# American Institute of Mathematical Sciences

March 2018, 7(1): 153-182. doi: 10.3934/eect.2018008

## Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay

 1 Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA 2 Department of Mathematics and Economics, Virginia State University, Petersburg, VA 23806, USA

* Corresponding author: Roberto Triggiani

The first author is supported by NSF grant DMS-1713506

Received  May 2017 Revised  September 2017 Published  January 2018

We consider a heat-plate interaction model where the 2-dimensional plate is subject to viscoelastic (strong) damping. Coupling occurs at the interface between the two media, where each components evolves. In this paper, we apply "low", physically hinged boundary interface conditions, which involve the bending moment operator for the plate. We prove three main results: analyticity of the corresponding contraction semigroup on the natural energy space; sharp location of the spectrum of its generator, which does not have compact resolvent, and has the point $\lambda = -1/ρ$ in its continuous spectrum; exponential decay of the semigroup with sharp decay rate. Here analyticity cannot follow by perturbation.

Citation: Roberto Triggiani, Jing Zhang. Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay. Evolution Equations & Control Theory, 2018, 7 (1) : 153-182. doi: 10.3934/eect.2018008
##### References:
 [1] G. Avalos, I. Lasiecka and R. Triggiani, Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system, Georgian Mathematical Journal, 15 (2008), 403-437; dedicated to the memory of J. [2] G. Avalos and R. Triggiani, The coupled PDE-system arising in fluid-structure interaction. Part Ⅰ: Explicit semigroup generator and its spectral properties, AMS Contemporary Mathematics, Fluids and Waves, 440 (2007), 15-54. [3] G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface, Discr. Cont. Dynam. Systems, 22 (2008), 817-833 (invited paper). doi: 10.3934/dcds.2008.22.817. [4] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system, Discr., & Cont. Dynam.Systems DCDS-S, 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417. [5] G. Avalos and R. Triggiani, A coupled parabolic-hyperbolic Stokes-Lamé PDE system: Limit behavior of the resolvent operator on the imaginary axis, Applicable Analysis, 88 (2009), 1357-1396. doi: 10.1080/00036810903278513. [6] G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system, J. Evol. Eqns., 9 (2009), 341-370. doi: 10.1007/s00028-009-0015-9. [7] G. Avalos and R. Triggiani, Rational decay rates for a PDE heat-structure interaction: A frequency domain approach, Evolution Equations and Control Theory, 2 (2013), 233-253. doi: 10.3934/eect.2013.2.233. [8] G. Avalos and R. Triggiani, Fluid-structure interaction with and without internal dissipation of the structure: A contrast in stability, Evolution Equations and Control Theory, 2 (2013), 563-598, special issue by invitation on the occasion of W. doi: 10.3934/eect.2013.2.563. [9] V. Barbu, Z. Grujic, I. Lasiecka and A. Tuffaha, Weak solutions for nonlinear fluid-structure interaction, AMS Contemporary Mathematics: Recent Trends in Applied Analysis, 440 (2007), 55-82. [10] V. Barbu, Z. Grujic, I. Lasiecka and A. Tuffaha, Smoothness of Weak Solutions to a nonlinear fluid-structure interaction model, Indiana Journal of Mathematics, 57 (2008), 1173-1207. doi: 10.1512/iumj.2008.57.3284. [11] G. Chen and D. L. Russel, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math., 39 (1982), 433-454. [12] S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems: The case α = 1/2, Springer-Verlag Lecture Notes in Mathematics, 1354 (1988), 234–256, Proceedings of Seminar on Approximation and Optimization, University of Havana, Cuba (January 1987). [13] S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems: The case $\text{1/2}\le \alpha \le \text{1}$), Pacific J. Math., 136 (1989), 15-55. doi: 10.2140/pjm.1989.136.15. [14] S. Chen and R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Diff. Eqns., 