Existence, uniqueness and stability of travelling waves in a discrete reaction–diffusion monostable equation with delay

Abstract

In this paper, we study the existence, uniqueness and asymptotic stability of travelling wavefronts of the following equation: u t ( x , t ) = D [ u ( x + 1 , t ) + u ( x - 1 , t ) - 2 u ( x , t ) ] - du ( x , t ) + b ( u ( x , t - r ) ) , where x ∈ R , t > 0 , D , d > 0 , r ⩾ 0 , b ∈ C 1 ( R ) and b ( 0 ) = dK - b ( K ) = 0 for some K > 0 under monostable assumption. We show that there exists a minimal wave speed c * > 0 , such that for each c > c * the equation has exactly one travelling wavefront U ( x + ct ) (up to a translation) satisfying U ( - ∞ ) = 0 , U ( + ∞ ) = K and lim sup ξ → - ∞ U ( ξ ) e - Λ 1 ( c ) ξ < + ∞ , where λ = Λ 1 ( c ) is the smallest solution to the equation c λ - D [ e λ + e - λ - 2 ] + d - b ′ ( 0 ) e - λ cr = 0 . Moreover, the travelling wavefront is strictly monotone and asymptotically stable with phase shift in the sense that if an initial data ϕ ∈ C ( R × [ - r , 0 ] , [ 0 , K ] ) satisfies lim inf x → + ∞ ϕ ( x , 0 ) > 0 and lim x → - ∞ max s ∈ [ - r , 0 ] | ϕ ( x , s ) e - Λ 1 ( c ) x - ρ 0 e Λ 1 ( c ) cs | = 0 for some ρ 0 ∈ ( 0 , + ∞ ) , then the solution u ( x , t ) of the corresponding initial value problem satisfies lim t → + ∞ sup R | u ( · , t ) / U ( · + ct + ξ 0 ) - 1 | = 0 for some ξ 0 = ξ 0 ( U , ϕ ) ∈ R .