## Periodic traveling waves and locating oscillating patterns in multidimensional domains

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- by Nicholas D. Alikakos, Peter W. Bates and Xinfu Chen PDF
- Trans. Amer. Math. Soc.
**351**(1999), 2777-2805 Request permission

## Abstract:

We establish the existence and robustness of layered, time-periodic solutions to a reaction-diffusion equation in a bounded domain in $\mathbb {R}^n$, when the diffusion coefficient is sufficiently small and the reaction term is periodic in time and bistable in the state variable. Our results suggest that these patterned, oscillatory solutions are stable and locally unique. The location of the internal layers is characterized through a periodic traveling wave problem for a related one-dimensional reaction-diffusion equation. This one-dimensional problem is of independent interest and for this we establish the existence and uniqueness of a heteroclinic solution which, in constant-velocity moving coodinates, is periodic in time. Furthermore, we prove that the manifold of translates of this solution is globally exponentially asymptotically stable.## References

- Nicholas D. Alikakos and K. C. Shaing,
*On the singular limit for a class of problems modelling phase transitions*, SIAM J. Math. Anal.**18**(1987), no. 5, 1453–1462. MR**902344**, DOI 10.1137/0518105 - Nicholas D. Alikakos and Henry C. Simpson,
*A variational approach for a class of singular perturbation problems and applications*, Proc. Roy. Soc. Edinburgh Sect. A**107**(1987), no. 1-2, 27–42. MR**918891**, DOI 10.1017/S0308210500029334 - Nicholas D. Alikakos and Peter Hess,
*Liapunov operators and stabilization in strongly order preserving dynamical systems*, Differential Integral Equations**4**(1991), no. 1, 15–24. MR**1079608** - S. B. Angenent and B. Fiedler,
*The dynamics of rotating waves in scalar reaction diffusion equations*, Trans. Amer. Math. Soc.**307**(1988), no. 2, 545–568. MR**940217**, DOI 10.1090/S0002-9947-1988-0940217-X - S. B. Angenent, J. Mallet-Paret, and L. A. Peletier,
*Stable transition layers in a semilinear boundary value problem*, J. Differential Equations**67**(1987), no. 2, 212–242. MR**879694**, DOI 10.1016/0022-0396(87)90147-1 - Martino Bardi and Benoît Perthame,
*Exponential decay to stable states in phase transitions via a double log-transformation*, Comm. Partial Differential Equations**15**(1990), no. 12, 1649–1669. MR**1080616**, DOI 10.1080/03605309908820742 - Henri Berestycki and Louis Nirenberg,
*Travelling fronts in cylinders*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**9**(1992), no. 5, 497–572 (English, with English and French summaries). MR**1191008**, DOI 10.1016/S0294-1449(16)30229-3 - Xinfu Chen,
*Generation and propagation of interfaces for reaction-diffusion equations*, J. Differential Equations**96**(1992), no. 1, 116–141. MR**1153311**, DOI 10.1016/0022-0396(92)90146-E - E. N. Dancer and P. Hess,
*Behaviour of a semilinear periodic-parabolic problem when a parameter is small*, Functional-analytic methods for partial differential equations (Tokyo, 1989) Lecture Notes in Math., vol. 1450, Springer, Berlin, 1990, pp. 12–19. MR**1084598**, DOI 10.1007/BFb0084895 - P. Faĭf and U. Grinli,
*Interior transition layers for elliptic boundary value problems with a small parameter*, Uspehi Mat. Nauk**29**(1974), no. 4 (178), 103–130 (Russian). Translated from the English by L. P. Volevič; Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ, II. MR**0481510** - Paul C. Fife and Ling Hsiao,
*The generation and propagation of internal layers*, Nonlinear Anal.**12**(1988), no. 1, 19–41. MR**924750**, DOI 10.1016/0362-546X(88)90010-7 - Paul C. Fife and J. B. McLeod,
*The approach of solutions of nonlinear diffusion equations to travelling wave solutions*, Bull. Amer. Math. Soc.**81**(1975), no. 6, 1076–1078. MR**380111**, DOI 10.1090/S0002-9904-1975-13922-X - Paul C. Fife and J. B. McLeod,
