Multidimensional stability of planar travelling waves

Kapitula, Todd

doi:10.1090/S0002-9947-97-01668-1

Multidimensional stability
of planar travelling waves

Author: Todd Kapitula
Journal: Trans. Amer. Math. Soc. 349 (1997), 257-269
MSC (1991): Primary 35B40, 35C15, 35K57
DOI: https://doi.org/10.1090/S0002-9947-97-01668-1
MathSciNet review: 1360225
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The multidimensional stability of planar travelling waves for systems of reaction-diffusion equations is considered in the case that the diffusion matrix is the identity. It is shown that if the wave is exponentially orbitally stable in one space dimension, then it is stable for $x\in % \mathbf {R}^n,\,n\ge 2$ . Furthermore, it is shown that the perturbation of the wave decays like $t^{-(n-1)/4}$ as $t\to \infty$ . The result is proved via an application of linear semigroup theory.

1. Peter W. Bates and Christopher K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics reported, Vol. 2, Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, Wiley, Chichester, 1989, pp. 1–38. MR 1000974
2. I-Liang Chern and Tai-Ping Liu, Convergence to diffusion waves of solutions for viscous conservation laws, Comm. Math. Phys. 110 (1987), no. 3, 503–517. MR 891950
3. J. Goodman, personal communication.
4. Jonathan Goodman, Stability of viscous scalar shock fronts in several dimensions, Trans. Amer. Math. Soc. 311 (1989), no. 2, 683–695. MR 978372, https://doi.org/10.1090/S0002-9947-1989-0978372-9
5. Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
6. Todd Kapitula, On the nonlinear stability of plane waves for the Ginzburg-Landau equation, Comm. Pure Appl. Math. 47 (1994), no. 6, 831–841. MR 1280990, https://doi.org/10.1002/cpa.3160470603
7. Todd Kapitula, On the stability of travelling waves in weighted 𝐿^{∞} spaces, J. Differential Equations 112 (1994), no. 1, 179–215. MR 1287557, https://doi.org/10.1006/jdeq.1994.1100
8. Shuichi Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), no. 1-2, 169–194. MR 899951, https://doi.org/10.1017/S0308210500018308
9. C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. II, Comm. Partial Differential Equations 17 (1992), no. 11-12, 1901–1924. MR 1194744, https://doi.org/10.1080/03605309208820908
10. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
11. A.E. Taylor, Introduction to functional analysis, Wiley, New York, 1961.
12. J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. I, Comm. Partial Differential Equations 17 (1992), no. 11-12, 1889–1899. MR 1194743, https://doi.org/10.1080/03605309208820907