Optimal polynomial decay for a coupled system of wave with past history

1. Introduction

In this paper we consider a coupled system of wave with past history given by

where \(\Omega\) is an open bounded set of \(\mathbb{R}^n\) with smooth boundary \(\Gamma\).

The above model can be used to describe the evolution of a system consisting of two elastic membranes subject to an elastic force that attracts one membrane to the other with coefficient \(\alpha >0\). Note that the term \(\int^{\infty}_0g(s)\Delta u(t-s)\;ds\), acts on the first membrane as a stabilizer.

Many interesting physical phenomena such as viscoelasticity, hereditary polarization in dielectrics, population dynamics or heat flow in real conductors, to name some, are modeled by differential equations which are influenced by the past values of one or more variables in play so-called equations with memory. The main problem in the analysis of equations of this kind lies in their nonlocal character, due to the presence of the memory term given by the time convolution of the unknown function against a suitable memory kernel. The memory term may can produce loss of exponential stability for the system, [1]. The history of nonlocal problems with integral conditions for partial differential equations is recent and goes back to [2]. In [3], a review of the progress in the nonlocal models with integral type was given with many discussions related to physical justifications, advantages, and numerical applications.

Coupled wave system has been considered in various contexts. In [4] both wave equations are damped on the boundary and the coupling is effected by compact operator and exponential stability is obtained when the boundary damping is linear. Boundary damping is also considered in [5, 6]. On exact boundary controllability for linearly coupled wave equations, we refer [7]. Uniform exponential stability was given in [8] for wave equations coupled in parallel with coupling distributed springs and viscous dampers due to different boundary conditions and wave propagation speeds.

For weak damping acting only one equation, the optimal polynomial decay to coupled wave equations was studied in [9]. In [10], it was proved that the energy of associated coupled system weakly dissipative decays polynomially with explicit polynomial decay rates for sufficiently smooth solutions. In [11], under new compatibility assumptions, the authors proved polynomial decay for the energy of solutions and optimized previous results by interpolation techniques introduced in [10].

On the asymptotic behavior of the coupled system (1)-(5) we refer the work [12] where the authors proved by method introduced in [11] that the solution has a polynomial rate of decay. The central question of this work is to analyze what is the best decay rate of the system (1)-(5). In this direction, we prove that the associated semigroup decays with rate \(t^{-\frac{1}{2}}\). Moreover we show that the rate is optimal. For what we know in the literature the optimal rate of polynomial decay for coupled wave systems with memory was not previously considered.

The mathematical structure of the paper is organized as follows: In Section 2 we discuss the existence, regularity and uniqueness of strong solutions of the system (1)-(5) by semigroup technique, see [13]. In Section 3 we study the lack of exponential decay using Prüss's results [14]. Finally in section 4 we show that the system is polynomially stable giving an optimal decay rate. That is, this rate cannot be improved. For this we use the recent result due to Borichev and Tomilov [15].

2. Semigroup Setup

Following the approach of Dafermos [16] and Fabrizio and Morro [17], we consider \(\eta =\eta ^{t}(s)\), the relative history of \(u\), defined as Hence, putting \[ \beta_0=1-\int_0^{\infty}g(s)\;ds>0, \] the system (1)-(5) turns into the system

where the third equation is obtained differentiating (6) with respect to \(s\) and the condition (13) means that the history is considered as an initial value.

We study the existence and uniqueness of solutions for the system (7)-(13) using the semigroup techniques. As in [18], we use the following hypotheses on \(g\)

