On the validity of the linear speed selection mechanism for fronts of the nonlinear diffusion equation

Details On the validity of the linear speed selection mechanism for fronts of the nonlinear diffusion equation On the validity of the linear speed selection mechanism for fronts of the nonlinear diffusion equation

1994-03-08

arxiv.org/abs/patt-sol/9403001v1

Phys. Rev. Lett., 73 (1994) 2272

Nonlinear Sciences

Pattern Formation and Solitons

9 pages, REVTEX

Scientific paper

10.1103/PhysRevLett.73.2272

We consider the problem of the speed selection mechanism for the one dimensional nonlinear diffusion equation $u_t = u_{xx} + f(u)$. It has been rigorously shown by Aronson and Weinberger that for a wide class of functions $f$, sufficiently localized initial conditions evolve in time into a monotonic front which propagates with speed $c^*$ such that $2 \sqrt{f'(0)} \leq c^* < 2 \sqrt{\sup(f(u)/u)}$. The lower value $c_L = 2 \sqrt{f'(0)}$ is that predicted by the linear marginal stability speed selection mechanism. We derive a new lower bound on the the speed of the selected front, this bound depends on $f$ and thus enables us to assess the extent to which the linear marginal selection mechanism is valid.

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