88 (1990), 279-293. doi: 10.1016/0022-0396(90)90100-4. [15] S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle perturbation, Proceedings Amer. Math. Soc., 110 (1990), 401-415. [16] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem, Discr. Dynam. Sys., 9 (2003), 633-650. doi: 10.3934/dcds.2003.9.633. [17] M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness for a freeboundary fluid-structure model, Journal of Mathematical Physics, 53 (2012), 115624, 13 pp. [18] M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness and small data global existence for an interface damped free boundary fluid-structure model, Nonlinearity, 27 (2014), 467-499. doi: 10.1088/0951-7715/27/3/467. [19] M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, Small data global existence for a fluid structure model with moving boundary, Nonlinearity, 30 (2017), 848-898. doi: 10.1088/1361-6544/aa4ec4. [20] I. Kukavica and A. Tuffaha, Regularity of solutions to a free boundary problem of fluid-structure interaction, Indiana Univ. Math. J., 61 (2012), 1817-1859. doi: 10.1512/iumj.2012.61.4746. [21] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system, J. Differential Equations, 247 (2009), 1452-1478. doi: 10.1016/j.jde.2009.06.005. [22] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid structure interaction system, Adv. Differential Equations, 15 (2010), 231-254. [23] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a Navier-Stokes-Lamé system on a domain with a non-flat boundary, Nonlinearity, 24 (2011), 159-176. doi: 10.1088/0951-7715/24/1/008. [24] J. Lagnese, Uniform boundary stabilization of homogeneous isotropic plates, Springer-Verlag Lecture Notes in Control and Information Sciences, 50 (1987), 204-215. doi: 10.1007/BFb0041992. [25] J. Lagnese, Boundary stabilization of thin plate, SIAM Studies in Applied Mathematics, 1989. [26] I. Lasiecka and Y. Lu, Stabilization of a fluid structure interaction with nonlinear damping, Control Cybernet, 42 (2013), 155-181. [27] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I, Abstract Parabolic Systems, Encyclopedia of Mathematics and Its Applications Series, Cambridge University Press, January 2000. [28] I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers, Communications in Pure and Applied Analysis, 15 (2016), 1513-1543. doi: 10.3934/cpaa.2016001. [29] J. L. Lions, Quelques Methods de Resolution des Problemes aux Limits Nonlinearies, Dunod. Paris, 1969. [30] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I, Springer-Verlag, 1972,357 pp. [31] Y. Lu, Uniform stabilization to equilibrium of a nonlinear fluid-structure interaction model, Nonlinear Anal. Real World Appl., 25 (2015), 51-63. doi: 10.1016/j.nonrwa.2015.02.006. [32] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, 1983. [33] J. Pruss, On the spectrum of $C_0$ semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857. [34] A. Taylor and D. Lay, Introduction to Functional Analysis, 2 $^nd$ edition, 1980, Wiley. [35] R. Triggiani, A heat-viscoelastic structure interaction model with Neumann or Dirichlet boundary control at the interface: Optimal regularity, control theoretic implications, Applied Mathematics and Optimization, special issue in memory of A.V.Balakrishnan, 73 (2016), 571-594. doi: 10.1007/s00245-016-9348-2. [36] R. Triggiani, Domain of fractional powers of the heat-structure operator with visco-elastic damping: Regularity and control-theoretical implications, J. Evol. Eqns., Special issue in honor of J. Prüss, 17 (2017), 573–597. doi: 10.1007/s00028-016-0359-x. [37] R. Triggiani, A matrix-valued generator $A$ with strong boundary coupling: A critical subspace of $\mathcal{D}((-A)^{1/2})$ and $\mathcal{D}((-A^*)^{1/2})$ and implications, Evolution Equations and Control Theory, 5 (2016), 185-199. doi: 10.3934/eect.2016.5.185. [38] J. Zhang, The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework, Evolution Equations and Control Theory, 6 (2017), 135-154.