In view of (14), let \(L^2_g(\mathbb{R}^+;H^1_0(\Omega))\) be the Hilbert space of \(H^1_0(\Omega)\)-value functions on \(\mathbb{R}^+\), endowed with the inner product \[ (f,h)_{L^2_g(\mathbb{R}^+,H^1_0(\Omega))}=\int^{\infty}_0g (s)\int_{\Omega}\nabla f(x,s) \cdot \nabla\overline{h(x,s)} \; dx \; ds. \] To give an accurate formulation of the evolution problem we introduce the product Hilbert spaces \[ \mathcal{H}=H^1_0(\Omega)\times L^2(\Omega)\times H^1_0(\Omega)\times L^2(\Omega)\times L^2_g(\mathbb{R}^+;H^1_0(\Omega)) \] endowed with the following inner product where \(U=(u_1,u_2,u_3,u_4,u_5)^T\), \(V=(v_1,v_2,v_3,v_4,v_5)^T\in \mathcal{H}\).
Let \(U=(u,u_t,v,v_t,\eta)^T\) be and we define the operator \(\mathcal{A}:D(\mathcal{A})\subset \mathcal{H}\rightarrow \mathcal{H}\) given by \begin{eqnarray*} \begin{array}{c} \mathcal{A}=\left[ \begin{array}{ccccc} \ 0 & I & 0 & 0 & 0 \\ \beta_0 \Delta & 0 & -\alpha I & 0 & \mathcal{T} \\ \ 0 & 0 & 0 & I & 0 \\ -\alpha I & 0 & \Delta & 0 & 0 \\ \ 0 & I & 0 & 0 & -(\cdot)_s \\ \end{array} \right] \end{array} \end{eqnarray*} with domain \begin{eqnarray*} D(\mathcal{A})&=&\{ (u,\varphi,v,\psi,\eta)^T\in \mathcal{H}; \quad \beta_0 u-\int^{\infty}_0g(s)\eta(s) \; ds \in H^1_0(\Omega)\cap H^2(\Omega),\\ &&\varphi \in H^1_0(\Omega), v\in H^1_0(\Omega)\cap H^2(\Omega), \psi \in H^1_0(\Omega), \eta \in D(\mathcal{T}) \} \end{eqnarray*} where \[ \mathcal{T}\eta = \int^{\infty}_0g(s)\Delta \eta (s) \; ds, \quad \forall \eta \in D(\mathcal{T}) \] with \[ D(\mathcal{T})=\{ \eta \in L^2_g(\mathbb{R}^+;H^1_0(\Omega));\eta_s\in L^2_g(\mathbb{R}^+;H^1_0(\Omega)), \eta(0)=0\}, \] where \(\eta_s\) is the distributional derivative of \(\eta\) with respect to the internal variable \(s\). Therefore, the system (7)-(13) is equivalent to with \(U=(u,u_t,v,v_t,\eta)^T\), \(U_0=(u_0,u_1,v_0,v_1,\eta_0)^T\). With the above notations, we have the following result.

Theorem 1. The operator \(\mathcal{A}\) generate a C\(_0\)-semigroup \(S(t)\) of contraction on \(\mathcal{H}\). Thus, for any initial data \(U_0\in \mathcal{H}\), the problem (7)-(13) has a unique weak solution \(U(t)\in C^0([0,\infty[, \mathcal{H})\). Moreover, if \(U_0\in D(\mathcal{A})\), then \(U(t)\) is strong solution of (7)-(13), that is, \(U(t)\in C^1([0,\infty[,\mathcal{H})\cap C^0([0,\infty[,D(\mathcal{A}))\).

Proof. It is easy to see that \(D(\mathcal{A})\) is dense in \(\mathcal{H}\). Now, for \(U=(u, u_t,v, v_t,\eta)^T\in D(\mathcal{A})\) and using the inner product (15), we get \begin{align*} &\langle {\mathcal{A}}U,U\rangle =\beta_0\int_{\Omega}\nabla u_t\cdot \nabla \overline{u}\;dx+\int_{\Omega}(\beta_0\Delta u-\alpha v+\int^{\infty}_0g(s) \Delta \eta(s)\;ds)\overline{u_t}\;dx\\ &\;\;\;+\int_{\Omega}\nabla v_t\cdot \nabla \overline{v}\;dx+\int_{\Omega}(\Delta v-\alpha u)\overline{v_t}\;dx+\alpha \int_{\Omega}(u_t\overline{v}+v\overline{u_t})\;dx+\int^{\infty}_0g(s)\int_{\Omega}\nabla(u_t-\eta_s(s))\cdot\nabla \overline{\eta} (s)\;dx\;ds \end{align*} from where it follows that \begin{eqnarray*} \langle {\mathcal{A}}U,U\rangle=-\int^{\infty}_0g(s)\int_{\Omega}\nabla \eta_s(s)\cdot \nabla \overline{\eta}(s)\;dx\;ds. \end{eqnarray*} Integrating by parts and using (14), we have \begin{eqnarray*} {\mathcal{R}}e\langle {\mathcal{A}}U,U\rangle=\frac{1}{2}\int^{\infty}_0g'(s)\int_{\Omega}|\nabla \eta(s)|^2\;dx\;ds\leq -\frac{q_1}{2}\int^{\infty}_0g(s)\int_{\Omega}|\nabla \eta(s)|^2\;dx\;ds\leq 0. \end{eqnarray*} Therefore, \(\mathcal{A}\) is a dissipative operator.
Next, we show that \((I-\mathcal{A})\) is maximal. For this, let us consider the equation \[ (I-\mathcal{A})U=F \] where \(U=(u,\varphi,v,\psi,\eta)^T\) and \(F=(f^1,f^2,f^3,f^4,f^5)^T\in \mathcal{H}\). Then, in terms of its components, we can write

