Global existence of solutions for a class of thermoelastic plate systems

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1 Liu Boundary Value Problems (18) 18:161 R E S E A R C H Open Access Global existence of solutions for a class of thermoelastic plate systems Yang Liu 1,* * Correspondence: liuyangnufn@163.com 1 College of Mathematics and Computer Science, Northwest Minzu University, Lanzhou, People s Republic of China College of Mathematics, Sichuan University, Chengdu, People s Republic of China Abstract This paper is concerned with the initial-boundary value problem for a class of thermoelastic plate systems. Under some appropriate assumptions, the global existence of solutions is obtained. AMS Subject Classification: Primary 35A1; 35D3; 35G61 Keywords: Thermoelastic plate systems; Initial-boundary value problem; Global existence 1 Introduction In this paper, we study the following initial-boundary value problem for a class of thermoelastic plate systems: u tt + u g( u L ) u u () tt + ν θ = f (u), (x, t) R +, θ t ω θ (1 ω) (1.1) k(τ) θ(t τ)dτ ν u t =, (x, t) R +, with boundary conditions u = u =, (x, t) [, ), θ =, (x, t) R, (1.) and initial conditions u(x,)=u (x), u t (x,)=u 1 (x), x, θ(x, t)=θ (x, t), (x, t) (,], (1.3) where is a bounded domain of R N (N 1) with a smooth boundary, ω <1,and ν >. The function g,externalforcef and memory kernel k will be specified later. It is well known that temperature gradients in a plate will contribute to plate deformation. Problem (1.1) (1.3) can be used to describe the deformation and the temperature distribution of a homogeneous, isotropic and thermoelastic thin material with memory, see [1, ] forthe details. Functions u(x, t) and θ(x, t) represent the displacement and temperature variation field relative to the equilibrium reference value, respectively. The cases The Author(s) 18. This article is distributed under the terms of the Creative Commons Attribution 4. International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 LiuBoundary Value Problems (18) 18:161 Page of 11 ω =and<ω <1in(1.1) are usually referred to as the Gurtin Pipkin model [3]andthe Coleman Gurtin model [4], respectively. In the absence of thermal effects, (1.1) reduces to an extensible plate equation. This class of equations model the vibrations of extensible elastic beams (when N = 1) and plates (when N = ), and have been extensively investigated (see, e.g., [5 11] and the references therein). Wu [1] studied u tt + u u t + θ + f (u)=, (x, t) R +, θ t k(τ) θ(t τ)dτ u t =, (x, t) R +, subject to (1.) and(1.3). Under the assumption f C, the author obtained the global existence and uniqueness of solutions, as well as the existence of global attractors. Moreover, when f is assumed real analytic, the convergence of global solutions to a single steady state, as time goes to infinity, was proved and also an estimate of the convergence rate was provided. Barbosa and Ma [13] investigated problem(1.1) (1.3) by adding an extra external force h to (1.1) 1. Using the assumptions g, f C 1, the authors derived the global well-posedness of solutions, the existence of global attractors with finite fractal dimension, and the existence of exponential attractors. In the present paper, our purpose is to tackle the global existence of solutions to problem (1.1) (1.3) under weaker assumptions on g and f.asin[1, 13], we employ the past history approach [14, 15], so that problem (1.1) (1.3) can be transformed into an equivalent system in the history phase space. By means of the potential well theory [16, 17], we establish the theorems on global existence of solutions by discussing the level of initial energy. This paper is organized as follows. In Sect., some assumptions on g, f and k are displayed. Moreover, problem (1.1) (1.3) is transformed into an equivalent system, and the main results of this paper are stated. In Sect. 3, the global existence of solutions with subcritical initial energy is established. In Sect. 4,the global existence ofsolutions with critical initial energy is derived. Preliminaries and main results.1 Notations and assumptions Throughout the paper, for simplicity, we denote p := L p (), :=. Moreover, (, ) denotes either the L -inner product or a duality pairing between a space and its dual space. We make the following assumptions on g, f and k,respectively. (A 1 ) g C(R), g(z)>, and there exists a constant α >such that αg(z) zg(z),where G(z)= z g(s)ds. (A ) f C(R). There exists a constant β >such that f (u) β u p 1,where p < if N 4, p < N N 4 if N >4.

3 LiuBoundary Value Problems (18) 18:161 Page 3 of 11 Moreover, there exists a constant γ > α such that uf (u) γ F(u), where α := max{1, α} and F(u)= u f (s)ds. (A 3 ) k C (R + ), k(τ), k (τ) and k (τ) for all τ R +. In addition, μ(τ) := (1 ω)k (τ).. Reformulation of the problem We define the auxiliary variable Thus τ t (x, τ)= θ(x, t s)ds, (x, τ) R +, t. (1 ω) k(τ) θ(t τ)dτ = μ(τ) t (τ)dτ. Consequently, in view of [15, p. 165], problem (1.1) (1.3) is transformed into the following equivalent system: u tt + u g( u ) u u tt + ν θ = f (u), (x, t) R +, θ t ω θ μ(τ) t (τ)dτ ν u t =, (x, t) R +, t t(x, τ)=θ(x, t) τ t(x, τ), (x, τ) R+, t >, (.1) with boundary conditions u = u =, (x, t) [, ), θ =, (x, t) [, ), t (x, τ)=, (x, τ) R +, t, (.) and initial conditions u(x,)=u (x), u t (x,)=u 1 (x), θ(x,)=θ (x), (x, τ)= (x, τ), (.3) where θ (x)=θ (x,), x, (x, τ)= τ θ(x, s)ds, (x, τ) R +.

