In this lecture we will consider traveling wave solutions to partial

differential equations in -dimensions. These are solutions

of the form

, to a system of reaction-diffusion equations

.

Here denotes the profile and denotes

the velocity of the wave.

First we will consider the question of existence of such solutions.

After that we will look at the question of stability, i.e. whether a

small perturbation of a traveling wave solution converges to the

traveling wave solution as time tends to . Since the equation

is space independent, one can only expect stability with asymptotic

phase:

for some suitable .

Good knowledge of Analysis I-III, good knowledge of Linear Algebra, some knowledge of Functional Analysis, some knowledge of Spectral theory, some knowledge of PDEs.

Furthermore, Bochner-Integration might be helpful but is not necessary.

# Exercises

Exercise sheet 1 |

Exercise sheet 2 |

Exercise sheet 3 |

Exercise sheet 4 |

Exercise sheet 5 |

Exercise sheet 6 |

Exercise sheet 7 |

# References

W. Arendt, Charles J. K. Batty, M. Hieber, and F. Neubrander: Vector-valued Laplace transforms and Cauchy problems. Volume 96 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 2001.

D. G. Aronson and H. F. Weinberger: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974), Volume 446 of Lecture Notes in Mathematics, pages 5-49. Springer, Berlin, 1975.

L. C. Evans: Partial differential equations. Volume 19 of Graduate Studies in Mathematics, AMS, Providence, RI, 1998.

Roger Knobel: An introduction to the mathematical theory of waves. Providence, RI: American Mathematical Society, 2000.

Heinz-Otto Kreiss and Jens Lorenz: Initial-boundary value problems and the Navier-Stokes equations. Volume 136 of Pure and Applied Mathematics, Academic Press Inc., Boston, MA, 1989.

Björn Sandstede: Stability of travelling waves. Handbook of dynamical systems, Vol. 2, pages 983-1055, North-Holland, Amsterdam, 2002.