Integrating (22), we have Substituting \(\varphi\) and \(\eta\) from (18) and (23) into (19), we get where \[ \beta_g=\beta_0+\int^{\infty}_0g(s)(1-e^{-s})\;ds. \] Note that \(\beta_g\) is a positive constant in virtue of (14). Moreover, it can be shown that the right-had side of (24) is in \(H^{-1}(\Omega)\).
On the other hand, the substitution of \(\psi\) given in (20) into (21) gives us First we prove that \(u, v \in H^1_0(\Omega)\). To do this, let us consider the bilinear form where \(\Phi_1=(u_1,v_1)\) and \(\Phi_2=(u_2,v_2)\). Then, Lax-Milgram theorem (see [19]) provides existence and uniqueness of the solutions \[ u,v\in H^1_0(\Omega). \] From (18) and (20), we have \(\varphi,\psi \in H^1_0(\Omega)\). Now, from (23), we obtain from where it follows that \[ \eta \in L^2_g(\mathbb{R}^+;H^1_0(\Omega)). \] From (19), we get \[ \beta_0 \Delta u+ \int^{\infty}_0g(s)\Delta \eta(s)\;ds \in L^2(\Omega). \] On the other hand, from (22), we obtain \begin{eqnarray*} ||\eta_s||^2_{L^2_g(\mathbb{R}^+;H^1_0(\Omega))}\leq C\left(||\varphi||^2_{H^1_0(\Omega)}+||f^5||^2_{L^2_g(\mathbb{R}^+;H^1_0(\Omega))} +||\eta||^2_{L^2_g(\mathbb{R}^+;H^1_0(\Omega))}\right)\,. \end{eqnarray*} From where it follows that \[ \eta_s\in L^2_g(\mathbb{R}^+;H^1_0(\Omega)). \] Again from (23), we have \[ \eta(0)=0. \] Thus, \(I-\mathcal{A}\) is maximal. Then, thanks to the Lumer-Phillips theorem (see [13], Theorem 4.3), the operator \(\mathcal{A}\) generates a C\(_0\)-semigroup of contractions \(e^{t\mathcal{A}}\) on \(\mathcal{H}\). The proof is now complete.

3. Lack of exponential decay

Our starting point is to show that the semigroup associated to the system (7)-(13) is not exponential stable. To show this, we assume that \(g(t)=e^{-\mu t}\), with \(t\in \mathbb{R}^+\) and \(\mu>1\). We will use the Prüss's theorem [14] to prove the lack of exponential stability.

Theorem 2. Let \(S(t)=e^{\mathcal{A}t}\) be a C\(_0\)-semigroup of contractions on Hilbert space. Then \(S(t)\) is exponentially stable if and only if \[ \rho(\mathcal{A})\supseteq \{i\beta:\beta\in \mathbb{R}\}\equiv i\mathbb{R} \] and \[ \overline{\lim_{|\beta|\rightarrow \infty}}\|(i\beta-\mathcal{A})^{-1}\|< \infty \] hold, where \(\rho(\mathcal{A})\) is the resolvent set of \(\mathcal{A}\).

To do this, let us consider the spectral problem: where \[ \lim_{m \rightarrow \infty}\lambda_{m}=+\infty. \] The following theorem describes the main results of this section.

