NEW GENERAL DECAY RESULT FOR A FOURTH-ORDER MOORE-GIBSON-THOMPSON EQUATION WITH MEMORY

. In this paper, we consider the fourth-order Moore-Gibson-Thompson equation with memory recently introduced by (Milan J. Math. 2017, 85: 215-234) that proposed the fourth-order model. We discuss the well-posedness of the solution by using Faedo-Galerkin method. On the other hand, for a class of relaxation functions satisfying g (cid:48) ( t ) ≤ − ξ ( t ) M ( g ( t )) for M to be increasing and convex function near the origin and ξ ( t ) to be a non- increasing function, we establish the explicit and general energy decay result, from which we can improve the earlier related results.


Introduction
The Moore-Gibson-Thompson (MGT) equation is one of the equations of nonlinear acoustics describing acoustic wave propagation in gases and liquids [13,15,30] and arising from modeling high frequency ultrasound waves [9,18] accounting for viscosity and heat conductivity as well as effect of the radiation of heat on the propagation of sound. This research field is highly active due to a wide range of applications such as the medical and industrial use of high intensity ultrasound in lithotripsy, thermotherapy, ultraound cleaning, etc. The classical nonlinear acoustics models include the Kuznetson's equation, the Westervelt equation and the Kokhlov-Zabolotskaya-Kuznetsov equation.
In order to gain a better understanding of the nonlinear MGT equation, we shall begin with the linearized model. In [15], Kaltenbacher, Lasiecka and Marchand investigated the following linearized MGT equation τ u ttt + αu tt + c 2 Au + bAu t = 0. (1.1) For equation (1.1), they disclosed a critical parameter γ = α − c 2 τ b and showed that when γ > 0, namely in the subcritical case, the problem is well-posed and its solution is exponentially stable; while γ = 0, the energy is conserved. Since its appearance, an increasing interest has been developed to study the MGT equation, see [4,5,8,10,14]. Caixeta, Lasiecka and Cavalcanti [4] considered the following nonlinear equation τ u ttt + αu tt + c 2 Au + bAu t = f (u, u t , u tt ). (1.2) They proved that the underlying PDE generates a well-posed dynamical system which admits a global and finite dimensional attractor. They also overcomed the difficulty of lacking the Lyapunov function and the lack of compactness of the trajectory. Now, we concentrate on the stabilization of MGT equation with memory which has received a considerable attention recently. For instance, Lasiecka and Wang [17] studied the following equation: τ u ttt + αu tt + bAu t + c 2 Au − t 0 g(t − s)Aw(s)ds = 0, (1.3) where α − c 2 τ b ≥ 0 and the form of w classifies the memory into three types. By imposing the assumption on the relaxation function g, for a positive constant c 0 , as g (t) ≤ −c 0 g(t), (1.4) they discussed the effect of memory described by three types on decay rates of the energy when α − c 2 τ b > 0. Moreover, in the critical case α − c 2 τ b = 0, they proved an exponential rate of decay for the solution of (1.3) under "the right mixture" of memory. Lasiecka and Wang [18] showed the general decay result of the equation  Alves et al. [1] investigated the uniform stability of equation (1.5) encompassing three different types of memory in a history space set by the linear semigroup theory. Moreover, we refer the reader to [3,6,7,12,24,25,26,28] for other works of the equation(s) with memory.
More recently, Filippo and Vittorino [11] considered the fourth-order MGT equation u tttt + αu ttt + βu tt + γAu tt + δAu t + Au = 0. (1.6) They investigated the stability properties of the related solution semigroup. And, according to the values of certain stability numbers depending on the strictly positive parameters α, β, γ, δ, , they established the necessary and sufficient condition for exponential stability. For other related results on the higher-order equations, please see [20,27,34,35,36,37] and the references therein.
Motivated by the above results, we intend to study the following abstract version of the fourth-order Moore-Gibson-Thompson (MGT) equation with a memory term where α, β, γ, δ, are strictly positive constants, A is a strictly positive self-adjoint linear operator defined in a real Hilbert space H where the (dense) embedding D(A) ⊂ H need not to be compact. And we consider the following initial conditions A natural question that arised in dealing with the general decay of fourth-order MGT equation with memory: • Can we get a general decay result for a class of relaxation functions satisfying g (t) ≤ −ξ(t)M (g(t)) for M to be increasing and convex function near the origin and ξ(t) to be a nonincreasing function?
Mustafa answered this question for viscoelastic wave equations in [31,32]. Messaoudi and Hassan [29] considered the similar question for memory-type Timoshenko system in the cases of equal and non-equal speeds of wave propagation. Moreover, they extended the range of polynomial decay rate optimality from p ∈ 1, 3 2 to p ∈ [1, 2) when g satisfies g (t) ≤ −ξ(t)g p (t). We refer to [19] for the non-equal wave speeds case. And, Liu et al. [22,23] also concerned with the similar question for third-order MGT equations with memory term.
The aim of this paper is to establish the well-posedness and answer the above mention question for fourth-order MGT equation with memory (1.7). We first use the Faedo-Galerkin method to prove the well-posedness result. We then use the idea developed by Mustafa in [31,32], taking into consideration the nature of fourthorder MGT equation, to prove new general decay results for the case γ − δ α > 0 and β − α δ > 0, based on the perturbed energy method and on some properties of convex functions. Our result substantially improves and generalizes the earlier related results in previous literature.
The rest of our paper is organized as follows. In Section 2, we give some assumptions and state our main results. In Section 3, we give the proof of well-posedness. In Section 4, we state and prove some technical lemmas that are relevant in the entire work. In Section 5, we prove the general decay result.

