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Register a new userQualitative properties of pushed fronts of reaction-diffusion-advection equations
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Authors
Hongjun Guo
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In this paper, we prove some qualitative properties of pushed fronts for the periodic reaction-diffusion-equation with general monostable nonlinearities. Especially, we prove the exponential behavior of pushed fronts when they are approaching their unstable state which has been left open so far. Through this property, we also prove the stability of pushed fronts.
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Using pointwise semigroup techniques, we establish sharp rates of decay in space and time of a perturbed reaction diffusion front to its time-asymptotic limit. This recovers results of Sattinger, Henry and others of time-exponential convergence in weighted $L^p$ and Sobolev norms, while capturing the new feature of spatial diffusion at Gaussian rate. Novel features of the argument are a point-wise Green function decomposition reconciling spectral decomposition and short-time Nash-Aronson estimates and an instantaneous tracking scheme similar to that used in the study of stability of viscous shock waves.
We study the minimal speed of propagating fronts of convection reaction diffusion equations of the form $u_t + mu phi(u) u_x = u_{xx} +f(u)$ for positive reaction terms with $f(0 >0$. The function $phi(u)$ is continuous and vanishes at $u=0$. A variational principle for the minimal speed of the waves is constructed from which upper and lower bounds are obtained. This permits the a priori assesment of the effect of the convective term on the minimal speed of the traveling fronts. If the convective term is not strong enough, it produces no effect on the minimal speed of the fronts. We show that if $f(u)/sqrt{f(0)} + mu phi(u) < 0$, then the minimal speed is given by the linear value $2 sqrt{f(0)}$, and the convective term has no effect on the minimal speed. The results are illustrated by applying them to the exactly solvable case $u_t + mu u u_x = u_{xx} + u (1 -u)$. Results are also given for the density dependent diffusion case $u_t + mu phi(u) u_x = (D(u)u_x)_x +f(u)$.
We give an explicit formula for the change of speed of pushed and bistable fronts of the reaction diffusion equation when a small cutoff is applied at the unstable or metastable equilibrium point. The results are valid for arbitrary reaction terms and include the case of density dependent diffusion.
We study the asymptotic speed of traveling fronts of the scalar reaction diffusion for positive reaction terms and with a diffusion coefficient depending nonlinearly on the concentration and on its gradient. We restrict our study to diffusion coefficients of the form $D(u,u_x) = m u^{m-1} u_x^{m(p-2)}$ for which existence and convergence to traveling fronts has been established. We formulate a variational principle for the asymptotic speed of the fronts. Upper and lower bounds for the speed valid for any $mge0, pge 1$ are constructed. When $m=1, p=2$ the problem reduces to the constant diffusion problem and the bounds correspond to the classic Zeldovich Frank-Kamenetskii lower bound and the Aronson-Weinberger upper bound respectively. In the special case $m(p-1) = 1$ a local lower bound can be constructed which coincides with the aforementioned upper bound. The speed in this case is completely determined in agreement with recent results.
We study the change in the speed of pushed and bistable fronts of the reaction diffusion equation in the presence of a small cut-off. We give explicit formulas for the shift in the speed for arbitrary reaction terms f(u). The dependence of the speed shift on the cut-off parameter is a function of the front speed and profile in the absence of the cut-off. In order to determine the speed shift we solve the leading order approximation to the front profile u(z) in the neighborhood of the leading edge and use a variational principle for the speed. We apply the general formula to the Nagumo equation and recover the results which have been obtained recently by geometric analysis. The formulas given are of general validity and we also apply them to a class of reaction terms which have not been considered elsewhere.
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