Theorem 3. Let \(S(t)\) be C\(_0\)-semigroup of contractions generated by \(\mathcal{A}\). Then \(S(t)\) is not exponentially stable.

Proof. Here we will use the Theorem 2. That is, we will show that there exists a sequence of values \(\lambda_{m}\) such that

It is equivalent to prove that there exist a sequence of data \(F_m\in {\mathcal{H}}\) and a sequence of complex numbers \(\lambda_m\in i\mathbb{R}\), with \(||F_m||_{\mathcal{H}}\leq1\) such that where with \(U_m\) not bounded. To simplify the notation we will omit the subindex \(m\). Then, the Equation (31) becomes Let us consider \(f1=f^3=f^5=0\) and \(f^2=f^4=w_m\) to obtain \(\varphi=i\lambda u\) e \(\psi=i\lambda v\). Then, the system (32) becomes We look for solutions of the form \[ u=aw_m, \quad v=bw_m, \quad \varphi=cw_m,\quad \psi=dw_m,\quad \eta(x,s)=\gamma(s)w_m \] with \(a,b,c,d\in \mathbb{C}\) and \(\gamma(s)\) depend on \(\lambda\) and will be determined explicitly in the sequel. From (33), we get \(a\) and \(b\) satisfy Solving (34)\(_3\) we get Since \(\eta(0)=0\) then \(C=-a\), and (35) becomes Then, from (36) we have where \[ b_0=\int^{\infty}_0g(s)\;ds. \] Now, choosing \(\lambda=\sqrt{\lambda_m}\), using the equation (34)\(_1\) and (34)\(_2\) we obtain \begin{eqnarray*} &&a=\frac{1}{\alpha}, \\ &&b=\frac{\lambda_m(1-\beta_0)}{\alpha^2}-\frac{\lambda_m}{\alpha}\int^{\infty}_0g(s)\gamma(s)\;ds+\frac{1}{\alpha}, \\ &&c=i\frac{\sqrt{\lambda_m}}{\alpha},\\ &&d=i\sqrt{\lambda_m}(\frac{\lambda_m(1-\beta_0)}{\alpha^2} -\frac{\lambda_m}{\alpha}\int^{\infty}_0g(s)\gamma(s)\;ds+\frac{1}{\alpha}). \end{eqnarray*} Recalling that \[ \varphi=cw_m=i\frac{\sqrt{\lambda_m}}{\alpha}w_m \] we get \[ ||\varphi||^2_{L^2(\Omega)}=\frac{\lambda_m}{\alpha^2}. \] Therefore we have \[ \lim_{m\rightarrow \infty}||U_m||^2_{\mathcal{H}}\geq \lim_{m\rightarrow \infty}||\varphi||^2_{L^2(\Omega)}=\lim_{m\rightarrow \infty}\frac{\lambda_m}{\alpha^2}= \infty. \] Using theorem 3 follows that \(S(t)\) is not exponentially stable. The proof is now complete.

4. Polynomial decay and optimally result

In this section we study the polynomial decay associated to the system (7)-(13) and subsequently we find the optimal rate of decay. Then, let us consider the resolvent equation \[ (i\lambda I- \mathcal{A})U=F,\quad \mbox{with}\quad \lambda \in \mathbb{R} \quad \mbox{and}\quad F\in \mathcal{H}, \] that is, In the next step we shall show three lemmas important to proof the main result.

Lemma 1. The solutions of the system (7)-(13), given by the Theorem \ref{teo2.1}, satisfies \begin{eqnarray*} \int_{\Omega}\int^{\infty}_0g(s)|\nabla \eta|^2\; ds \; dx\leq K |\lambda|^2 ||U||_{\mathcal{H}}||F||_{\mathcal{H}} \end{eqnarray*} where \(K\) is a positive constant and \(|\lambda|> 1\).

Proof. Multiplying the equality (39) by \(\overline{\varphi}\) and integrating by parts on \(\Omega\), we get

Substituting \(\varphi\) given in (38) into \(I_1\) and \(I_2\), we have and Now, substituting \(\varphi\) given in (42) into \(I_3\) and integrating by parts, we obtain Substituting (44), (45) and (46) into (43), we get Taking the real part on the left side of the above equality and using the hypotheses (14) on \(g\), our conclusion follows.