*The approach of solutions of nonlinear diffusion equations to travelling front solutions*, Arch. Rational Mech. Anal.**65**(1977), no. 4, 335–361. MR**442480**, DOI 10.1007/BF00250432 - Paul C. Fife and J. B. McLeod,
*A phase plane discussion of convergence to travelling fronts for nonlinear diffusion*, Arch. Rational Mech. Anal.**75**(1980/81), no. 4, 281–314. MR**607901**, DOI 10.1007/BF00256381 - R. A. Fisher,
*The wave of advance of advantageous genes,*Ann. Eugenics**7**(1937),355–369. - Yasumasa Nishiura and Hiroshi Fujii,
*SLEP method to the stability of singularly perturbed solutions with multiple internal transition layers in reaction-diffusion systems*, Dynamics of infinite-dimensional systems (Lisbon, 1986) NATO Adv. Sci. Inst. Ser. F: Comput. Systems Sci., vol. 37, Springer, Berlin, 1987, pp. 211–230. MR**921913** - R. A. Gardner,
*On the structure of the spectra of periodic travelling waves*, J. Math. Pures Appl. (9)**72**(1993), no. 5, 415–439. MR**1239098** - Daniel Henry,
*Geometric theory of semilinear parabolic equations*, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR**610244**, DOI 10.1007/BFb0089647 - Jack K. Hale,
*Ordinary differential equations*, 2nd ed., Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. MR**587488** - Jack K. Hale and Paul Massatt,
*Asymptotic behavior of gradient-like systems*, Dynamical systems, II (Gainesville, Fla., 1981) Academic Press, New York, 1982, pp. 85–101. MR**703689** - Peter Hess,
*Periodic-parabolic boundary value problems and positivity*, Pitman Research Notes in Mathematics Series, vol. 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. MR**1100011** - Ja. I. Kanel′,
*Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory*, Mat. Sb. (N.S.)**59 (101)**(1962), no. suppl., 245–288 (Russian). MR**0157130** - A. N. Kolmogorov, I. G. Petrovski & N. S. Piscounov,
*A study of the equation of diffusion with increase in quantity of matter, and its application to a biological problem,*Biul. Moskovskogo Gos. Univ.**17**(1937), 1-72. - O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva,
*Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa*, Izdat. “Nauka”, Moscow, 1967 (Russian). MR**0241821** - G. Barles and J. Burdeau,
*The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit time control problems*, Comm. Partial Differential Equations**20**(1995), no. 1-2, 129–178. MR**1312703**, DOI 10.1080/03605309508821090

## Additional Information

**Nicholas D. Alikakos**- Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300; Department of Mathematics, University of Athens, Panestimiopolis, Greece 15784
- Email: alikakos@utk.edu, nalikako@atlas.uoa.gr
**Peter W. Bates**- Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
- MR Author ID: 32495
- Email: peter@math.byu.edu
**Xinfu Chen**- Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
- MR Author ID: 261335
- Email: xinfu+@pitt.edu
- Received by editor(s): March 23, 1995
- Received by editor(s) in revised form: February 18, 1997
- Published electronically: March 1, 1999
- Additional Notes: The first author was partially supported by the National Science Foundation Grant DMS–9306229, the Science Alliance, and the NATO Scientific Affairs Division CRG930791.

The second author was partially supported by the National Science Foundation Grant DMS–9305044, and the NATO Scientific Affairs Division CRG 930791.

The third author partially supported by the National Science Foundation Grant DMS–9404773, and the Alfred P. Sloan Research Fellowship. - © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**351**(1999), 2777-2805 - MSC (1991): Primary 35B10, 35B25, 35B40, 35K57
- DOI: https://doi.org/10.1090/S0002-9947-99-02134-0
- MathSciNet review: 1467460