Preliminaries and main results
In this section, we consider the following assumptions and state our main results. We use c > 0 to denote a positive constant which does not depend on the initial data.
First, we consider the following assumptions as in [11] for (A1), in [18] for (A3), (A5) and in [31] for (A2), (A4): (A1) γ − δ α > 0 and β − α δ > 0. (A2) g : R + → R + is a non-increasing differentiable function such that There exists a non-increasing differentiable function ξ : R + → R + and a C 1 function M : [0, ∞) → [0, ∞) which is either linear or strictly increasing and strictly convex C 2 function on (0, r], r ≤ g(0), with M (0) = M (0) = 0, such that Remark 1. ([31, Remark 2.8]) (1) From assumption (A2), we deduce that Furthermore, from the assumption (A4), we obtain that there exists t 0 ≥ 0 large enough such that The non-increasing property of g(t) and ξ(t) gives A combination of these with the continuity of H, for two constants a, d > 0, yields Consequently, for any t ∈ [0, t 0 ], we get and, hence, (2) If M is a strictly increasing and strictly convex C 2 function on (0, r], with M (0) = M (0) = 0, then it has an extension M , which is strictly increasing and strictly convex C 2 function on (0, ∞). For example, if we set M (r) = A, M (r) = B, M (r) = C, we can define M , for any t > r, by Then, inspired by the notations in [11], we define the Hilbert spaces In order to simplify the notation, we denote the usual space H 0 by H. The phase space of our problem is Moreover, we will denote the inner product of H by (·, ·) and its norm by · . After that, we introduce the following energy functional where G(t) = t 0 g(s)ds and for any v ∈ L 2 loc R + ; L 2 (Ω) , As in [31], we set, for any 0 < ν < 1, The following lemmas play an important role in the proof of our main results.
Lemma 2.1. ( [31]) Assume that condition (A2) holds. Then for any u ∈ L 2 loc (R + ; )(The generalized Young inequality) If f is a convex function defined on a real vector space X and ite convex conjugate is denoted by f * , then We are now in a position to state the well-posedness and the general decay result for problem (1.7)-(1.8).   Remark 2. Assume that M (s) = s p , 1 ≤ p < 2 in (A4), then by simple calculations, we see that the decay rate of E(t) is given by, for constants k, k and C, (2.6)

Proof of the well-posedness
In this section, we will prove the global existence and uniqueness of the solution of problem (1.7)-(1.8). Firstly, we give the following lemmas.
which is trivially true.
Proof. From the definition of E(t), we have Then, we estimate the sixth term of the above equality A combination of the above results, we complete the proof of lemma. Now, we prove the well-posedness result of problem (1.7)-(1.8).
Proof of Theorem 2.1. The proof is given by Faedo-Galerkin method and combines arguments from [16,39,38]. We present only the main steps.
Step 1. Approximate problem We construct approximations of the solution u by the Faedo-Galerkin method as follows. For every m ≥ 1, let W m = span{w 1 , · · · , w m } be a Hilbertian basis of the space We define now the approximations: where u m (t) are solutions to the finite dimensional Cauchy problem (written in normal form): According to the standard theory of ordinary differential equation, the finite dimensional problem (3.2)-(3.3) has a local solution (u m (t), u m t (t), u m tt (t), u m ttt (t)) in some interval [0, T m ) with 0 < T m ≤ T , for every m ∈ N. Next, we present some estimates that allow us to extend the local solutions to the interval [0, T ], for any given T > 0.
The proof now can be completed arguing as in [21].