Lemma 2. For any \(\epsilon>0\), there exists a positive constant \(K_{\epsilon}\) such that \begin{eqnarray*} &&\beta_0\int_{\Omega}|\nabla u|^2\; dx+\int_{\Omega}|\nabla v|^2\;dx+\alpha \int_{\Omega}(u\overline{v}+v\overline{u})\;dx\\ &&\leq \int_{\Omega}|\varphi|^2\;dx+\int_{\Omega}|\psi|^2\;dx+\epsilon \int_{\Omega}|\nabla u|^2\;dx+K_{\epsilon} |\lambda|^2 ||U||_{\mathcal{H}}||F||_{\mathcal{H}}+K||U||_{\mathcal{H}}||F||_{\mathcal{H}} \end{eqnarray*} where \(K\) is a positive constant.

Proof. Multiplying the equalities (39) and (41) by \(\overline{u}\) and \(\overline{v}\), respectively, integrating by parts on \(\Omega\) and summing up the result, we get

Substituting \(\overline{i\lambda u}\) given in (38) into \(I_4\) and \(\overline{i\lambda v}\) given in (40) into \(I_5\), we find Now, using Poincaré and Young inequalities, we have Using the Lemma 1, our conclusion follows.

Lemma 3. Under the conditions of the previous lemma, we have \begin{eqnarray*} \frac{b_0}{2}\int_{\Omega}|\varphi|^2\; dx&\leq &\epsilon\int_{\Omega}(|\nabla u|^2+|\nabla v|^2)\;dx+ K_{\epsilon} |\lambda|^2 ||U||_{\mathcal{H}}||F||_{\mathcal{H}}+K ||U||_{\mathcal{H}}||F||_{\mathcal{H}} \end{eqnarray*} and \begin{eqnarray*} \left(\frac{1}{2}-\frac{K}{|\lambda|^2}\right)\int_{\Omega}|\psi|^2\;dx\leq K_{\epsilon} |\lambda|^2 ||U||_{\mathcal{H}}||F||_{\mathcal{H}}+K ||U||_{\mathcal{H}}||F||_{\mathcal{H}} \end{eqnarray*} with \(|\lambda|>1\) large enough.

Proof. Multiplying the equation (42) by \(\int^{\infty}_0g(s)\;ds\overline{\varphi}\) and integrating on \(\Omega\), we find \begin{eqnarray*} \underbrace{i\lambda \int^{\infty}_0g(s)\int_{\Omega}\eta(s)\overline{\varphi}\;dx\;ds}_{:=I_6}-b_0\int_{\Omega}|\varphi|^2\;dx +\int^{\infty}_0g(s)\int_{\Omega}\eta_s(s)\overline{\varphi}\;dx\;ds =\int^{\infty}_0g(s)\int^{\infty}_0f^5(s)\overline{\varphi}\;dx\;ds \end{eqnarray*} where \(b_0=\int^{\infty}_0g(s)\;ds\). On the other hand, noting that \[ \int^{\infty}_0g(s)\int_{\Omega}\eta_s(s)\overline{\varphi}\;dx\;ds=-\int^{\infty}_0g'(s)\int_{\Omega}\eta(s)\overline{\varphi}\;dx\;ds \] and substituting \(\overline{i\lambda \varphi}\) given in (39) into \(I_6\), we get

Using the hypotheses on \(g\) given in (14) and taking into account the Poincaré and Young inequalities, we have \begin{eqnarray*} \frac{b_0}{2}\int_{\Omega}|\varphi|^2\;dx \leq \epsilon \int_{\Omega}(|\nabla u|^2+|\nabla v|^2)\;dx+K_{\epsilon}||\eta||^2_{L^2_g(\mathbb{R}^+;H^1_0(\Omega))}+K ||U||_{\mathcal{H}}||F||_{\mathcal{H}}. \end{eqnarray*} Using the Lemma 1, follows the first inequality. To show the second inequality, we substitute the equation (38) into (42). This gives, Now, we substitute \(u\) given in (41) into (52). Then, we obtain, Multiplying the equation (53) by \(\overline{\int^{\infty}g(s)\psi}\), integrating by parts on \(\Omega\) and proceeding as to obtain the first estimate, we have \begin{eqnarray*} \frac{1}{2}\int_{\Omega}|\psi|^2\;dx\leq \frac{K}{|\lambda|^2}\int_{\Omega}|\psi|^2\;dx+K|\lambda|^2||U||_{\mathcal{H}}||F||_{\mathcal{H}}+K||F||^2_{\mathcal{H}}. \end{eqnarray*} From where follows the second inequality. The proof is now complete.

