We will us the following assumptions: (A1) The memory kernelg has typical properties g(0)>0, l= 1−µ1 Z ∞ 0 g(s)ds >0, (1.4) where µ1 &gt

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

DECAY OF SOLUTIONS FOR A PLATE EQUATION WITH p-LAPLACIAN AND MEMORY TERM

WENJUN LIU, GANG LI, LINGHUI HONG

Abstract. In this note we show that the assumption on the memory termg in Andrade [1] can be modified to beg⁰(t)≤ −ξ(t)g(t), whereξ(t) satisfies

ξ⁰(t)≤0, Z +∞

0

ξ(t)dt=∞.

Then we show that rate of decay for the solution is similar to that of the memory term.

1. Introduction

Consider a bounded domain Ω inR^N with smooth boundary Γ =∂Ω, and study the solutions to the problem

utt+ ∆²u−∆pu+ Z t

0

g(t−s)∆u(s)ds−∆ut+f(u) = 0 in Ω×R⁺, (1.1)

u= ∆u= 0 on Γ×R⁺, (1.2)

u(·,0) =u₀, u_t(·,0) =u₁ in Ω, (1.3) where ∆pu= div(|∇u|^p−2∇u) is thep-Laplacian operator.

This problem without the memory term models elastoplastic flows. We refer to [1] for a motivation and references concerning the study of problem (1.1)-(1.3). We will us the following assumptions:

(A1) The memory kernelg has typical properties g(0)>0, l= 1−µ1

Z ∞

0

g(s)ds >0, (1.4)

where µ₁ > 0 is the embedding constant for k∇uk²₂ ≤ µ₁k∆uk²₂. There exists a constantk₁>0 such that

g⁰(t)≤ −k1g(t), ∀ t≥0. (1.5) (A2) The forcing termf satisfies

f(0) = 0, |f(u)−f(v)| ≤k2(1 +|u|^ρ+|v|^ρ)|u−v|, ∀u, v∈R, (1.6) 0≤fb(u)≤f(u)u, ∀ u∈R, (1.7)

2000Mathematics Subject Classification. 35L75, 35B40.

Key words and phrases. Rate of decay; plate equation;p-Laplacian; memory term.

c

2012 Texas State University - San Marcos.

Submitted April 20, 2012. Published August 15, 2012.

1

wherek2 is a positive constant,fb(z) =Rz

0 f(s)ds, and 0< ρ≤ 4

N−4 ifN≥5 and ρ >0 if 1≤N ≤4.

(A3) The constantpsatisfies 2≤p≤ 2N−2

N−2 ifN ≥3 and p≥2 if N= 1,2. (1.8) Theorem 1.1 ([1, Theorem 2.1]). Assume that(A1)–(A3)hold.

(i) If the initial data(u0, u1)∈(H²(Ω)∩H₀¹(Ω))×L²(Ω), then problem (1.1)- (1.3)has a unique weak solution

u∈C(R⁺;H²(Ω)∩H₀¹(Ω))∩C¹(R⁺;L²(Ω)).

(ii) If the initial data (u₀, u₁)∈H_Γ³(Ω)×H₀¹(Ω), where H_Γ³(Ω) ={u∈H³(Ω)|u= ∆u= 0on Γ},

then problem (1.1)-(1.3)has a unique strong solution satisfying u∈L^∞(R⁺;H_Γ³(Ω)), u_t∈L^∞(R⁺;H₀¹(Ω)), u_tt∈L²(0, T;H⁻¹(Ω)).

(iii) In both cases, the energyE(t)of problem(1.1)-(1.3)satisfies the decay rate E(t)≤CE(0)e^−γt, t≥0,

for someC, γ >0, where E(t) = 1

2kut(t)k²₂+1

2k∆u(t)k²₂+1

pk∇u(t)k^p_p+ Z

Ω

fb(u(t))dx. (1.9) In this note, we shall extend the above exponential rate of decay to the general case, which is similar to that ofg. We use the following assumption which is weaker than (1.5).

(A4) There exists a positive differentiable functionξ(t) such that g⁰(t)≤ −ξ(t)g(t), ∀t≥0,

andξ(t) satisfies

ξ⁰(t)≤0, ∀t >0, Z +∞

0

ξ(t)dt=∞.

Then, we can prove the following main result.

Theorem 1.2. Assume that(A2)–(A4)and (1.4)hold. If the initial data(u0, u1)∈ (H²(Ω)∩H₀¹(Ω))×L²(Ω) or (u0, u1)∈H_Γ³(Ω)×H₀¹(Ω), then the energy E(t) of problem (1.1)-(1.3)satisfies the inequality

E(t)≤KE(0)e^{−kR0tξ(s)ds}, t≥0, (1.10) for someK, k >0.

Remark 1.3. We note that a similar decay rate was given in [5, Theorem 3.5].

However, unlike [5, (G2)] and [6, (A1)], we do not use the condition of |^ξ_ξ(t)^0(t)| ≤k here.

Remark 1.4. For ξ(t) ≡ k1, (1.10) recaptures the exponential decay rate in [1, Theorem 2.1]. Forξ(t) =a(1 +t)⁻¹, we can get polynomial decay rate, which is nt addressed in [1].

