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Asymptotic Periodicity for Flexible Structural Systems and Applications

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Abstract

We study the existence of pseudo \(S\)-asymptotically \(\omega\)-periodic mild solutions for an abstract version of the damped wave equation \(u''+\alpha u'''=\beta\Delta u +\gamma\Delta u'+f\) which model the vibrations of flexible structures possessing internal material damping and external force \(f\). We show concrete applications of our methods to flexible structures, heat conduction, fractional equations and viscoelasticity.

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Notes

  1. The motivation for incorporating \(g\) as an input disturbance in the governing differential equation arises from the fact that very small amount of these, are always present in real materials (see e.g. [32]) as long as the system vibrates.

  2. A measurable set \(C\subseteq[0,\infty)\) is an ergodic zero set if \(\frac{\lambda (C\cap[0,t])}{t}\rightarrow0\), \(t\rightarrow\infty\), where \(\lambda \) denotes the Lebesgue measure.

  3. A continuous function \(f:\mathbb{R}^{+}\times X\rightarrow X\) is called asymptotically almost periodic if it admits a decomposition \(f=g+\phi \), where \(\phi:\mathbb{R}^{+}\times X\rightarrow X\) is a continuous function so that \(\lim_{t\rightarrow\infty}\phi(t,x)=0\) uniformly for \(x\) in any compact subset of \(X\) and \(g\) is almost periodic, that is, \(g\) is a continuous map, and for each \(\epsilon>0\) and any compact \(K\subseteq X\) there exists \(l(\epsilon)>0\) such that every interval \(I\) of length \(l(\epsilon)\) contains a number \(\tau\) with the property that \(\|f(t+\tau,x)-f(t,x)\|\leq\epsilon\) for all \(t\in \mathbb{R}\), \(x\in K\).

  4. A continuous function \(u:\mathbb {R}^{+}\rightarrow X\) satisfying the integral equation \(u(t)=\alpha\mathcal{R}(t)z+\int_{0}^{t}\mathcal {R}(t-s)f(s,u(s),u'(s))ds\), \(\forall t\geq0\), where \(z\in D(A)\) is called a mild solution to the problem (3.6)–(3.7) (see [10]).

  5. \(PAP_{0}(X)=\{\phi\in C_{b}([0,\infty );X):\lim_{T\rightarrow\infty}\frac{1}{T}\int^{T}_{0}\|\phi(t)\| _{X}dt=0\}\). It is well known that a function \(u\) belongs to \(PAP_{0}(X)\) if and only if for each \(\epsilon>0\), the set \(C_{\epsilon}=\{t\in [0,\infty): \|u(t)\|_{X}\geq\epsilon\}\) is an ergodic zero set (see [93]).

  6. The existence of the pseudo S-asymptotically \(\omega\)-periodic solution is obtained through the Banach fixed point theorem and this theorem does give uniqueness in the space where the fixed point argument is set. In this case the space is the ball \(B_{r}(PSAP_{\omega}^{0}(X))\), which is not the total space. This is the reason why uniqueness is not obtained. One may wonder whether is there any example where uniqueness of \(PSAP_{\omega}^{0}(X)\) solutions does not hold? Unfortunately we do not known the answer.

  7. See [25].

  8. \(\overline{\mathit{co}(E)}\) denotes the closure of the convex hull of \(E\). As the name suggests the convex hull is always convex. It is the smallest convex set that contain \(E\).

  9. To be more precise, recall that a continuous function \(f:[0,\infty)\times X\rightarrow X\) is called uniformly continuous on bounded sets of \(X\) if for every \(\varepsilon>0\) and every bounded subset \(K\) of \(X\) there is \(\delta=\delta_{\varepsilon,K}>0\) such that \(\|f(t,x)-f(t,y)\|_{X}\leq\varepsilon\) for all \(t\geq0\) and all \(x,y\in K\) so that \(\|x-y\|_{X}\leq\delta\). This definition coming essentially from [1].

  10. To see this, let \(\varepsilon\) be any positive real number, put \(\rho:=M(\alpha+1)\| y\|_{X}+\alpha M\|z\|_{X}+\sigma\). Since \(f\) is uniformly continuous on bounded sets of \(X\), there is \(\delta>0\) such that \(\|f(t,x)-f(t,y)\| _{X}\leq\frac{\mu}{M}\varepsilon\) for all \(t\geq0\) and \(\|x-y\| _{X}\leq\delta\) with \(\|x\|_{X}\leq\rho\), and \(\|y\|_{X}\leq\rho\). We can choose \(n_{0}\in\mathbb{N}\) such that for each \(n\geq n_{0}\) implies \(\|u_{n}-u\|_{C_{b}([0,\infty);X)}<\delta\). From this we immediately get \(\|\varLambda^{\odot}u^{n}-\varLambda^{\odot}u\|_{\infty}\leq\varepsilon\).

