Topol. Methods Nonlinear Anal. 5 (1995), no. 1, 141–175. Contributions dedicated to Ky Fan on the occasion of his 80th birthday. MR 1350350, DOI https://doi.org/10.12775/TMNA.1995.009

E. N. Dancer and Shusen Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math. 189 (1999), no. 2, 241–262. MR 1696122, DOI https://doi.org/10.2140/pjm.1999.189.241

E. N. Dancer and Shusen Yan, A singularly perturbed elliptic problem in bounded domains with nontrivial topology, Adv. Differential Equations 4 (1999), no. 3, 347–368. MR 1671254

Ha Dang, Paul C. Fife, and L. A. Peletier, Saddle solutions of the bistable diffusion equation, Z. Angew. Math. Phys. 43 (1992), no. 6, 984–998. MR 1198672, DOI https://doi.org/10.1007/BF00916424

Paul C. Fife, Hansjörg Kielhöfer, Stanislaus Maier-Paape, and Thomas Wanner, Perturbation of doubly periodic solution branches with applications to the Cahn-Hilliard equation, Phys. D 100 (1997), no. 3-4, 257–278. MR 1429160, DOI https://doi.org/10.1016/S0167-2789%2896%2900190-X

G. Fusco and J. K. Hale, Stable equilibria in a scalar parabolic equation with variable diffusion, SIAM J. Math. Anal. 16 (1985), no. 6, 1152–1164. MR 807902, DOI https://doi.org/10.1137/0516085

G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynam. Differential Equations 1 (1989), no. 1, 75–94. MR 1010961, DOI https://doi.org/10.1007/BF01048791

David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190

N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), no. 3, 481–491. MR 1637919, DOI https://doi.org/10.1007/s002080050196

Changfeng Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J. 84 (1996), no. 3, 739–769. MR 1408543, DOI https://doi.org/10.1215/S0012-7094-96-08423-9

Changfeng Gui and Juncheng Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations 158 (1999), no. 1, 1–27. MR 1721719, DOI https://doi.org/10.1016/S0022-0396%2899%2980016-3

Changfeng Gui and Juncheng Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math. 52 (2000), no. 3, 522–538. MR 1758231, DOI https://doi.org/10.4153/CJM-2000-024-x

Changfeng Gui, Juncheng Wei, and Matthias Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 1, 47–82 (English, with English and French summaries). MR 1743431, DOI https://doi.org/10.1016/S0294-1449%2899%2900104-3

Timothy J. Healey and Hansjörg Kielhöfer, 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 1079191, DOI https://doi.org/10.1007/BF00374696

Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244

Hansjörg Kielhöfer, Pattern formation of the stationary Cahn-Hilliard model, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 6, 1219–1243. MR 1489434, DOI https://doi.org/10.1017/S0308210500027037

Robert V. Kohn and Peter Sternberg, Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), no. 1-2, 69–84. MR 985990, DOI https://doi.org/10.1017/S0308210500025026

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

Michał Kowalczyk, 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 1687412, DOI https://doi.org/10.1215/S0012-7094-99-09802-2

Yan Yan Li, On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differential Equations 23 (1998), no. 3-4, 487–545. MR 1620632, DOI https://doi.org/10.1080/03605309808821354

C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988), no. 1, 1–27. MR 929196, DOI https://doi.org/10.1016/0022-0396%2888%2990147-7

Hiroshi Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci. 15 (1979), no. 2, 401–454. MR 555661, DOI https://doi.org/10.2977/prims/1195188180

Hiroshi Matano, 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 672070

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

Zhen Mei, 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 1226613, DOI https://doi.org/10.1017/S0308210500025841

Noriko Mizoguchi and Takashi Suzuki, Equations of gas combustion: $S$-shaped bifurcation and mushrooms, J. Differential Equations 134 (1997), no. 2, 183–215. MR 1432094, DOI https://doi.org/10.1006/jdeq.1996.3221

Wei-Ming Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc. 45 (1998), no. 1, 9–18. MR 1490535

Wei-Ming Ni and Izumi Takagi, 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 849484, DOI https://doi.org/10.1090/S0002-9947-1986-0849484-2

Wei-Ming Ni and Izumi Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), no. 7, 819–851. MR 1115095, DOI https://doi.org/10.1002/cpa.3160440705

Wei-Ming Ni and Izumi Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), no. 2, 247–281. MR 1219814, DOI https://doi.org/10.1215/S0012-7094-93-07004-4

Tiancheng Ouyang and Junping Shi, Exact multiplicity of positive solutions for a class of semilinear problems, J. Differential Equations 146 (1998), no. 1, 121–156. MR 1625731, DOI https://doi.org/10.1006/jdeq.1998.3414

Tiancheng Ouyang and Junping Shi, Exact multiplicity of positive solutions for a class of semilinear problem. II, J. Differential Equations 158 (1999), no. 1, 94–151. MR 1721723, DOI https://doi.org/10.1016/S0022-0396%2899%2980020-5

Pablo Padilla and Yoshihiro Tonegawa, On the convergence of stable phase transitions, Comm. Pure Appl. Math. 51 (1998), no. 6, 551–579. MR 1611144, DOI https://doi.org/10.1002/%28SICI%291097-0312%28199806%2951%3A6%3C551%3A%3AAID-CPA1%3E3.0.CO%3B2-6

Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513. MR 0301587, DOI https://doi.org/10.1016/0022-1236%2871%2990030-9

Renate Schaaf, Global solution branches of two-point boundary value problems, Lecture Notes in Mathematics, vol. 1458, Springer-Verlag, Berlin, 1990. MR 1090827

Junping Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal. 169 (1999), no. 2, 494–531. MR 1730558, DOI https://doi.org/10.1006/jfan.1999.3483

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

Junping Shi, Blow up points of solution curves for a semilinear problem, Topol. Methods Nonlinear Anal. 15 (2000), no. 2, 251–266. MR 1784141, DOI https://doi.org/10.12775/TMNA.2000.019

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

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

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

Junping Shi and Junping Wang, Morse indices and exact multiplicity of solutions to semilinear elliptic problems, Proc. Amer. Math. Soc. 127 (1999), no. 12, 3685–3695. MR 1694880, DOI https://doi.org/10.1090/S0002-9939-99-05542-2

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. ;

Masaharu Taniguchi, 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 1317464

Zheng Yuan Li and Qi Xiao Ye, On multiple positive periodic solutions for parabolic system, Gaoxiao Yingyong Shuxue Xuebao 7 (1992), no. 3, 367–380. MR 1193569

Juncheng Wei, On the boundary spike layer solutions to a singularly perturbed Neumann problem, J. Differential Equations 134 (1997), no. 1, 104–133. MR 1429093, DOI https://doi.org/10.1006/jdeq.1996.3218

Juncheng Wei and Matthias Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), no. 4, 459–492 (English, with English and French summaries). MR 1632937, DOI https://doi.org/10.1016/S0294-1449%2898%2980031-0

Juncheng Wei and Matthias Winter, Multi-peak solutions for a wide class of singular perturbation problems, J. London Math. Soc. (2) 59 (1999), no. 2, 585–606. MR 1709667, DOI https://doi.org/10.1112/S002461079900719X