May  2022, 15(5): 983-993. doi: 10.3934/dcdss.2021106

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 Published  May 2022 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 and Continuous Dynamical Systems - S, 2022, 15 (5) : 983-993. doi: 10.3934/dcdss.2021106
References:
[1]

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

[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.

[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.

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[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.

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

show all references

References:
[1]

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

[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.

[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.

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[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.

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