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doi: 10.3934/dcdss.2021106
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Well-posedness and direct internal stability of coupled non-degenrate Kirchhoff system via heat conduction

UR Analysis and Control of PDE's, UR 13ES64, Higher Institute of transport and Logistics of Sousse, University of Sousse, Tunisia

Received  April 2021 Revised  July 2021 Early access September 2021

In the paper under study, we consider the following coupled non-degenerate Kirchhoff system
$\begin{equation} \left \{ \begin{aligned} & y_{tt}-\mathtt{φ}\Big(\int_\Omega | \nabla y |^2\,dx\Big)\Delta y +\mathtt{α} \Delta \mathtt{θ} = 0, &{\rm{ in }}&\; \Omega \times (0, +\infty)\\ & \mathtt{θ}_t-\Delta \mathtt{θ}-\mathtt{β} \Delta y_t = 0, &{\rm{ in }}&\; \Omega \times (0, +\infty)\\ & y = \mathtt{θ} = 0,\; &{\rm{ on }}&\;\partial\Omega\times(0, +\infty)\\ & y(\cdot, 0) = y_0, \; y_t(\cdot, 0) = y_1,\;\mathtt{θ}(\cdot, 0) = \mathtt{θ}_0, \; \; &{\rm{ in }}&\; \Omega\\ \end{aligned} \right. \end{equation} \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$
where
$ \Omega $
is a bounded open subset of
$ \mathbb{R}^n $
,
$ \mathtt{α} $
and
$ \mathtt{β} $
be two nonzero real numbers with the same sign and
$ \mathtt{φ} $
is given by
$ \mathtt{φ}(s) = \mathfrak{m}_0+\mathfrak{m}_1s $
with some positive constants
$ \mathfrak{m}_0 $
and
$ \mathfrak{m}_1 $
. So we prove existence of solution and establish its exponential decay. The method used is based on multiplier technique and some integral inequalities due to Haraux and Komornik[5,8].
Citation: Akram Ben Aissa. Well-posedness and direct internal stability of coupled non-degenrate Kirchhoff system via heat conduction. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021106
References:
[1]

R. A. Adams, Sobolev Spaces, Academic press, Pure and Applied Mathematics, vol. 65, 1975.  Google Scholar

[2]

P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system, Electron. J. Differential Equations, 2000 (2000), No. 22, 15 pp.  Google Scholar

[3]

A. Benaissa and A. Guesmia, Global existence and general decay estimates of solutions for degenerate or non-degenerate Kirchhoff equation with general dissipation, J. Evol. Equation, 11 (2011), 1399-1424.  doi: 10.1007/s00028-010-0076-9.  Google Scholar

[4]

B. Gilbert, A. Ben Aissa and S. Nicaise, Same decay rate of second order evolution equations with or without delay, Systems Control Lett., 141 (2020), 104700, 8 pp. doi: 10.1016/j.sysconle.2020.104700.  Google Scholar

[5]

A. Haraux, Two remarks on dissipative hyperbolic problems, in Lions, J. L. and Brezis, H. (Eds): Nonlinear Partial Differential Equations and Their Applications, College de France Seminar Volume XVIII (Research Notes in Mathematics, Vol. 122), Pitman: Boston, MA, (1985), 161–179.  Google Scholar

[6]

V. KeyantuoL. Tebou and M. Warma, A gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models, Discrete Contin. Dyn. Syst., 40 (2020), 2875-2889.  doi: 10.3934/dcds.2020152.  Google Scholar

[7]

G. Kirchhoff, Vorlesungen uber Mechanik, Teubner, Leipzig, 1897. Google Scholar

[8]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson Wiley, Paris (1994).  Google Scholar

[9]

I. LasieckaS. Maad and A. Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 689-715.  doi: 10.1007/s00030-008-0011-8.  Google Scholar

[10]

I. LasieckaM. Pokojovy and X. Wan, Long-time behavior of quasilinear thermoelastic Kirchhoff/Love plates with second sound, Nonlinear Analysis, 186 (2019), 219-258.  doi: 10.1016/j.na.2019.02.019.  Google Scholar

[11]

