TY - JOUR
T1 - The reactive-telegraph equation and a related kinetic model
AU - Henderson, Christopher
AU - Souganidis, Panagiotis E.
N1 - Funding Information: We wish to thank the anonymous referees for a close reading of the manuscript and their very helpful comments. CH would like to thank Jacek Jendrej for pointing out the refence [26] regarding regularity of hyperbolic equations. CH was partially supported by the National Science Foundation Research Training Group Grant DMS-1246999. PS was partially supported by the National Science Foundation Grants DMS-1266383 and DMS-1600129 and the Office for Naval Research Grant N00014-17-1-2095. Publisher Copyright: © 2017, Springer International Publishing AG, part of Springer Nature.
PY - 2017/12/1
Y1 - 2017/12/1
N2 - We study the long-range, long-time behavior of the reactive-telegraph equation and a related reactive-kinetic model. The two problems are equivalent in one spatial dimension. We point out that the reactive-telegraph equation, meant to model a population density, does not preserve positivity in higher dimensions. In view of this, in dimensions larger than one, we consider a reactive-kinetic model and investigate the long-range, long-time limit of the solutions. We provide a general characterization of the speed of propagation and we compute it explicitly in one and two dimensions. We show that a phase transition between parabolic and hyperbolic behavior takes place only in one dimension. Finally, we investigate the hydrodynamic limit of the limiting problem.
AB - We study the long-range, long-time behavior of the reactive-telegraph equation and a related reactive-kinetic model. The two problems are equivalent in one spatial dimension. We point out that the reactive-telegraph equation, meant to model a population density, does not preserve positivity in higher dimensions. In view of this, in dimensions larger than one, we consider a reactive-kinetic model and investigate the long-range, long-time limit of the solutions. We provide a general characterization of the speed of propagation and we compute it explicitly in one and two dimensions. We show that a phase transition between parabolic and hyperbolic behavior takes place only in one dimension. Finally, we investigate the hydrodynamic limit of the limiting problem.
KW - 35D40
KW - 35F21
KW - 35L15
KW - 35L70
KW - Primary 35F25
KW - Secondary 92D25
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U2 - 10.1007/s00030-017-0488-0
DO - 10.1007/s00030-017-0488-0
M3 - Article
SN - 1021-9722
VL - 24
JO - Nonlinear Differential Equations and Applications
JF - Nonlinear Differential Equations and Applications
IS - 6
M1 - 66
ER -