Periodic solutions for the Allen-Cahn equation. (English) Zbl 1422.35004
Summary: In this paper we consider the time-periodic Allen-Cahn equation subject to homogeneous boundary value condition and time-periodic condition. For the case of a smooth bounded domain with spatial dimension \(N\leq 3\), we prove the existence of classical nontrivial periodic solutions. For the case of a star shaped domain with \(N\geq4\), we prove the nonexistence of nontrivial periodic solutions. For the case of an annulus domain with \(N \geq3 \), we prove the existence of nontrivial radial periodic solutions. Some numerical simulations are also presented to illustrate our results.
MSC:
| 35B10 | Periodic solutions to PDEs |
| 35K55 | Nonlinear parabolic equations |
| 35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
| 35K65 | Degenerate parabolic equations |
| 35B45 | A priori estimates in context of PDEs |
References:
| [1] | Allen, S, Cahn, JW: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1084-1095 (1979) · doi:10.1016/0001-6160(79)90196-2 |
| [2] | Cirilloa, ENM, Ianirob, N, Sciarra, G: Allen-Cahn and Cahn-Hilliard-like equations for dissipative dynamics of saturated porous media. J. Mech. Phys. Solids 61(2), 629-651 (2013) · Zbl 1258.74064 · doi:10.1016/j.jmps.2012.08.014 |
| [3] | Shen, J, Yang, F: Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst. 28(4), 1669-1691 (2010) · Zbl 1201.65184 · doi:10.3934/dcds.2010.28.1669 |
| [4] | Murray, JD: Mathematical Biology, 2nd edn. Springer, Berlin (1993) · Zbl 0779.92001 · doi:10.1007/b98869 |
| [5] | Cahn, JW; Novick-Cohen, A., Limiting motion for an Allen-Cahn/Cahn-Hilliard system, Zakopane, 1995, Harlow · Zbl 0865.35145 |
| [6] | de Mottoni, P, Schatzman, M: Geometrical evolution of developed interfaces. Trans. Am. Math. Soc. 347(5), 1533-1589 (1995) · Zbl 0840.35010 · doi:10.1090/S0002-9947-1995-1672406-7 |
| [7] | de la Llave, R, Valdinoci, E: Multiplicity results for interfaces of Ginzburg-Landau-Allen-Cahn equations in periodic media. Adv. Math. 215, 379-426 (2007) · Zbl 1152.35038 · doi:10.1016/j.aim.2007.03.013 |
| [8] | Kowalczyk, M, Liu, Y, Wei, J: Singly periodic solutions of the Allen-Cahn equation and the Toda lattice. Commun. Partial Differ. Equ. 40(2), 329-356 (2015) · Zbl 1321.35054 · doi:10.1080/03605302.2014.947379 |
| [9] | Matano, H, Nara, M: Large time behavior of disturbed planar fronts in the Allen-Cahn equation. J. Differ. Equ. 251, 3522-3557 (2011) · Zbl 1252.35050 · doi:10.1016/j.jde.2011.08.029 |
| [10] | Novaga, M, Valdinoci, E: Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations. ESAIM Control Optim. Calc. Var. 15, 914-933 (2009) · Zbl 1196.35202 · doi:10.1051/cocv:2008058 |
| [11] | Novaga, M, Valdinoci, E: Bump solutions for the mesoscopic Allen-Cahn equation in periodic media. ESAIM Control Optim. Calc. Var. 40, 37-49 (2011) · Zbl 1206.35119 |
| [12] | Gurtin, ME: Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D 92, 178-192 (1996) · Zbl 0885.35121 · doi:10.1016/0167-2789(95)00173-5 |
| [13] | Esteban, MJ: On periodic solutions of superlinear parabolic problems. Trans. Am. Math. Soc. 293, 171-189 (1986) · Zbl 0619.35058 · doi:10.1090/S0002-9947-1986-0814919-8 |