G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal., 141 (1998), 297–329. doi: 10.1007/s002050050078.  Google Scholar

[12]

J.-L. Lions, Quelques Méthodes De Résolution Des Problémes Aux Limites Nonlinéaires, Dund Gautier-Villars, Paris, 1969.  Google Scholar

[13]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM Control Optim. Calc. Var., 4 (1999), 419-444.  doi: 10.1051/cocv:1999116.  Google Scholar

[14]

K. Nishihara and Y. Yamada, On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms, Funkcial. Ekvac., 33 (1990), 151-159.   Google Scholar

[15]

K. Ono, On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Anal. Appl., 216 (1997), 321-342.  doi: 10.1006/jmaa.1997.5697.  Google Scholar

[16]

L. Tebou, Stabilization of some coupled hyperbolic/parabolic equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1601-1620.  doi: 10.3934/dcdsb.2010.14.1601.  Google Scholar

[17] P. Villaggio, Mathematical Models for Elastic Structures, Cambridge Univ. Press, 1997.  doi: 10.1017/CBO9780511529665.  Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic press, Pure and Applied Mathematics, vol. 65, 1975.  Google Scholar

[2]

P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system, Electron. J. Differential Equations, 2000 (2000), No. 22, 15 pp.  Google Scholar

[3]

A. Benaissa and A. Guesmia, Global existence and general decay estimates of solutions for degenerate or non-degenerate Kirchhoff equation with general dissipation, J. Evol. Equation, 11 (2011), 1399-1424.  doi: 10.1007/s00028-010-0076-9.  Google Scholar

[4]

B. Gilbert, A. Ben Aissa and S. Nicaise, Same decay rate of second order evolution equations with or without delay, Systems Control Lett., 141 (2020), 104700, 8 pp. doi: 10.1016/j.sysconle.2020.104700.  Google Scholar

[5]

A. Haraux, Two remarks on dissipative hyperbolic problems, in Lions, J. L. and Brezis, H. (Eds): Nonlinear Partial Differential Equations and Their Applications, College de France Seminar Volume XVIII (Research Notes in Mathematics, Vol. 122), Pitman: Boston, MA, (1985), 161–179.  Google Scholar

[6]

V. KeyantuoL. Tebou and M. Warma, A gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models, Discrete Contin. Dyn. Syst., 40 (2020), 2875-2889.  doi: 10.3934/dcds.2020152.  Google Scholar

[7]

G. Kirchhoff, Vorlesungen uber Mechanik, Teubner, Leipzig, 1897. Google Scholar

[8]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson Wiley, Paris (1994).  Google Scholar

[9]

I. LasieckaS. Maad and A. Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 689-715.  doi: 10.1007/s00030-008-0011-8.  Google Scholar

[10]

I. LasieckaM. Pokojovy and X. Wan, Long-time behavior of quasilinear thermoelastic Kirchhoff/Love plates with second sound, Nonlinear Analysis, 186 (2019), 219-258.  doi: 10.1016/j.na.2019.02.019.  Google Scholar

[11]

G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal., 141 (1998), 297–329. doi: 10.1007/s002050050078.  Google Scholar

[12]

J.-L. Lions, Quelques Méthodes De Résolution Des Problémes Aux Limites Nonlinéaires, Dund Gautier-Villars, Paris, 1969.  Google Scholar

[13]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM Control Optim. Calc. Var., 4 (1999), 419-444.  doi: 10.1051/cocv:1999116.  Google Scholar

[14]

K. Nishihara and Y. Yamada, On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms, Funkcial. Ekvac., 33 (1990), 151-159.   Google Scholar

[15]

K. Ono, On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Anal. Appl., 216 (1997), 321-342.  doi: 10.1006/jmaa.1997.5697.  Google Scholar

[16]

L. Tebou, Stabilization of some coupled hyperbolic/parabolic equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1601-1620.  doi: 10.3934/dcdsb.2010.14.1601.  Google Scholar

[17] P. Villaggio, Mathematical Models for Elastic Structures, Cambridge Univ. Press, 1997.  doi: 10.1017/CBO9780511529665.  Google Scholar
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