WELL-POSEDNESS AND BLOW UP OF SOLUTIONS FOR LAPLACIAN WITH NONOLOCAL
MEMORY UNDER ROBIN BOUNDARY CONDITION

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.