A spectral condition for Liouville-type result of monostable KPP equation in periodic shear flows

Abstract

This paper is devoted to providing a simple condition, in term of spectral theory, that characterizes existence/nonexistence and uniqueness of positive bounded solution to

$$\begin{aligned} \nabla \cdot [n(y)\nabla u(x,y)] +\alpha (y)\partial _x u+\beta (y)\cdot \nabla _y u+f(x,y,u)=0 \quad (x,y)\in \mathbb {R}\times \mathbb {R}^{N-1},\qquad \end{aligned}$$

(0.1)

where f is of monostable KPP type nonlinearity and periodic in y. Our contribution answers a conjecture raised by Prof. H. Berestycki: which suitable assumption can impose at infinity that characterizes existence/nonexistence and uniqueness of (0.1) instead of the followings $\liminf _{|z|\rightarrow \infty }\partial _u f(z,0)>0$ as in Berestycki et al. (Ann Mat Pura Appl 186(4):469–507, 2007) and $\limsup _{|z|\rightarrow \infty }\partial _u f(z,0)<0$ as in Berestycki et al. (Bull Math Biol 71:399, 2008) and Berestycki and Rossi (Discret Contin Dyn Syst Ser B 21:41–67, 2008) but allow $\partial _u f(z,0)$ to change sign all the way as $|z|\rightarrow \infty$? Our result is simply based on maximum principle and complements to those in Berestycki et al. (Ann Mat Pura Appl 186:469–507, 2007; Bull Math Biol 71:399, 2008), Berestycki and Rossi (Discret Contin Dyn Syst Ser B 21:41–67, 2008) and Vo (J Differ Equ 259:4947–4988, 2015).