  11. That is, for each \(T>0\) the functions in \(\varLambda^{\odot}(B_{\sigma}(C_{b}^{0}))\) restricted to the domain \([0,T]\) are equicontinuous. We see that some authors would call this equicontinuity, not local equicontinuity (see [25]).

  12. A function \(u:[0,1]\rightarrow X\) is called a mild solution of (3.29)–(3.30) if it satisfies the integral equation

    $$ u(t)=\alpha\mathcal{R}(t)g(u)+ \int_{0}^{t}\mathcal {R}(t-s)f\bigl(s,u(s),u'(s) \bigr)ds,\quad t\in[0,1]. $$

  13. As the name suggests, measure of noncompactness give an idea of the “lack of compactness” of a given set or operator. Such situation arises in many fields: integral equations with strongly singular kernels, non-linear superposition operators between various function spaces and much more (see [9]).

  14. \(f(\cdot,x,y)\) is mensurable for all \(x,y\in X\) and \(f(t,\cdot):X\times X\rightarrow X\) is continuous for all \(t\in[0,1]\).

  15. We note that similar conditions to those made in Theorem 3.8 where previous considered in the literature (cf. [86]).

  16. The equicontinuity of the set \(\widetilde {\varLambda}(\mathcal{B})'\) is a consequence of the following decomposition

    $$\begin{aligned} (\widetilde{\varLambda}u )'(t+s)- (\widetilde{\varLambda }u )'(t) = & \alpha \bigl(\mathcal{R}'(t+s)-\mathcal {R}'(t) \bigr)g(u) + \int_{t}^{t+s}\mathcal{R}'(t+s-\xi)f \bigl(\xi,u(\xi),u'(\xi)\bigr)d\xi \\ &{}+ \int_{0}^{t} \bigl(\mathcal{R}'(t+s- \xi)-\mathcal{R}'(t-\xi) \bigr)f\bigl(\xi,u(\xi),u'( \xi)\bigr)d\xi, \quad u\in\mathcal{B}. \end{aligned}$$

  17. To handle it, we suppose that (3.35) holds. One can show that

    $$\begin{aligned} \xi \bigl(\widetilde{\varLambda}^{n+1}(\mathcal{B})'(t) \bigr) \leq &M_{\mathcal{R}\mathcal{R}'} \int_{0}^{t}H(s)\xi \bigl(\widetilde { \varLambda}^{n}(\mathcal{B})'(s) \bigr)ds \\ \leq& \Biggl(\sum_{j=0}^{n}C_{j}^{n}a^{n+1-j} \frac{(bt)^{j}}{j!} \Biggr)\widehat{\xi}(\mathcal{B}) + \Biggl(\sum _{j=1}^{n+1}C_{j+1}^{n}a^{n+1-j} \frac{(bt)^{j}}{j!} \Biggr)\widehat{\xi}(\mathcal{B}) \\ = & \Biggl(a^{n+1}+\sum_{j=1}^{n} \bigl(C_{j-1}^{n}+C_{j}^{n} \bigr)a^{n+1-j}\frac {(bt)^{j}}{j!}+\frac{(bt)^{n+1}}{(n+1)!} \Biggr)\widehat{\xi}( \mathcal {B}) \\ = & \Biggl(\sum_{j=0}^{n+1}C_{j}^{n+1}a^{n+1-j} \frac {(bt)^{j}}{j!} \Biggr)\widehat{\xi}(\mathcal{B}). \end{aligned}$$

  18. The case \(\lambda=0\) is the so called Voigt model of viscoelasticity (see [53]).

  19. We denote by \(H^{m}(\varOmega)\) the Sobolev space of functions which are in \(L^{2}(\varOmega)\), together with all their derivatives of order \(\leq m\). \(H_{0}^{1}(\varOmega)\) is the Hilbert subspace of \(H^{1}(\varOmega)\), made of functions vanishing on \(\partial\varOmega\).

