On the \(L_{p}\) analytic semigroup associated with the linear thermoelastic plate equations in the half-space. (English) Zbl 1184.35062

The authors study the following system of linear equations, which describe a linear thermoelastic plate, \[ \begin{aligned} u_{tt}(x,t)+\Delta^2u(x,t)+\Delta\theta(x,t)&=0,\\ \theta_t(x,t)-\Delta\theta(x,t)-\Delta u_t(x,t)&=0, \end{aligned}\tag{1} \] where \((x,t)\in\mathbb R^n_+\times\mathbb R_+\), \(\mathbb R^n_+:=\{x=(x_1,\dots,x_n)\in\mathbb R^n\;|\;x_n>0\}\), the unknown function \(u\) denotes a mechanical variable denoting the vertical displacement of the plate and the unknown function \(\theta\) denotes a thermal variable. The system (1) is subject to the initial conditions \[ u(x,0)=u_0(x),\quad u_t(x,0)=v_0(x),\quad \theta(x,0)=\theta_0(x)\tag{2} \] and the boundary conditions \[ u|_{x_n=0}=D_nu|_{x_n=0}=\theta|_{x_n=0}=0.\tag{3} \] It is proved in the paper that the initial boundary value problem (1)–(3) generates an analytic semigroup in the \(L_p(\mathbb R^n_+)\) framework, \(1<p<\infty\), and the corresponding estimates for the solution of the problem are established. In particular, for the solution \((u,\theta)\), for \(t>0\), \[ \|\nabla^j(\nabla^2u(\cdot,t),D_t u(\cdot,t),\theta(\cdot,t))\|_{L_q(\mathbb R^n_+)}\leq C_{p,q}t^{-\frac j2-\frac n2\left(\frac 1p-\frac 1q\right)}\|(\nabla^2 u_0,v_0,\theta_0)\|_{L_p(\mathbb R^n_+)}, \] for \(j=0,1,2\), provided that \(1<p\leq q\leq\infty\) when \(j=0,1\) and that \(1<p\leq q<\infty\) when \(j=2\), where \(\nabla^j\) stands for space gradient of order \(j\). In order to get the results, the authors, first, study the corresponding resolvent equation and prove the corresponding resolvent estimates.


35B40 Asymptotic behavior of solutions to PDEs
74F05 Thermal effects in solid mechanics
47A10 Spectrum, resolvent
47D06 One-parameter semigroups and linear evolution equations
Full Text: DOI Link


[1] M. Agranovich and M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type, Russian Math. Surveys, 19 (1964), 53-157. · Zbl 0137.29602
[2] F. Ammar Khodja and A. Benabdallah, Sufficient conditions for uniform stabilization of second order equations by dynamical controllers, Dynam. Contin. Discrete Impuls. Systems, 7 (2000), 207-222. · Zbl 0946.35013
[3] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1996), sppl., (1997), 1-28. · Zbl 0896.73028
[4] R. Denk, R. Mennicken and L. Volevich, The Newton polygon and elliptic problems with parameter, Math. Nachr., 192 (1998), 125-157. · Zbl 0917.47023
[5] R. Denk and R. Racke, \(L^{p}\)-resolvent estimate and time decay for generalized thermoelastic plate equations, Electronic J. Differential Equations, 48 (2006), 1-16. · Zbl 1114.35019
[6] J. Lagnese, Boundary stabilization of thin plates, SIAM Studies in Applied Mathematics, 10 , Society for Industrial and Applied Mathematics, Philadelphia, PA, 1989. · Zbl 0696.73034
[7] J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., 23 (1992), 889-899. · Zbl 0755.73013
[8] I. Lasiecka and R. Triggiani, Two direct proofs on the analyticity of the S.C. semigroup arizing in abstract thermoelastic equations, Adv. Differential Equations, 3 (1998), 387-416. · Zbl 0949.35135
[9] I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermo-elastic semigroups, ESAIM, Proc., 4 , Soc. Math. Appl. Indust., Paris, 1998, pp. 199-222. · Zbl 0906.35018
[10] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with coupled hinged/Neumann boundary conditions, Abstr. Appl. Anal., 3 (1998), 153-169. · Zbl 0973.35184
[11] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4), 27 (1998), 457-482. · Zbl 0954.74029
[12] K. Liu and Z. Liu, Exponential stability and analyticity of abstract linear thermoelastic systems, Z. Angew. Math. Phys., 48 (1997), 885-904. · Zbl 0955.74019
[13] Z. Liu and M. Renardy, A note on the equations of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6. · Zbl 0826.35048
[14] Z. Liu and J. Yong, Qualitative properties of certain \(C_{0}\) semigroups arising in elastic systems with various dampings, Adv. Differential Equations, 3 (1998), 643-686. · Zbl 0962.47020
[15] Z. Liu and S. Zheng, Exponential stability of the Kirchhoff plate with thermal or viscoelastic dampings, Quart. Appl. Math., 55 (1997), 551-564. · Zbl 0889.35015
[16] Z. Liu and S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC Research Notes in Mathematics, 398 , Chapman & Hall/CRC, Boca Raton, FL, 1999. · Zbl 0924.73003
[17] J. E. Munõz Rivera and R. Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal., 26 (1995), 1547-1563. · Zbl 0842.35039
[18] J. E. Munõz Rivera and R. Racke, Large solutions and smoothing properties for nonlinear thermoelastic systems, J. Diff. Eqns., 127 (1996), 454-483. · Zbl 0852.73018
[19] Y. Shibata, On the exponential decay of the energy of a linear thermoelastic plate, Mat. Apl. Comput., 13 (1994), 81-102. · Zbl 0811.73006
[20] S. G. Mikhlin, Multidimensional singular integrals and integral equations, Pergamon Press, 1965. · Zbl 0129.07701
[21] Y. Shibata and R. Shimada, On a generalized resolvent estimate for the Stokes system with Robin boundary condition, J. Math. Soc. Japan, 59 (2007), 469-519. · Zbl 1127.35043
[22] Y. Shibata and S. Shimizu, On a resolvent estimate for the Stokes system with Neumann boundary condition, Differential Integral Equations, 16 (2003), 385-426. · Zbl 1054.35056
[23] Y. Shibata and S. Shimizu, On the \(L_{p}\)-\(L_{q}\) maximal regularity of the Neumann problem for the Stokes equations in a bounded domain, J. Reine Angew. Math., 615 (2008), 157-209. · Zbl 1145.35053
[24] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, 30 , Princeton University Press, Princeton, NJ, 1970. · Zbl 0207.13501
[25] H. Triebel, Interpolation theory, function spaces, differential operators, second edition, Johann Ambrosius Barth, Heidelberg, 1995. · Zbl 0830.46028
[26] I. I. Vrabie, \(C_{0}\)-semigroups and applications, North-Holland Mathematics Studies, 191 , North-Holland Publishing Co., Amsterdam, 2003. · Zbl 1119.47044
[27] L. R. Volevich, Solvability of boundary value problems for general elliptic systems, Mat. Sb., 68 (110), No.3 (1965), 373-416 (in Russian); English transl. Amer. Math. Soc. Transl., Ser. 2., 67 (1968), 182-225. · Zbl 0177.37401
[28] L. Volevich Newton polygon and general parameter-elliptic (parabolic) systems, Russ. J. Math. Phys., 8 (2001), 375-400. · Zbl 1186.35095
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.