2. Proof of Theorem 1.2

Let us first prove the decay property for the strong solutionuof problem (1.1)- (1.3). We modify the perturbed energy method in [1] by using the idea of [4, 5].

Assume that condition (A4) holds and define the modified energy, as in [1], F(t) = 1

2ku_t(t)k²₂+1

2k∆u(t)k²₂+1

pk∇u(t)k^p_p+ Z

Ω

fb(u(t))dx

−1 2

Z t

0

g(s)ds

k∇u(t)k²₂+1

2(g◦ ∇u)(t), where

(g◦ ∇u)(t) = Z t

0

g(t−s)k∇u(t)− ∇u(s)k²₂ds.

Then we obtain

E(t)≤1 lF(t), andF(t) is decreasing because

F⁰(t) =−k∇ut(t)k²₂+1

2(g⁰◦ ∇u)(t)−1

2g(t)k∇u(t)k²₂

≤ −k∇ut(t)k²₂−1

2ξ(t)(g◦ ∇u)(t)≤0.

(2.1)

Let

Ψ(t) = Z

Ω

ut(t)u(t)dx and

Fε(t) =F(t) +εΨ(t), ∀ε >0.

To obtain the decay result, we use the following lemmas which are of crucial im- portance in the proof.

Lemma 2.1 ([1, Lemma 4.1]). There existsC1>0 such that

|Fε(t)−F(t)| ≤εC1F(t), ∀t≥0, ∀ ε >0.

Lemma 2.2 ([1, (27) in Lemma 4.2]). There exist positive constants C2, C3 such that

Ψ⁰(t)≤ −F(t) +C₂k∇ut(t)k²₂+C₃(g◦ ∇u)(t). (2.2) Now, we conclude the proof of the decay property. Let

ε₀= min 1 2C₁, 1

C₂ . It follows from Lemma 2.1 that, forε < ε0,

1

2F(t)≤F_ε(t)≤ 3

2F(t), t≥0. (2.3)

By the definition ofFε(t), (2.1) and (2.2), we obtain ξ(t)F_ε⁰(t) =ξ(t)F⁰(t) +εξ(t)Ψ⁰(t)

≤ −ξ(t)k∇ut(t)k²₂−ξ²(t)

2 (g◦ ∇u)(t)−εξ(t)F(t) +εC₂ξ(t)k∇ut(t)k²₂+εC₃ξ(t)(g◦ ∇u)(t)

≤ −(1−εC₂)ξ(t)k∇ut(t)k²₂−εξ(t)F(t) +εC₃ξ(t)(g◦ ∇u)(t)

≤ −εξ(t)F(t) +εC3ξ(t)(g◦ ∇u)(t)

≤ −εξ(t)F(t)−2εC3F⁰(t).

(2.4)

We set

L(t) =ξ(t)Fε(t) + 2εC3F(t).

Then,L(t) is equivalent toF(t). In fact, we have L(t)≤ξ(0)Fε(t) + 2εC3F(t)≤3

2ξ(0) + 2εC3

F(t) and

L(t)≥ 1

2ξ(t)F(t) + 2εC3F(t)≥2εC3F(t).

SinceF(t)≥lE(t)≥0 andξ⁰(t)≤0, from (2.3) and (2.4) we obtain L⁰(t) =ξ⁰(t)Fε(t) +ξ(t)F_ε⁰(t) + 2εC3F⁰(t)

≤ξ(t)F_ε⁰(t) + 2εC₃F⁰(t)

≤ −εξ(t)F(t)≤ −εkξ(t)L(t),

(2.5)

where we have used (2.4) andkis a positive constant.

A simple integration of (2.5) leads to

L(t)≤L(0)e^{−kR0tξ(s)ds}, ∀t≥0. (2.6) This proves the decay property for strong solutions inH_Γ³(Ω).

The result can be extended to weak solutions by standard density arguments, as in Cavalcanti et al. [2, 3].

Acknowledgements. This work was partly supported by the Tianyuan Fund of Mathematics (Grant No. 11026211) and the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant No. 09KJB110005).

References

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[2] M. M. Cavalcanti, V. N. Domingos Cavalcanti, T. F. Ma; Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains,Differential Integral Equations17(2004), no. 5-6, 495–510.

[3] M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano; Global existence and asymp- totic stability for the nonlinear and generalized damped extensible plate equation,Commun.

Contemp. Math.6(2004), no. 5, 705–731.

[4] W. J. Liu, J. Yu; On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms,Nonlinear Anal.74(2011), no. 6, 2175–2190.

[5] S. A. Messaoudi; General decay of the solution energy in a viscoelastic equation with a non- linear source,Nonlinear Anal.69(2008), no. 8, 2589–2598.

[6] S.-T. Wu; General decay of solutions for a viscoelastic equation with nonlinear damping and source terms,Acta Math. Sci. Ser. B Engl. Ed.31(2011), no. 4, 1436–1448.

Wenjun Liu

College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

E-mail address:[email protected]

Gang Li

College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

E-mail address:[email protected]

Linghui Hong

College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

E-mail address:[email protected]