  20. It follows from the Poincare’s inequality [19]

    $$ \|\alpha\mathcal{R}(t)z\|_{L^{2}(\varOmega)}\leq\|v\|_{L^{2}(\varOmega )}+\lambda \|u'\|_{L^{2}(\varOmega)} \leq C_{v}\|\nabla v \|_{L^{2}(\varOmega)} + C_{u'}\lambda\|\nabla u' \|_{L^{2}(\varOmega)} \leq N^{*}(C_{v}+\lambda C_{u'})e^{-\mu t} $$

    and \(\|\alpha\mathcal{R}'(t)z\|_{L^{2}(\varOmega)} = \|u'(t)\|_{L^{2}(\varOmega)} \leq C_{u'}\|\nabla u'\|_{L^{2}(\varOmega)} \leq C_{u'}N^{*}e^{-\mu t}\). Using the preceding estimates and the uniform boundedness principle we have that \(\mathcal{R}(t)\) satisfies (4.3).

  21. \(W^{m,2}(\varOmega)=H^{m}(\varOmega)\).

  22. The following estimate is responsible for the fact that \(\widetilde{K}\) is relatively compact in \(L^{2}(\varOmega)\),

    $$ \sup_{t\leq0,\|\varphi\|_{L^{2}(\varOmega)}\leq r}\|f(t,\varphi)\| _{W^{1,2}(\varOmega)}\leq\|L \|_{C_{b}([0,\infty);\mathbb{R})} \bigl(\varPsi \bigl(v(\varOmega)^{1/2}r\bigr)+|g(0)| \bigr)\|\varPhi_{0}\|_{W^{1,2}(\varOmega)}. $$

  23. The resolvent is used to obtain a variation of parameters formula. In the convolution case it is quite effective for dealing with perturbations because it employs the solution of the unforced equation about which we frequently know a great deal. The non convolution case presents many difficulties. One is often inclined to believe that more progress can be made by attacking the original equation directly without going through a variation of parameters argument. In the case of an integro-differential equation, one may use differential inequalities and Liapunov functional to bypass the resolvent (see [20]).

  24. \(S_{\omega}(\mathbb{R})\) denotes the subspace of \(C_{b}([0,\infty);\mathbb{R})\) formed for all the functions \(u(\cdot)\) such that \(\lim_{t\rightarrow\infty }(u(t+n\omega)-u(t))=0\), uniformly for \(n\in\mathbb{N}\) endowed with the norm of the uniform convergence. It is easy to see that \(S_{\omega}(\mathbb{R})\) is a closed subspace of \(C_{b}([0,\infty);\mathbb{R})\). If \(u\in S_{\omega}(\mathbb{R})\) it follows from [61] that \(u\) is asymptotically \(\omega\)-periodic.

  25. We notice that

    $$\begin{aligned} \varGamma f(t+n\omega)-\varGamma f(t) =& \int_{0}^{n\omega}R(t+n\omega -s)f(s)ds+ \int_{0}^{L}R(t-s) \bigl(f(s+n\omega)-f(s)\bigr)ds\\ &{}+ \int _{L}^{t}R(t-s) \bigl(f(s+\omega)-f(s)\bigr)ds. \end{aligned}$$

  26. Note that \(\lim_{t\rightarrow \infty}\frac{(t+\omega)^{3/2}-t^{3/2}}{t}=\lim_{t\rightarrow\infty }\frac{3t^{2}\omega+3t\omega^{2}+\omega^{3}}{t^{5/2}}\frac{1}{ (1+\frac{\omega}{t} )^{3/2}+1}=0\).

  27. In writing the function \(q\) is assumed for simplicity and without loss of generality that the history of the temperature \(u\) is prescribed as zero for \(t<0\). For details and references to the underlying physical theory, we refer the reader to Nohel [83].

  28. For any \(T>0\) we have the critical property that \(\int _{0}^{T}C(s)ds<\infty\). \(C(t)\) is completely monotone on \((0,\infty)\) in the sense that \((-1)^{k}C^{(k)}(t)\geq0\) for \(k=0,1,2,\ldots \) and \(t\in (0,\infty)\).

  29. Note that \(R\) satisfies \(0\leq R(t)\leq C(t)\) for all \(t>0\), so \(R(t)\rightarrow0\) as \(t\rightarrow \infty\). Since \(C \notin L^{1}[0,\infty)\) we get \(\int_{0}^{\infty}R(s)ds=1\). By [78, Theorem 7.2] we can see that \(R\) is a completely monotone function on \(0< t<\infty\).

  30. The Stojanovic and Gorenflo’s method used in [90] is based on the successive application of the Laplace and Fourier transform in order to arrive to a linear Abel Volterra type equation. The Gronwall’s inequality yields the super viscosity solution assuming a Lipschitz condition for the force term.

  31. See [67, Example 6.1], and [34, Example 3.8].

  32. Let \(\{ T(t)\}_{t\geq0}\subseteq\mathcal{B}(X)\) be a strongly mensurable family of operators, i.e. \(T(t)x\) is Bochner-mensurable in \(X\), for each \(x\in X\). Then \(T(t)\) is called uniformly integrable if \(T(\cdot )\in L^{1}(\mathbb{R}^{+},\mathcal{B}(X))\).

