On the large time behaviour of the multi-dimensional Fisher–KPP equation with compactly supported initial data

SynopsisUsing the method of generalized characteristics, we discuss the regularity and large time behaviour of admissible weak solutions of a single conservation law, in one space variable, with one inflection point.We show that when the initial data are C∞ then, generically, the solution is C∞ except: (a) on a finite set of C∞ arcs across which it experiences jump discontinuities (genuine shocks or left contact discontinuities); (b) on a finite set of straight line characteristic segments across which its derivatives of order m, m = 1, 2,…, experience jump discontinuities (weak waves of order m); and (c) on the finite set of points of interaction of shocks and weak waves. Weak waves of order 1 are triggered by rays grazing upon contact discontinuities. Weak waves of order m, m ≥ 2, are generated by the collision of a weak wave of order m − 1 with a left contact discontinuity.We establish sharp decay rates for solutions with initial data of the following types: (a) with bounded primitive; (b) with primitive having sublinear growth; (c) in L1; (d) of compact support; and (e) periodic.

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Decay rates for the solutions of model equations for bore propagation

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500028080 ◽

1995 ◽

Vol 125 (2) ◽

pp. 371-398 ◽

Cited By ~ 3

Author(s):

S. V. Rajopadhye

Keyword(s):

Initial Data ◽

Large Time ◽

Burgers Equation ◽

Decay Rates ◽

Time Behaviour ◽

Large Time Behaviour ◽

Temporal Decay ◽

De Vries ◽

Model Equations

We study the large-time behaviour of solutions to the Korteweg-de Vries-Burgers equation with bore-like initial data. This work relies on the methods of Amick, Bona and Schonbeck to obtain sharp rates of temporal decay of certain norms of the solution, thus obtaining an improvement over results of Naumkin and Shishmarev.

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Large time behaviour of solutions of balance laws with periodic initial data

Nonlinear Differential Equations and Applications NoDEA ◽

10.1007/bf01194015 ◽

1995 ◽

Vol 2 (1) ◽

pp. 111-131 ◽

Cited By ~ 14

Author(s):

Carlo Sinestrari

Keyword(s):

Initial Data ◽

Large Time ◽

Balance Laws ◽

Time Behaviour ◽

Large Time Behaviour ◽

Periodic Initial Data

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Large-time behaviour of solutions of Burgers' equation

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500001906 ◽

2002 ◽

Vol 132 (4) ◽

pp. 843-878 ◽

Cited By ~ 2

Author(s):

Daniel B. Dix

Keyword(s):

Initial Data ◽

Asymptotic Behaviour ◽

Initial Value Problem ◽

Large Time ◽

Burgers Equation ◽

Time Behaviour ◽

Initial Value ◽

Large Time Behaviour ◽

Image Position ◽

Sharp Error

The large-time asymptotic behaviour of real-valued solutions of the pure initial-value problem for Burgers' equation ut + uuxuxx = 0, is studied. The initial data satisfy u0(x) ~ nx as |x| , where n R. There are two constants of the motion that affect the large-time behaviour: Hopf considered the case n = 0 (i.e. u0L1(R)), and the case sufficiently small was considered by Dix. Here we completely remove that smallness condition. When n < 1, we find an explicit function U(), depending only on and n, such that uniformly in . When n 1, there are two different functions U() that simultaneously attract the quantity t12u(t12, t), and each one wins in its own range of . Thus we give an asymptotic description of the solution in different regions and compute its decay rate in L. Sharp error estimates are proved.

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