show all references

##### References:
 [1] G. Avalos, I. Lasiecka and R. Triggiani, Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system, Georgian Mathematical Journal, 15 (2008), 403-437; dedicated to the memory of J. [2] G. Avalos and R. Triggiani, The coupled PDE-system arising in fluid-structure interaction. Part Ⅰ: Explicit semigroup generator and its spectral properties, AMS Contemporary Mathematics, Fluids and Waves, 440 (2007), 15-54. [3] G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface, Discr. Cont. Dynam. Systems, 22 (2008), 817-833 (invited paper). doi: 10.3934/dcds.2008.22.817. [4] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system, Discr., & Cont. Dynam.Systems DCDS-S, 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417. [5] G. Avalos and R. Triggiani, A coupled parabolic-hyperbolic Stokes-Lamé PDE system: Limit behavior of the resolvent operator on the imaginary axis, Applicable Analysis, 88 (2009), 1357-1396. doi: 10.1080/00036810903278513. [6] G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system, J. Evol. Eqns., 9 (2009), 341-370. doi: 10.1007/s00028-009-0015-9. [7] G. Avalos and R. Triggiani, Rational decay rates for a PDE heat-structure interaction: A frequency domain approach, Evolution Equations and Control Theory, 2 (2013), 233-253. doi: 10.3934/eect.2013.2.233. [8] G. Avalos and R. Triggiani, Fluid-structure interaction with and without internal dissipation of the structure: A contrast in stability, Evolution Equations and Control Theory, 2 (2013), 563-598, special issue by invitation on the occasion of W. doi: 10.3934/eect.2013.2.563. [9] V. Barbu, Z. Grujic, I. Lasiecka and A. Tuffaha, Weak solutions for nonlinear fluid-structure interaction, AMS Contemporary Mathematics: Recent Trends in Applied Analysis, 440 (2007), 55-82. [10] V. Barbu, Z. Grujic, I. Lasiecka and A. Tuffaha, Smoothness of Weak Solutions to a nonlinear fluid-structure interaction model, Indiana Journal of Mathematics, 57 (2008), 1173-1207. doi: 10.1512/iumj.2008.57.3284. [11] G. Chen and D. L. Russel, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math., 39 (1982), 433-454. [12] S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems: The case α = 1/2, Springer-Verlag Lecture Notes in Mathematics, 1354 (1988), 234–256, Proceedings of Seminar on Approximation and Optimization, University of Havana, Cuba (January 1987). [13] S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems: The case $\text{1/2}\le \alpha \le \text{1}$), Pacific J. Math., 136 (1989), 15-55. doi: 10.2140/pjm.1989.136.15. [14] S. Chen and R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Diff. Eqns., 88 (1990), 279-293. doi: 10.1016/0022-0396(90)90100-4. [15] S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle perturbation, Proceedings Amer. Math. Soc., 110 (1990), 401-415. [16] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem, Discr. Dynam. Sys., 9 (2003), 633-650. doi: 10.3934/dcds.2003.9.633. [17] M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness for a freeboundary fluid-structure model, Journal of Mathematical Physics, 53 (2012), 115624, 13 pp. [18] M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness and small data global existence for an interface damped free boundary fluid-structure model, Nonlinearity, 27 (2014), 467-499. doi: 10.1088/0951-7715/27/3/467. [19] M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, Small data global existence for a fluid structure model with moving boundary, Nonlinearity, 30 (2017), 848-898. doi: 10.1088/1361-6544/aa4ec4. [20] I. Kukavica and A. Tuffaha, Regularity of solutions to a free boundary problem of fluid-structure interaction, Indiana Univ. Math. J., 61 (2012), 1817-1859. doi: 10.1512/iumj.2012.61.4746. [21] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system, J. Differential Equations, 247 (2009), 1452-1478. doi: 10.1016/j.jde.2009.06.005. [22] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid structure interaction system, Adv. Differential Equations, 15 (2010), 231-254. [23] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a Navier-Stokes-Lamé system on a domain with a non-flat boundary, Nonlinearity, 24 (2011), 159-176. doi: 10.1088/0951-7715/24/1/008. [24] J. Lagnese, Uniform boundary stabilization of homogeneous isotropic plates, Springer-Verlag Lecture Notes in Control and Information Sciences, 50 (1987), 204-215. doi: 10.1007/BFb0041992. [25] J. Lagnese, Boundary stabilization of thin plate, SIAM Studies in Applied Mathematics, 1989. [26] I. Lasiecka and Y. Lu, Stabilization of a fluid structure interaction with nonlinear damping, Control Cybernet, 42 (2013), 155-181. [27] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I, Abstract Parabolic Systems, Encyclopedia of Mathematics and Its Applications Series, Cambridge University Press, January 2000. [28] I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers, Communications in Pure and Applied Analysis, 15 (2016), 1513-1543. doi: 10.3934/cpaa.2016001. [29] J. L. Lions, Quelques Methods de Resolution des Problemes aux Limits Nonlinearies, Dunod. Paris, 1969. [30] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I, Springer-Verlag, 1972,357 pp. [31] Y. Lu, Uniform stabilization to equilibrium of a nonlinear fluid-structure interaction model, Nonlinear Anal. Real World Appl., 25 (2015), 51-63. doi: 10.1016/j.nonrwa.2015.02.006. [32] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, 1983. [33] J. Pruss, On the spectrum of $C_0$ semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857. [34] A. Taylor and D. Lay, Introduction to Functional Analysis, 2 $^nd$ edition, 1980, Wiley. [35] R. Triggiani, A heat-viscoelastic structure interaction model with Neumann or Dirichlet boundary control at the interface: Optimal regularity, control theoretic implications, Applied Mathematics and Optimization, special issue in memory of A.V.Balakrishnan, 73 (2016), 571-594. doi: 10.1007/s00245-016-9348-2. [36] R. Triggiani, Domain of fractional powers of the heat-structure operator with visco-elastic damping: Regularity and control-theoretical implications, J. Evol. Eqns., Special issue in honor of J. Prüss, 17 (2017), 573–597. doi: 10.1007/s00028-016-0359-x. [37] R. Triggiani, A matrix-valued generator $A$ with strong boundary coupling: A critical subspace of $\mathcal{D}((-A)^{1/2})$ and $\mathcal{D}((-A^*)^{1/2})$ and implications, Evolution Equations and Control Theory, 5 (2016), 185-199. doi: 10.3934/eect.2016.5.185. [38] J. Zhang, The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework, Evolution Equations and Control Theory, 6 (2017), 135-154.
The Fluid–Structure Interaction
The set ${{\mathcal{K}}_{\rho }}$, $0 < \rho\mu < 1$
The Triangular Sector $\Sigma_{\theta_1}$ and its Complement $\Sigma^c_{\theta_1}$. The Disk $\mathcal{S}_{r_0}\subset \rho(\mathcal{A})$
Admissible points $\{\alpha, \omega\}$ in the proof of Theorem 2.4, ; lie in shaded region, $r_1>0$, $\varepsilon >0$ arbitrarily small
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