  33. If we assume that \(\beta(t)\equiv0\), then (4.69) is the equation governing one-dimensional longitudinal motions of a viscoelastic bar with density 1 (see [60]).

  34. Properties of the solutions of Eq. (4.76) have been studied in several contexts, for example well-posedness [45, 57], mild solutions [8, 71], and asymptotic periodicity [26, 43, 47, 48].

  35. See Sect. 4.2 for the definition of \(S_{\omega}(X)\). Note that we must to change \(X\) instead of ℝ.

  36. The elastic materials store all the energy due to deformation. However viscoelastic materials do not store all the energy under deformation, actually dissipate some of this energy. The ability to dissipate energy is one of the main reasons for using viscoelastic materials for isolating vibration, dampening noise, and absorbing shock(pertinent references in this regard are [17, 32, 51, 75]).

  37. Which means that \(S(t)D(A)\subseteq D(A)\) and \(AS(t)x=S(t)Ax\) for all \(x\in D(A)\) and \(t\geq0\).

  38. Recall that a resolvent family \(\{S(t)\}_{t\geq0}\) is called uniformly stable if \(\lim_{t\rightarrow\infty}\|S(t)\| _{\mathcal{B}(X)}=0\).

  39. Let \(a\in L^{1}_{loc}(\mathbb{R}^{+})\) be of subexponential growth and \(k\in\mathbb{N}\), the function \(a(t)\) is called \(k\)-regular if there is a constant \(c>0\) such that \(|\lambda^{n}\hat {a}^{n}(\lambda)|\leq c|\hat{a}(\lambda)|\) for all \(\operatorname{Re}(\lambda)>0\) and \(1\leq n\leq k\).

  40. We observe that from parabolicity and 1-regularity of \(a(t)\) we have by [87, Theorem 3.1] the existence of a resolvent family \(S(t)\), continuous and bounded for \(t\geq0\), hence \(S(\cdot)\in L^{1}_{loc}(\mathbb{R}^{+},\mathcal {B}(X))\).

  41. The condition \(\nabla\circ u=0\) means that the fluid is homogeneous and incompressible. In this case we call \(u\) divergence-free or solenoidal.

  42. By [89, Lemma 2.5.3] we get \(L^{2}_{0}(\varOmega;\mathbb{R}^{n})=\{f\in L^{2}(\varOmega; \mathbb{R}^{n}):\nabla\circ f=0, N\cdot f\mid_{\partial\varOmega}=0\} \), \(N\cdot f\mid_{\partial\varOmega}\) means the generalized trace.

  43. Note that \(P\nabla p=0\) by the definition of \(P\) and \(\int_{0}^{t}\int_{0}^{s}\Delta u(s-\tau,x)da(\tau)ds=\int _{0}^{t}a(t-\tau)\Delta u(\tau,x)d\tau\).

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Acknowledgements

The authors would like to express their appreciation to professor T.A. Burton for supplying references [23, 25] concerning recent work, which enable us to make Remark 3.11 and Sect. 4.3. This subsection was written during a visit of C. Cuevas to State University of Campinas (UNICAMP). C. Cuevas is grateful to professor Lucas Ferreira for its generous hospitality. This work was completed when C. Cuevas and C. Silva were visiting the University of La Frontera, Temuco, Chile. They are grateful to the Evolution Equations and Applications Group and the Department of Mathematics and Statistics for providing a stimulating atmosphere to work. B. de Andrade and C. Cuevas are partially financed by CNPQ/Brazil under Grant 478053/2013-4. C. Silva has been supported by CNPQ/Brazil. H. Soto is partially financed by DIUFRO under Grant DI15-0050. The authors are very grateful to the referee for pointing out omissions and providing nice comments.

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  1. Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão, SE, Brazil

    Bruno de Andrade

  2. Departamento de Matemática, Universidade Federal de Pernambuco, Recife, PE, CEP 50540-740, Brazil

    Claudio Cuevas & Clessius Silva

  3. Departamento de Matemática y Estadística, Universidad de La Frontera, Casilla, 54-D, Temuco, Chile

    Herme Soto

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  1. Bruno de Andrade
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Correspondence to Claudio Cuevas.

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de Andrade, B., Cuevas, C., Silva, C. et al. Asymptotic Periodicity for Flexible Structural Systems and Applications. Acta Appl Math 143, 105–164 (2016). https://doi.org/10.1007/s10440-015-0032-3

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