Accelerating fronts in autocatalysis (Q2713375)

In this paper the authors consider the system of reaction-diffusion equations modelling an autocatalytic reaction, NEWLINE\[NEWLINE \psi_t = \psi_{xx} + (1-\psi)f(\theta), \qquad \theta_t = l \theta_{xx} + (1-\psi)f(\theta) NEWLINE\]NEWLINE on the real line, \(t \geq 0\), where \(\theta(x,t)\) is the concentration of the autocatalyst, \(1-\psi(x,t)\) is the concentration of the reagent that gives rise to \(\theta\), \(l\) is the Lewis number, and the autocatalytic nature of the process is expressed by taking \(f(\theta)=\theta^m\), \(m \geq 2\). The above system is supplemented with an initial condition \(u(x,0)=(\psi(x,0),\theta(x,0))^T\), both components of which go to \(1\) as \(x \rightarrow -\infty\) and to \(0\) as \(x \rightarrow \infty\). The authors derive sufficient conditions on the decay rate of the initial profile that ensure that fronts do not accelerate, meaning that their speed does not grow without bound. NEWLINENEWLINENEWLINESince in this case methods based on comparison principles for parabolic systems are not applicable, the authors use energy-type estimates to produce bounds on solutions in weighted \(L^2\), \(L^{2n}\) for large enough \(n\), and \(H^1\) spaces. The main result, contained in Theorem 3.3 for exponentially decaying initial profiles, and in Theorem 3.4 for algebraically decaying ones, states, roughly, that if the profile decays exponentially or algebraically with exponent \(\mu > 1/(m-1)\), the front does not accelerate. NEWLINENEWLINENEWLINEThe methods used in the paper leave open the question of acceleration if \(\mu < 1/(m-1)\) and the behaviour in the critical case \(\mu=1/(m-1)\).