| [14] | Esteban, MJ: A remark on the existence of positive periodic solutions of superlinear parabolic problems. Proc. Am. Math. Soc. 102, 131-136 (1988) · Zbl 0653.35039 · doi:10.1090/S0002-9939-1988-0915730-7 |
| [15] | Húska, J: Periodic solutions in superlinear parabolic problems. Acta Math. Univ. Comen. 19, 19-26 (2002) · Zbl 1049.35026 |
| [16] | Húska, J: Periodic solutions in semilinear parabolic problems. Thesis, Comenius University, Slovakia (2002) |
| [17] | Hirano, N, Mizoguchi, N: Positive unstable periodic solutions for superlinear parabolic equations. Proc. Am. Math. Soc. 123, 1487-1495 (1995) · Zbl 0828.35063 · doi:10.1090/S0002-9939-1995-1234627-2 |
| [18] | Quittner, P: Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems. NoDEA Nonlinear Differ. Equ. Appl. 11, 237-258 (2004) · Zbl 1058.35120 · doi:10.1007/s00030-003-1056-3 |
| [19] | Seidman, TI: Periodic solutions of a nonlinear parabolic equation. J. Differ. Equ. 19, 242-257 (1975) · Zbl 0281.35005 · doi:10.1016/0022-0396(75)90004-2 |
| [20] | Wang, YF, Yin, JX, Wu, ZQ: Periodic solutions of evolution p-Laplacian equations with nonlinear sources. J. Math. Anal. Appl. 219, 76-96 (1998) · Zbl 0932.35127 · doi:10.1006/jmaa.1997.5783 |
| [21] | Yin, JX, Jin, CH: Periodic solutions of the evolutionary p-Laplacian with nonlinear sources. J. Math. Anal. Appl. 368, 604-622 (2010) · Zbl 1207.35203 · doi:10.1016/j.jmaa.2010.03.006 |
| [22] | Zhou, Q, Huang, R, Ke, Y, Wang, YF: Existence of the nontrivial nonnegative periodic solutions for the quasilinear parabolic equation with nonlocal terms. Comput. Math. Appl. 50(8-9), 1293-1302 (2005) · Zbl 1090.35022 · doi:10.1016/j.camwa.2005.06.005 |
| [23] | Deng, J: Existence of positive radial solutions for a nonlinear elliptic problem in annulus domains. Math. Methods Appl. Sci. 35, 1594-1600 (2012) · Zbl 1250.35082 · doi:10.1002/mma.2547 |
| [24] | Bartsch, T, Poláčik, P, Quittner, P: Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations. J. Eur. Math. Soc. 13, 219-247 (2011) · Zbl 1215.35041 · doi:10.4171/JEMS/250 |
| [25] | Hess, P: Periodic-Parabolic Boundary Value Problems and Positivity. Pitman Research Notes in Mathematics Series, vol. 247. Longman, Harlow (1991) · Zbl 0731.35050 |
| [26] | Wang, YF, Yin, JX: Periodic solutions for a class of degenerate parabolic equations with Neumann boundary conditions. Nonlinear Anal., Real World Appl. 12(4), 2069-2076 (2011) · Zbl 1219.35017 · doi:10.1016/j.nonrwa.2010.12.022 |
| [27] | Wang, YF, Yin, JX: Coexistence periodic solutions of a doubly nonlinear parabolic system with Neumann boundary conditions. J. Math. Anal. Appl. 396(2), 704-714 (2012) · Zbl 1248.35215 · doi:10.1016/j.jmaa.2012.07.022 |
| [28] | Hameed, RA, Sun, JB, Wu, BY: Existence of periodic solutions of a p-Laplacian-Neumann problem. Bound. Value Probl. 2013, 171 (2013) · Zbl 1295.35293 · doi:10.1186/1687-2770-2013-171 |
| [29] | Quittner, P, Souplet, P: Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States. Birkhäuser, Berlin (2007) · Zbl 1128.35003 |
| [30] | Quittner, P: Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems. Houst. J. Math. 29, 757-799 (2003) · Zbl 1034.35013 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
