TY - JOUR

T1 - Comparison theorems and variable speed waves for a scalar reaction-diffusion equation

AU - Kay, Alison L.

AU - Sherratt, Jonathan A.

AU - McLeod, J. B.

PY - 2001

Y1 - 2001

N2 - This paper concerns the reaction-diffusion equation ut = uxx + u2(1 - u). Previous numerical solutions of this equation have demonstrated various different types of wave front solutions, generated by different initial conditions. In this paper, the authors use a phase-plane form of comparison theorems for partial differential equations (PDEs) to confirm analytically these numerical results. In particular, they show that initial conditions with an exponentially decaying tail evolve to the unique exponentially decaying travelling wave, while initial conditions with algebraically decaying tails evolve either to an algebraically decaying travelling wave, or to the exponentially decaying wave, or to a perpetually accelerating wave, dependent upon the exact form of the decay of the initial conditions. We then focus on the case of accelerating waves and investigate their form in more detail, by approximating the full equation in this case with a hyperbolic PDE, which we solve using the method of characteristics. We use this approximate solution to derive a leading-order approximation to the wave speed. © 2001 The Royal Society of Edinburgh.

AB - This paper concerns the reaction-diffusion equation ut = uxx + u2(1 - u). Previous numerical solutions of this equation have demonstrated various different types of wave front solutions, generated by different initial conditions. In this paper, the authors use a phase-plane form of comparison theorems for partial differential equations (PDEs) to confirm analytically these numerical results. In particular, they show that initial conditions with an exponentially decaying tail evolve to the unique exponentially decaying travelling wave, while initial conditions with algebraically decaying tails evolve either to an algebraically decaying travelling wave, or to the exponentially decaying wave, or to a perpetually accelerating wave, dependent upon the exact form of the decay of the initial conditions. We then focus on the case of accelerating waves and investigate their form in more detail, by approximating the full equation in this case with a hyperbolic PDE, which we solve using the method of characteristics. We use this approximate solution to derive a leading-order approximation to the wave speed. © 2001 The Royal Society of Edinburgh.

UR - http://www.scopus.com/inward/record.url?scp=23044531135&partnerID=8YFLogxK

M3 - Literature review

VL - 131

SP - 1133

EP - 1161

JO - Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

SN - 0308-2105

IS - 5

ER -