Large time behavior of solutions for degenerate p-degree Fisher equation with algebraic decaying initial data

Abstract

This paper is concerned with the large time behavior of solutions to the onedimensional degenerate p-degree Fisher equation, where the initial data are assumed to be asymptotically front-like and to decay to zero non-exponentially at one end. By applying sub-super solution method we first prove the Lyapunov stability of all the wave fronts with noncritical speeds in some polynomially weighted space. Further, by using semigroup estimates, sub-super solution method and the nonlinear stability of the traveling waves, we obtain the asymptotic estimates of the solutions which indicate that the solution still moves like a wave front and the asymptotic spreading speed of the level set of the solution can be finite or infinite, which is solely determined by the detailed decaying rate of the initial data.

Acceleration of propagation in a chemotaxis‐growth system with slowly decaying initial data

“…By (1.8), the explicit upper and lower bounds locating $$E_\omega (t)$$ were obtained in [18] for different forms of $$u_0$$, for example, if $$u_0(x)=C x^{-p}$$ for large $$x$$ with $$p,C>0$$, then $$\ln (\min E_\omega (t))\sim \ln (\max E_\omega (t))\sim ap^{-1}t\nobreakspace \text{as}\nobreakspace t\rightarrow +\infty$$. We refer to [3, 19, 20] for the acceleration propagation results recently developed for more general reaction–diffusion equations. We also refer to [9, 11, 13] for the acceleration propagation of fractional diffusion equations and [4, 14, 15] for non‐local dispersal equations.…”

“…By (1.8), the explicit upper and lower bounds locating 𝐸 𝜔 (𝑡) were obtained in [18] for different forms of 𝑢 0 , for example, if 𝑢 0 (𝑥) = 𝐶𝑥 −𝑝 for large 𝑥 with 𝑝, 𝐶 > 0, then ln(min 𝐸 𝜔 (𝑡)) ∼ ln(max 𝐸 𝜔 (𝑡)) ∼ 𝑎𝑝 −1 𝑡 as 𝑡 → +∞. We refer to [3,19,20] for the acceleration propagation results recently developed for more general reaction-diffusion equations. We also refer to [9,11,13] for the acceleration propagation of fractional diffusion equations and [4,14,15] for non-local dispersal equations.…”

Acceleration of Propagation in a chemotaxis-growth system with slowly decaying initial data

“…where u −1 0 {A} = {x ∈ R, u 0 (x) ∈ A} denotes the inverse image of u 0 from the set A. We refer to [3,19,20] for the acceleration propagation results recently developed for more general reaction-diffusion equations. We also refer to [9,11,13] for the acceleration propagation in fractional diffusion equations and [4,14,15] for nonlocal dispersal equations.…”