Transactions of the American Mathematical Society

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ISSN 1088-6850(e) ISSN 0002-9947(p)

Multidimensional stability of planar travelling waves

Author(s): Todd Kapitula
Journal: Trans. Amer. Math. Soc. 349 (1997), 257-269.
MSC (1991): Primary 35B40, 35C15, 35K57
MathSciNet review: 1360225
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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.

References:

1.: P. Bates and C.K.R.T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Reported 2 (1989), 1-38. MR 90g:58017
2.: I-L. Chern and T.-P. Liu, Convergence to diffusion waves of solutions for viscous conservation laws, Comm. Math. Phys. 110 (1987), 503-517. MR 88g:35127
3.: J. Goodman, personal communication.
4.: -, Stability of viscous scalar shock fronts in several dimensions, Trans. AMS 311 (1989), 683-695. MR 89j:35059
5.: D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics 840, Springer-Verlag, New York, 1981. MR 83j:35084
6.: T. Kapitula, On the nonlinear stability of plane waves for the Ginzburg-Landau equation, Comm. Pure Appl. Math. 47 (1994), no. 6, 831-841. MR 95e:35200
7.: -, On the stability of travelling waves in weighted $L^\infty$ spaces, J. Diff. Eq. 112 (1994), no. 1, 179-215. MR 95h:35107
8.: S. Kawashima, Large-time behavior of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edin. 106A (1987), 169-194. MR 89d:35022
9.: C. D. Levermore and J. X. Xin, Multidimensional stability of travelling waves in a bistable reaction-diffusion equation, II, Comm. PDE 17 (1992), no. 11&12, 1901-1924. MR 94c:35105
10.: A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. MR 85g:47061
11.: A.E. Taylor, Introduction to functional analysis, Wiley, New York, 1961.
12.: J. X. Xin, Multidimensional stability of travelling waves in a bistable reaction-diffusion system, I, Comm. PDE 17 (1992), no. 11&12, 1889-1900. MR 94c:35104

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