Now, we are in the position the main result of this paper.

Theorem 4. The semigroup associated to the system (7)-(13) is polynomially stable and \[ ||S(t)U_0||_{\mathcal{H}}\leq \frac{K}{\sqrt{t}}||U_0||_{D(\mathcal{A})}. \] Moreover, this result is optimal.

Proof. From Lemmas 1, 2 and 3, choosing \(\epsilon >0\) small enough and for \(|\lambda|> 1\) large enough, we have \[ ||U||^2_{\mathcal{H}}\leq K|\lambda|^2||U||_{\mathcal{H}} ||F||_{\mathcal{H}}+K ||F||^2_{\mathcal{H}}. \] From where it follows that \[ ||U||^2_{\mathcal{H}}\leq K|\lambda|^4||F||^2_{\mathcal{H}} \] that can be written as \begin{eqnarray*} ||(\lambda I-\mathcal{A})^{-1}||\leq K|\lambda|^2, \end{eqnarray*} that is

Then using the Theorem of A. Borichev and Y. Tomilov, (see [15], Theorem 2.4), the condition (54) is equivalent to \begin{eqnarray*} ||S(t){\mathcal{A}}^{-1}||=K\mathcal{O}(t^{-\frac{1}{2}})\Rightarrow ||S(t){\mathcal{A}}^{-1}F||_{\mathcal{H}}\leq \frac{K}{\sqrt{t}}||F||_{\mathcal{H}}, \,\,t \rightarrow \infty. \end{eqnarray*} Then taking \(\mathcal{A}U_0=F\), we get \[ ||S(t)U_0||_{\mathcal{H}}\leq \frac{K}{\sqrt{t}}||U_0||_{D(\mathcal{A})}. \] Therefore the solution decays polynomially. To prove that the rate of decay is optimal, we will argue by contradiction. Suppose that the rate \(t^{-\frac{1}{2}}\) can be improved; for example that the rate is \(t^{-\frac{1}{2-\epsilon}}\) for some \(0< \epsilon < 2\). From Theorem 5.3 in [20], the operator \[ |\lambda|^{\displaystyle{-2+ \frac{\epsilon}{2}}}||(\lambda I-{\mathcal{A}})^{-1}|| \] should be limited, but this does not happen. For this, let us suppose that there exist a sequence \((\lambda_{\mu})\subset {\mathbb{R}}\) with \(\lim_{\mu \rightarrow \infty}|\lambda_{\mu}|=\infty\) and \((U_{\mu})\subset D(\mathcal{A})\) for \((F_{\mu})\subset {\mathcal{H}}\) such that \[ (i\lambda_{\mu}I-{\mathcal{A}})U_{\mu}=F_{\mu} \] is bounded in \({\mathcal{H}}\) and \[ \lim_{\mu \rightarrow \infty}|\lambda_{\mu}|^{\displaystyle{-2+ \frac{\epsilon}{2}}}||U_{\mu}||_{\mathcal{H}}=\infty. \] So, following the same steps as in the proof of Theorem 3 we can conclude that \[ |\lambda_{\mu}|^{\displaystyle{-2+ \frac{\epsilon}{2}}}||U_{\mu}||_{\mathcal{H}}\geq \mathcal{O}\left(\mu^{\displaystyle{\frac{\epsilon}{2}}}\right)\rightarrow \infty,\quad \mbox{as}\quad \mu \rightarrow \infty. \] Therefore the rate cannot be improved. The proof is now complete.

Acknowledgments

This research is partially supported by PNPD/UFBA/CAPES(Brazil).

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.