Asymptotic Energy Estimates for Nonlinear Petrovsky Plate Model Subject to Viscoelastic Damping

and Applied Analysis 3 point of view, these damping effects are modeled by integro-differential operators. Therefore, the dynamics of viscoelastic materials are of great importance and interest as they have wide applications in natural sciences. From the physical point of view, the problem 1.1 describes the positionw x, y, t of the material particle x, y at time t, which is clamped in the portion Γ0 of its boundary. Models of Petrovsky type are of interest in applications in various areas in mathematical physics, as well as in geophysics and ocean acoustics 1, 2 . The Petrovsky type models without memory were discussed in 3, 4 . Messaoudi 3 considered the initialboundary value problem utt Δ2u |ut|ut |u|p−2u, x ∈ Ω, t > 0, u x, t ∂νu x, t 0, x ∈ ∂Ω, t ≥ 0, u x, 0 u0 x , ut x, 0 u1 x , x ∈ Ω, 1.10 established an existence result for 1.10 , and showed that the solution continues to exist globally if m ≥ p, however if m < p and the initial energy is negative, the solution blows up in finite time. Chen and Zhou 4 proved that the solution of 1.10 blows up with positive initial energy. Moreover, she claimed that the solution blows up in finite time for vanishing initial energy under the condition m 2 by different method. The Petrovsky type equations with memory arouse the attention of mathematicians to study them. Alabau-Boussouira et al. 5 discussed the initial-boundary value problem of linear Petrovsky equation related to a plate model with memory, utt Δ2u − ∫ t 0 g t − s Δ2u t, s ds 0 in Ω × 0,∞ , u ∂νu 0 on ∂Ω × 0,∞ , u|t 0 u0, ut|t 0 u1 in Ω, 1.11 and showed that the solution decays exponentially or polynomially as t → ∞ if the initial data is sufficient small. Yang 6 considered the problem inN-dimensional space, utt Δ2u λut N ∑ i 1 ∂ ∂xi σi uxi in Ω × 0,∞ , u|∂Ω 0, ∂u ∂n ∣∣∣ ∂Ω 0 on 0,∞ , u x, 0 u0 x , ut x, 0 u1 x , x ∈ Ω, 1.12 and proved that under rather mild conditions on nonlinear terms and initial data the abovementioned problem admits a global weak solution and the solution decays exponentially to zero as t → ∞ in the states of large initial data and small initial energy. In particular, in the 4 Abstract and Applied Analysis case of space dimension N 1, the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. Muñoz Rivera et al. 7 considered the initial-boundary problem for viscoelastic plate equation, utt − γΔutt Δ2u t − ∫ t 0 g t − τ Δ2u τ dτ 0 in Ω × 0,∞ , u ( x, y, 0 ) u0 ( x, y ) , ut ( x, y, 0 ) u1 ( x, y ) in Ω, u ∂νu 0 on Γ0 × 0,∞ , B1u t B1 ∫∞ 0 g s u s ds 0 on Γ1 × 0,∞ , B2u t − γ∂νu′′ t B2 ∫∞ 0 g s u s ds 0 on Γ1 × 0,∞ , 1.13 and proved that the first and second order energies associated with its solution decay exponentially provided the kernel of the convolution also decays exponentially. When the kernel decays polynomially then the energy also decays polynomially. More precisely if the kernel g satisfies g t ≤ −c0g 1/p t , g, g1 1/p ∈ L1 R with p > 2, 1.14 then the energy decays as 1/ 1 t . On the recently related papers concerning the Petrovsky type models, the readers can see references 8–12 . In 13–15 , Li et al. proved the existence uniqueness, uniform rates of decay, and limit behavior of the solution to nonlinear viscoelastic Marguerre-von Karman shallow shells system, respectively. To our best knowledge, we do not find the research report on the problem 1.1 which is considered in this paper. Motivated by the above work, we obtain the energy functional associated with the equation decays exponentially or polynomially to zero as time goes to infinity. The main contribution of this paper are as follows. a The problem considered in this paper is nonlinear equation with integral dissipation, to our knowledge this model has not been considered; b the hypothesis on h and initial data are weaker; c we naturally define the energy by simple computation and only define simple auxiliary functionals to prove our result by precise priori estimates. The outline of this paper is the following. In Section 2, we present some material needed to be proved. Section 3 contains the statement and the proof of our results. Abstract and Applied Analysis 5 2. Notations and Preliminaries In this section, we will prepare some material needed in the proof of our main results. We use standard Lebesgue space L Ω and Sobolev spaceH Ω and adopt the following notations u, v : u, v L2 Ω , 〈u, v〉 : u, v L2 Γ1 , ‖u‖p : ‖u‖Lp Ω , ‖u‖ : ‖u‖L2 Ω . 2.1 We denote |A t | : ∑and Applied Analysis 5 2. Notations and Preliminaries In this section, we will prepare some material needed in the proof of our main results. We use standard Lebesgue space L Ω and Sobolev spaceH Ω and adopt the following notations u, v : u, v L2 Ω , 〈u, v〉 : u, v L2 Γ1 , ‖u‖p : ‖u‖Lp Ω , ‖u‖ : ‖u‖L2 Ω . 2.1 We denote |A t | : ∑ i,j aij t , d dt A t : A′ t ( aij t ) , A t · B t : ∑ i,j aij t bij t , A t ,B t : ∑ i,j ( aij t , bij t ) 2.2 for any pair of tensors A t aij t and B t bij t . We introduce the following space W Ω { w ∈ H2 Ω , w ∂νw 0 on Γ0 } . 2.3 Define the bilinear form a ·, · as follows