In this paper, we study the homogeneous Robin boundary value problem for the Laplacian equation with the nonlocal memory term { u t − △ u = ∫ 0 t g ( t − s ) △ u ( x , s ) ds + a ( x , t ) | u | σ − 2 u + h ( x , t ) , ( x , t ) ∈ Q T = Ω × ( 0 , T ) ∂u ∂η + k ( x , t ) u = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] u ( x , 0 )= u 0 ( x ) in Ω where Ω ⊂ R n ( n ≥ 2 ) is a bounded open domain with sufficiently smooth boundary ∂Ω, T>0, σ is real constant such that σ>1, ∆ is the n dimensional Laplace operator; a( x,t), k( x,t), h( x,t) are given functions, g( s) is a given memory kernel. We show that under appropriate conditions on a, k, σ, g the problem has a global and local in time solution. We established conditions of uniqueness. Lastly, by using the energy method, we obtain sufficient conditions that the solutions of this problem with non-positive initial energy blow up in finite time.