Technical lemmas
In this section, we state and prove some lemmas needed to establish our general decay result.
Proof. By differentiating F 2 (t) with respect to t, using (1.7) and integrating by parts, we obtain Now, we estimate the terms in the right-hand side of the above identity. Using Young's inequality, we obtain, for 0 < ε 1 < 1, Also, we have and − g(0) Exploiting Young's inequality and (A5), we get A combination of all the above estimates gives the desired result.
Proof. Using the equation (1.7), a direct computation leads to the following identity Now, the first and third terms in the right-hand side of (4.6) can be estimated as follows: where 0 < ε 2 < 1. Using Young's inequality and Lemma 2.1, we get Then, combining the above inequalities, we obtain the desired result.
Proof. Noting that f (t) = −g(t), we see that Exploiting Young's inequality and the fact t 0 g(s)ds ≤ − l, we obtain Moreover, taking account of f (t) ≤ f (0) = − l, we have Combining the above estimates, we arrive at the desired result.
satisfies, for a suitable choice of N, N 2 , N 3 , and the estimate, for all t ≥ t 0 , where t 0 has been introduced in Remark 2.1.
Proof. Combining Lemmas 4.1-4.4 and recalling that g = νg − h, we obtain, for all t ≥ t 0 , At this point, we need to choose our constants very carefully. First, we choose and The above choice yields Then, we choose N 2 large enough so that Next, we choose N 3 large enough so that Now, as ν 2 g(s) νg(s)−g (s) < g(s), it is easy to show, using the Lebesgue dominated convergence theorem, that Hence, there is 0 < ν 0 < 1 such that if ν < ν 0 , then Now, let us choose N large enough and choose ν satisfying So we arrive at, for positive constant c, On the other hand, from Lemma 3.2, we find that Therefore, we can choose N even large (if needed) so that (4.8) is satisfied.

Proof of the general decay result
In this section, we will give an estimate to the decay rate for the problem (1.7)-(1.8).
Proof of Theorem 2.2. Our proof starts with the observation that, for any t ≥ t 0 , t0 0 g(s) which are derived from (2.2) and Lemma 4.1 and can be used in (4.8).
Taking F(t) = L(t) + cE(t), which is obviously equivalent to E(t), we get, for all t ≥ t 0 , where m is a positive constant. Then, we obtain that We consider the following two cases relying on the ideas presented in [31]. We multiply (5.1) by ξ(t), then on account of (A1)-(A4) and Lemma 4.1, we obtain, for all t ≥ t 0 , Therefore, As ξ(t) is non-increasing and F(t) ∼ E(t), we have It follows immediately that We may now integrate over (t 0 , t) to conclude that, for two positive constants k 1 and k 2 By the continuity of E(t), we have (ii) M is nonlinear. First, we define the functional Obviously, L(t) is nonnegative. And, by Lemma 4.5 and Lemma 4.6, there exists b > 0 such that Therefore, integrating the above inequality over (t 0 , t), we see at once that It is sufficient to show that ∞ 0 E(s)ds < ∞ (5.2) and Now, we define a functional λ(t) by Clearly, we have After that, we define another functional I(t) by Now, the following inequality holds under Lemma 4.1 and (5.2) that    Let 0 < ε 0 < r, we define the functional F 1 (t) by Then, recalling that E (t) ≤ 0, M > 0 and M > 0 as well as making use of estimate (5.6), we deduce that F 1 (t) ∼ E(t) and also, for any t ≥ t 0 , we have Taking account of Lemma 2.3, we obtain So, combining (5.7), (5.8) and (5.9), we obtain From this, we multiply the above inequality by ξ(t) to get ξ(t)F 1 (t) ≤ −(mE(0) − cε 0 )ξ(t) E(t) E(0) M ε 0 E(t) E(0) + cqλ(t) + ξ(t)E (t).