dorsal/arxiv
View SchemaOn the validity of the linear speed selection mechanism for fronts of the nonlinear diffusion equation
| Authors | R. D. Benguria, M. C. Depassier |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9403001 |
| URL | https://arxiv.org/abs/patt-sol/9403001 |
| DOI | 10.1103/PhysRevLett.73.2272 |
| Journal | Phys. Rev. Lett., 73 (1994) 2272 |
Abstract
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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"abstract": "We consider the problem of the speed selection mechanism for the one\ndimensional nonlinear diffusion equation $u_t = u_{xx} + f(u)$. It has been\nrigorously shown by Aronson and Weinberger that for a wide class of functions\n$f$, sufficiently localized initial conditions evolve in time into a monotonic\nfront which propagates with speed $c^*$ such that $2 \\sqrt{f\u0027(0)} \\leq c^* \u003c 2\n\\sqrt{\\sup(f(u)/u)}$. The lower value $c_L = 2 \\sqrt{f\u0027(0)}$ is that predicted\nby the linear marginal stability speed selection mechanism. We derive a new\nlower bound on the the speed of the selected front, this bound depends on $f$\nand thus enables us to assess the extent to which the linear marginal selection\nmechanism is valid.",
"arxiv_id": "patt-sol/9403001",
"authors": [
"R. D. Benguria",
"M. C. Depassier"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1103/PhysRevLett.73.2272",
"journal_ref": "Phys. Rev. Lett., 73 (1994) 2272",
"title": "On the validity of the linear speed selection mechanism for fronts of the nonlinear diffusion equation",
"url": "https://arxiv.org/abs/patt-sol/9403001"
},
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