Spatial trajectories and convergence to traveling fronts for bistable reaction-diffusion equations

Peter Poláčik

doi:10.1007/978-3-319-19902-3_24

Spatial trajectories and convergence to traveling fronts for bistable reaction-diffusion equations

Peter Poláčik

School of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

We consider the semilinear parabolic equation u_t = u_xx + 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), u_x(x, t)): x∈ ℝ} of solutions of (1).

Original language	English (US)
Pages (from-to)	405-423
Number of pages	19
Journal	Progress in Nonlinear Differential Equations and Their Application
Volume	86
DOIs	https://doi.org/10.1007/978-3-319-19902-3_24
State	Published - 2017

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© 2015, Springer International Publishing Switzerland.

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abstract = "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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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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SP - 405

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JO - Progress in Nonlinear Differential Equations and Their Application

JF - Progress in Nonlinear Differential Equations and Their Application

ER -

Spatial trajectories and convergence to traveling fronts for bistable reaction-diffusion equations

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