Clustering layers and boundary layers in spatially inhomogeneous phase transition problems. (Couches limites dans des problèmes de transition de phase spatialement inhomogènes.) (English) Zbl 1114.35005

Summary: The authors study the existence of solutions with multiple clustered transition layers for the following inhomogeneous problem of Allen-Cahn type: \[ -\varepsilon^2u_{xx}+W_u(x,u)=0 \text{ in } (0,1),\quad u_x(0)=u_x(1)=0,\tag{1} \] where \(\varepsilon>0\) is a small parameter and \(W(x,u)\) is a double-well potential. A typical example of \(W(x,u)\) is \({1\over4}h(x)^2(u^2-1)^2\). In particular, they show the existence of solutions with clustered layers and layers.


35B25 Singular perturbations in context of PDEs
35Q35 PDEs in connection with fluid mechanics
34B15 Nonlinear boundary value problems for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
47J30 Variational methods involving nonlinear operators
76T99 Multiphase and multicomponent flows
Full Text: DOI Numdam EuDML


[1] S. Ai, S.P. Hastings, A shooting approach to layers and chaos in a forced duffing equation, I, Preprint · Zbl 1025.34015
[2] Ambrosetti, A.; Badiale, M.; Cingolani, S., Semiclassical states of nonlinear Schrödinger equations, Arch. rat. mech. anal., 140, 285-300, (1997) · Zbl 0896.35042
[3] Angenent, S.B.; Mallet-Paret, J.; Peletier, L.A., Stable transition layers in a semilinear boundary value problem, J. differential equations, 67, 212-242, (1987) · Zbl 0634.35041
[4] Chen, C.-N., Multiple solutions for a class of nonlinear sturm – liouville problems on the half line, J. differential equations, 85, 236-275, (1990) · Zbl 0703.34032
[5] del Pino, M.; Felmer, P., Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. var. PDE, 4, 121-137, (1996) · Zbl 0844.35032
[6] del Pino, M.; Felmer, P., Multi-peak bound states of nonlinear Schrödinger equations, Ann. IHP, analyse nonlinéaire, 15, 127-149, (1998) · Zbl 0901.35023
[7] Fleor, A.; Weinstein, A., Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. funct. anal., 69, 3, 397-408, (1986) · Zbl 0613.35076
[8] Gedeon, T.; Kokubu, H.; Mischaikow, K.; Oka, H., Chaotic solutions in slowly varying perturbations of Hamiltonian systems with applications to shallow water sloshing, J. dynam. differential equations, 14, 63-84, (2002) · Zbl 1005.37028
[9] Gui, C., Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Comm. partial differential equations, 21, 787-820, (1996) · Zbl 0857.35116
[10] Gui, C.; Wei, J., Multiple interior peak solutions for some singularly perturbed Neumann problems, J. differential equations, 158, 1-27, (1999) · Zbl 1061.35502
[11] Hemple, J.A., Multiple solutions for a class of nonlinear boundary value problems, Indiana univ. math. J., 20, 11, 983-996, (1971) · Zbl 0225.35045
[12] Kang, X.; Wei, J., On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. differential equations, 5, 899-928, (2000) · Zbl 1217.35065
[13] Kath, W.L., Slowly varying phase planes and boundary-layer theory, Stud. appl. math., 72, 3, 221-239, (1985) · Zbl 0586.76047
[14] Li, Y.-Y., On a singularly perturbed elliptic equation, Adv. differential equations, 2, 955-980, (1997) · Zbl 1023.35500
[15] K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation, J. Differential Equations, to appear · Zbl 1042.35575
[16] Nakashima, K., Stable transition layers in a balanced bistable equation, Differential integral equations, 13, 1025-1038, (2000) · Zbl 0981.34011
[17] Oh, Y.-G., Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V)a, Comm. partial differential equations, 13, 1499-1519, (1988) · Zbl 0702.35228
[18] Oh, Y.-G., Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials, Comm. math. phys., 121, 11-33, (1989) · Zbl 0693.35132
[19] Oh, Y.-G., On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. math. phys., 131, 223-253, (1990) · Zbl 0753.35097
[20] Rabinowitz, P.H., On a class of nonlinear Schrödinger equations, Z. angew. math. phys., 43, 270-291, (1992) · Zbl 0763.35087
[21] Wang, X., On concentration of positive bound states of nonlinear Schrödinger equations, Comm. math. phys., 153, 229-244, (1993) · Zbl 0795.35118
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.