Monotonicity of entire solutions to reaction-diffusion equations involving fractional p-Laplacian

Abstract

We obtain the one-dimensional symmetry and monotonicity of the entire positive solutions to some reaction-diffusion equations involving fractional p-Laplacian by virtue of the sliding method. More precisely, we consider the following problem { ∂ ⁡ u ∂ ⁡ t ⁢ ( x , t ) + ( - Δ ) p s ⁢ u ⁢ ( x , t ) = f ⁢ ( t , u ⁢ ( x , t ) ) , ( x , t ) ∈ Ω × ℝ , u ⁢ ( x , t ) > 0 , ( x , t ) ∈ Ω × ℝ , u ⁢ ( x , t ) = 0 , ( x , t ) ∈ Ω c × ℝ , \left\{\begin{aligned} \displaystyle{}\frac{\partial u}{\partial t}(x,t)+(-% \Delta)_{p}^{s}u(x,t)&\displaystyle=f(t,u(x,t)),&\hskip 10.0pt(x,t)&% \displaystyle\in\Omega\times\mathbb{R},\\ \displaystyle u(x,t)&\displaystyle>0,&\hskip 10.0pt(x,t)&\displaystyle\in% \Omega\times\mathbb{R},\\ \displaystyle u(x,t)&\displaystyle=0,&\hskip 10.0pt(x,t)&\displaystyle\in% \Omega^{c}\times\mathbb{R},\end{aligned}\right. where s ∈ ( 0 , 1 ) {s\in(0,1)} , p ≥ 2 {p\geq 2} , ( - Δ ) p s {(-\Delta)_{p}^{s}} is the fractional p-Laplacian, f ⁢ ( t , u ) {f(t,u)} is some continuous function, the domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} is unbounded and Ω c = ℝ n ∖ Ω {\Omega^{c}=\mathbb{R}^{n}\setminus\Omega} . Firstly, we establish a maximum principle involving the parabolic p-Laplacian operator. Then, under certain conditions of f, we prove the asymptotic behavior of solutions far away from the boundary uniformly in t ∈ ℝ {t\in\mathbb{R}} . Finally, the sliding method is implemented to derive the monotonicity and uniqueness of the bounded positive entire solutions. To our best knowledge, there has not been any results on the symmetry and monotonicity properties of solutions to the parabolic fractional p-Laplacian equations before.