Generalized Traveling Waves in Disordered Media: Existence, Uniqueness, and Stability

We prove existence, uniqueness, and stability of transition fronts (generalized traveling waves) for reaction-diffusion equations in cylindrical domains with general inhomogeneous ignition reactions. We also show uniform convergence of solutions with exponentially decaying initial data to time translations of the front. In the case of stationary ergodic reactions, the fronts are proved to propagate with a deterministic positive speed. Our results extend to reaction-advection-diffusion equations with periodic advection and diffusion.

[1]  Henri Berestycki,et al.  Generalized Transition Waves and Their Properties , 2010, 1012.0794.

[2]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[3]  J. Roquejoffre,et al.  Generalized fronts for one-dimensional reaction-diffusion equations , 2009 .

[4]  V. Volpert,et al.  Generalized travelling waves for perturbed monotone reaction-diffusion systems , 2001 .

[5]  J. Kingman,et al.  The Ergodic Theory of Subadditive Stochastic Processes , 1968 .

[6]  Andrej Zlatoš.  Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations , 2011, 1103.3094.

[7]  Hiroshi Matano,et al.  Bistable traveling waves around an obstacle , 2009 .

[8]  Henri Berestycki,et al.  Travelling fronts in cylinders , 1992 .

[9]  Henri Berestycki,et al.  Front propagation in periodic excitable media , 2002 .

[10]  L. Ryzhik,et al.  Traveling waves in a one-dimensional heterogeneous medium , 2009 .

[11]  R. Fisher.  THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[12]  Jack Xin,et al.  Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media , 1993 .

[13]  Emmanuel Grenier,et al.  Existence and Nonexistence of Traveling Wave Solutions for a Bistable Reaction-Diffusion Equation in an Infinite Cylinder Whose Diameter is Suddenly Increased , 2005 .

[14]  Jack Xin,et al.  Existence of planar flame fronts in convective-diffusive periodic media , 1992 .

[15]  Henri Berestycki,et al.  The speed of propagation for KPP type problems. II , 2010 .

[16]  Lenya Ryzhik,et al.  KPP pulsating front speed-up by flows , 2007 .

[17]  François Hamel,et al.  The speed of propagation for KPP type problems. I: Periodic framework , 2005 .

[18]  P. Lions,et al.  Multi-dimensional travelling-wave solutions of a flame propagation model , 1990 .

[19]  Wenxian Shen,et al.  Traveling Waves in Diffusive Random Media , 2004 .

[20]  T. Liggett.  An Improved Subadditive Ergodic Theorem , 1985 .

[21]  J. Roquejoffre,et al.  Stability of Generalized Transition Fronts , 2009 .

[22]  L. Ryzhik,et al.  Traveling waves in a one-dimensional random medium , 2007, 0710.1858.