On Asymptotic Behavior and Blow-Up of Solutions for a Nonlinear Viscoelastic Petrovsky Equation with Positive Initial Energy

where a, b > 0 and p,m > 2 and proved that the solution is global whenm ≥ p while the solution blows up in finite time with negative initial energy when m < p. Later, this blowup result was improved by Chen and Zhou [3] with positive initial energy. For a related study, we may see the work of Wu and Tsai [4]. In [5], Amroun and Benaissa studied (3) by generalizing the damping term into the form of g(u t )


Introduction
In this work, we investigate the following nonlinear viscoelastic Petrovsky problem: where Ω is a bounded domain in R  ( ≥ 1) with a smooth boundary Ω, ,  > 1; ] is the unit outer normal on Ω; and  is a nonnegative memory term.
Under suitable growth conditions on , the author established global existence, uniqueness, and decay results by using the semigroup method.Messaoudi [2] investigated a nonlinearly damped semilinear Petrovsky equation where ,  > 0 and ,  > 2 and proved that the solution is global when  ≥  while the solution blows up in finite time with negative initial energy when  < .Later, this blowup result was improved by Chen and Zhou [3] with positive initial energy.For a related study, we may see the work of Wu and Tsai [4].In [5], Amroun and Benaissa studied (3) by generalizing the damping term into the form of (  ) and obtained the global existence of the solutions by means of the stable set method combined with the Faedo-Galerkin procedure.Very recently, in the presence of the strong damping, Li et al. [6] considered the following Petrovsky equation: Without any interaction between  and , the authors obtained the global existence and uniform decay of solutions when the initial data are in some stable set.And a blow-up result was established when  <  and the initial energy is less than the potential well depth.
In the presence of the viscoelastic term (i.e.,  ̸ = 0), Muñoz Rivera et al. [7] studied the following equation: (5) They proved that the memory effect produces strong dissipation capable of making uniform rate of decay for the energy.Later, in the presence of strong damping term, M. M. Cavalcanti et al. [8] considered and obtained a global existence for  ≥ 0 and uniform exponential decay for  > 0. This work was extended by Messaoudi and Tatar [9] to a situation where a nonlinear source term is competing with the damping induced by −Δ  and the integral term.Then in the case of  = 0, the same authors [10] showed that the damping induced by the viscoelastic term is enough to ensure global existence and uniform decay of solutions provided that the initial data are in some stable set by introducing a new functional and using the potential well method.Recently, Wu [11] improved [10] and a general decay result was obtained.In the presence of strong damping term Δ  and dispersive term Δ  , Xu et al. [12] considered the initial boundary value problem for the following viscoelastic wave equation: By introducing a family of potential wells, the authors not only obtained the invariant sets, but also proved the existence and nonexistence of global weak solution under some conditions with low initial energy.Furthermore, they established a blow-up result for certain solutions with arbitrary positive initial energy (high energy case).Very recently, Tahamtani and Peyravi [13] considered problem (1) and obtained the exponential decay of the energy under some assumptions on  without any interaction between source term and damping term.Under an appropriate restriction on , they also proved that the  +1 norm of any solution grows as an exponential function if  <  and the initial energy is negative.For other related works, we refer the readers to [14][15][16][17][18][19][20][21][22][23][24] and the references therein.Motivated by the above works, in this paper, we intend to consider problem (1) and establish some asymptotic behavior and blow-up results for solutions with positive initial energy.For our purpose, we use the functional () = ‖‖ 2 2 + ‖∇‖ 2  2 and give a modified manner to estimate the term | ∫ Ω |  | −1   d| so that the appearance of the form like  =  − () (for constants , , and ) which has been used in many earlier works (e.g., in [3,19,25]) can be avoided.
The paper is organized as follows.In Section 2 we present some assumptions and known results and state the main results.Section 3 is devoted to proof the the blow-up result-Theorem 4.

Preliminaries and Main Results
In this section, we first present some assumptions and known results which will be used throughout this work.
Next, we define the following functionals: where Remark 3. A multiplication of (1) by   and integration over Ω easily yield since   () ≤ 0.
Our main results read as follows.

Proof of the Main Results
We denote then we can prove the following lemma.
Lemma 5.For  ≥ 0, we have Proof.Obviously, Straightforward computations yield then which leads to .
An elementary calculation shows Using (G1) and Lemma 1 we arrive at which implies that () ≥  1 .

Journal of Function Spaces and Applications
To get (19), straightforward computations lead to which implies that  2  ∈ N. Also, for any  ∈ N, we note that Therefore we have  2 () =  for all  ∈ N. Hence, we complete the proof.