Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMinimal speed of fronts of reaction-convection-diffusion equations
72
0
0.0
(
0
)
Added by
M. Cristina Depassier
Publication date
2001
fields
Physics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
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 $u_t + mu phi(u) u_x = (D(u)u_x)_x +f(u)$.
rate research
Read More
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.
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.
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.
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 establish an integral variational principle for the spreading speed of the one dimensional reaction diffusion equation with Stefan boundary conditions, for arbitrary reaction terms. This principle allows to obtain in a simple way the dependence of the speed on the Stefan constant. As an application a generalized Zeldovich-Frank-Kamenetskii lower bound for the speed, valid for monostable and combustion reaction terms, is given.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
