Entire solutions in reaction-advection-diffusion equations in cylinders

This paper is concerned with the existence of entire solutions of a reaction-advection-diffusion equation with monostable and ignition temperature nonlinearities in infinite-cylinders. Here the entire solutions are defined in the whole infinite cylinder and for all time t∈R. A comparison argument is used to prove the existence of entire solutions which behave as two traveling wave fronts coming from both directions. The main techniques are to characterize the asymptotic behavior of the solutions as t→−∞ in term of appropriate subsolutions and supersolutions. In order to illustrate our main results, a passive-reaction-diffusion equation model arising from propagation of fronts is considered. This is probably the first time the existence of entire solutions of reaction-diffusion equations in infinite-cylinders has been studied.

[1]  Yi Li,et al.  Travelling Fronts in Cylinders and Their Stability , 1997 .

[2]  Wan-Tong Li,et al.  Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity , 2008 .

[3]  François Hamel,et al.  Entire solutions of the KPP equation , 1999 .

[4]  Hiroki Yagisita,et al.  Backward Global Solutions Characterizing Annihilation Dynamics of Travelling Fronts , 2002 .

[5]  Wan-Tong Li,et al.  Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay , 2007 .

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

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

[8]  Jong-Shenq Guo,et al.  Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations , 2004 .

[9]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[10]  Henri Berestycki,et al.  Generalized travelling waves for reaction-diffusion equations , 2006 .

[11]  Hirokazu Ninomiya,et al.  Entire Solutions with Merging Fronts to Reaction–Diffusion Equations , 2006 .

[12]  Xinfu Chen,et al.  Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations , 1997, Advances in Differential Equations.

[13]  Vitaly Volpert,et al.  Traveling Wave Solutions of Parabolic Systems , 1994 .

[14]  H. Berestycki.  The Influence of Advection on the Propagation of Fronts in Reaction-Diffusion Equations , 2002 .

[15]  François Hamel,et al.  Travelling Fronts and Entire Solutions¶of the Fisher-KPP Equation in ℝN , 2001 .

[16]  J. Vega.  The Asymptotic Behavior of the Solutions of Some Semilinear Elliptic Equations in Cylindrical Domains , 1993 .

[17]  J. Roquejoffre.  Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders , 1997 .

[18]  J. Roquejoffre.  Convergence to Travelling Waves for Solutions of a Class of Semilinear Parabolic Equations , 1994 .

[19]  Wan-Tong Li,et al.  Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays , 2006 .

[20]  J. Roquejoffre.  Stability of travelling fronts in a model for flame propagation Part II: Nonlinear stability , 1992 .

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

[22]  K. P. Hadeler,et al.  Travelling fronts in nonlinear diffusion equations , 1975 .

[23]  Wan-Tong Li,et al.  Traveling Fronts in Monostable Equations with Nonlocal Delayed Effects , 2008 .

[24]  Luisa Malaguti,et al.  The influence of convective effects on front propagation in certain diffusive models , 2003 .

[25]  J. Roquejoffre,et al.  Stability of travelling fronts in a model for flame propagation part I: Linear analysis , 1992 .

[26]  Xinfu Chen,et al.  Entire solutions of reaction—diffusion equations with balanced bistable nonlinearities , 2006, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[27]  H. Bruce Stewart,et al.  Generation of analytic semigroups by strongly elliptic operators under general boundary conditions , 1980 .

[28]  Xinfu Chen,et al.  Existence and uniqueness of entire solutions for a reaction-diffusion equation , 2005 .

[29]  Luisa Malaguti,et al.  Travelling wavefronts in reaction-diffusion equations with convection effects and non-regular terms , 2002 .

[30]  Wan-Tong Li,et al.  Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity , 2008 .

[31]  Adam M. Oberman,et al.  Bulk Burning Rate in¶Passive–Reactive Diffusion , 1999, math/9907132.

[32]  D. Aronson,et al.  Multidimensional nonlinear di u-sion arising in population genetics , 1978 .

[33]  A. Kiselev,et al.  Enhancement of the traveling front speeds in reaction-diffusion equations with advection , 2000, math/0002175.

[34]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

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

[36]  Hirokazu Ninomiya,et al.  SOME ENTIRE SOLUTIONS OF THE ALLEN–CAHN EQUATION , 2004 .

[37]  J. McLeod,et al.  The approach of solutions of nonlinear diffusion equations to travelling front solutions , 1977 .