In this paper, we investigate a class of anisotropic initial-boundary value problems, involving the Finsler-Laplacian. Firstly, by using a first-order differential inequality technique, we provide some appropriate conditions on the data which guarantee the blow-up of the solution at some explicit finite time. Next, under different appropriate conditions on the data, we will make use of a comparison principle to prove the global boundedness of the solution. Finally, by using a maximum principle for an appropriate P-function, in the sense of L.E. Payne, we derive some explicit exponential time decay bounds for the solution and its derivatives..

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.