TY - JOUR
T1 - Spatial trajectories and convergence to traveling fronts for bistable reaction-diffusion equations
AU - Poláčik, Peter
N1 - Funding Information:
This work was supported in part by the NSF Grant DMS–1161923.
Publisher Copyright:
© 2015, Springer International Publishing Switzerland.
PY - 2017
Y1 - 2017
N2 - We consider the semilinear parabolic equation ut = uxx + f(u), x∈ ℝ, t > 0, where f is a bistable nonlinearity. It is well known that for a large class of initial data, the corresponding solutions converge to traveling fronts. We give a new proof of this classical result as well as some generalizations. Our proof uses a geometric method, which makes use of spatial trajectories {(u(x, t), ux(x, t)): x∈ ℝ} of solutions of (1).
AB - We consider the semilinear parabolic equation ut = uxx + f(u), x∈ ℝ, t > 0, where f is a bistable nonlinearity. It is well known that for a large class of initial data, the corresponding solutions converge to traveling fronts. We give a new proof of this classical result as well as some generalizations. Our proof uses a geometric method, which makes use of spatial trajectories {(u(x, t), ux(x, t)): x∈ ℝ} of solutions of (1).
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U2 - 10.1007/978-3-319-19902-3_24
DO - 10.1007/978-3-319-19902-3_24
M3 - Article
AN - SCOPUS:85035078427
SN - 1421-1750
VL - 86
SP - 405
EP - 423
JO - Progress in Nonlinear Differential Equations and Their Application
JF - Progress in Nonlinear Differential Equations and Their Application
ER -