4 LiuBoundary Value Problems (18) 18:161 Page 4 of 11.3 Statement of main results We introduce a weighted L -space L := L μ( R + ; H 1 ()) = { : R + H 1 () μ(τ) (τ) dτ < }, which is a Hilbert space equipped with the inner product (, ϕ) L = and the norm μ(τ)(, ϕ)dτ, L = μ(τ) dτ. Definition.1 (u(t), θ(t), v t ) is called a weak solution to problem (.1) (.3) ifu L (, T; H () H 1 ()), u t L (, T; H 1 ()), θ L (, T; L ()), t L (, T; L), u(x,)=u (x), u t (x,)=u 1 (x), θ(x,)=θ (x), (x, τ)= (x, τ), and (u tt, ϕ 1 )+( u, ϕ 1 )+ ( g ( u ) ) u, ϕ 1 +( utt, ϕ 1 ) ν( θ, ϕ 1 )= ( ) f (u), ϕ 1, (θ t, ϕ )+ω( θ, ϕ )+ ( t, ϕ )L + ν( u t, ϕ )=, ( ) t t, ϕ 3 L =(θ, ϕ 3) L ( τ t, ϕ ) 3 L, for any ϕ 1 H () H 1(), ϕ H 1(), ϕ 3 L and a.e. t (, T]. The energy associated with problem (.1) (.3)isgivenby E(t)= 1 u t + 1 u t + 1 u + 1 G( u ) + 1 θ + 1 t L F(u)dx. Furthermore, we define the energy functional J ( u, θ, t) = 1 u + 1 G( u ) + 1 θ + 1 t L F(u)dx, and the Nehari functional I ( u, θ, t) = u + g ( u ) u + θ + t L uf (u)dx. Thus, all nontrivial stationary solutions belong to the Nehari manifold defined by N = {( u, θ, t) H \ { (,,) } I ( u, θ, t) = }, where H := H () H 1 () L () L. Then the mountain pass level of J can be characterized as d = inf (u,θ, t ) N J ( u, θ, t).

5 LiuBoundary Value Problems (18) 18:161 Page 5 of 11 We introduce the potential well W = {( u, θ, t) H I ( u, θ, t) >,J ( u, θ, t) < d } { (,,) }, and let W := W W = {( u, θ, t) H I ( u, θ, t), J ( u, θ, t) d }. The main results of this paper are stated as follows. Theorem.1 Let assumptions (A 1 ) (A 3 ) be fulfilled, u H () H 1(), u 1 H 1(), θ L (), L. Assume that <E() < d, and I(u, θ, )>or (u, θ, )=(,,). Then problem (.1) (.3) admits a global solution (u, θ, t ) W. Moreover, E(t)+ω θ dτ E(). (.4) Theorem. Let assumptions (A 1 ) (A 3 ) be fulfilled, u H () H 1(), u 1 H 1(), θ L (), L. Assume that E() = dandi(u, θ, ). Then problem (.1) (.3) admits a global solution (u, θ, t ) W. 3 Proof of Theorem.1 Let {w j } j=1 be an orthogonal basis of H () H 1()andletanorthonormalbasisofL () be given by eigenfunctions of w = λw, in, w = w =, on. Then {v j } j=1 is an orthonormal basis of H1 (), where v j = w j.weselect{e j } j=1 as {l kv j } k,j=1, where {l k } k=1 is an orthonormal basis of L μ (R+ ). Then {e j } j=1 is an orthonormal basis of L. We construct the approximate solutions to problem (.1) (.3)as n n u n (x, t)= ξ jn (t)w j (x), θ n (x, t)= η jn (t)v j (x), j=1 j=1 n n t (x, τ)= ζ jn (t)e j (x, τ), n =1,,..., which satisfy j=1 1 λ 4 j (u ntt, w j )+( u n, w j )+(g( u n ) u n, w j ) +( u ntt, w j ) ν( θ n, w j )=(f(u n ), w j ), (θ nt, v j )+ω( θ n, v j )+(n t, v j) L + ν( u nt, v j )=, (nt t, e j) L =(θ n, e j ) L (nτ t, e j) L, j =1,,...,n, (3.1)

6 LiuBoundary Value Problems (18) 18:161 Page 6 of 11 with u n (x,)= n j=1 ξ jn()w j (x) u (x) inh () H 1(), u nt (x,)= n jn ()w j(x) u 1 (x) inh 1(), j=1 ξ θ n (x,)= n j=1 η jn()v j (x) θ (x) inl (), n (x, τ)= n j=1 ζ jn()e j (x, τ) (x, τ) inl. (3.) The approximate problem (3.1) (3.) can be reduced to an ordinary differential system in the variables ξ jn (t), η jn (t) andζ jn (t). In terms of standard theory for ODEs, there exists asolution(u n (t), θ n (t), t n ) on some interval [, T n)witht n T. The following estimates will allow us to extend the local solutions to [, T] for all T >. Multiplying (3.1) 1 by ξ jn (t), (3.1) by η jn (t), and (3.1) 3 by ζ jn (t), summing over j, and adding the two results, we obtain where d dt E n(t)+ω θ n = ( nτ t, n t ) L, (3.3) E n (t)= 1 u nt + 1 u nt + 1 u n + 1 G( u n ) + 1 θ n + 1 t n L Since t n (x, ) =, we deduce from (A 3)that ( t nτ, t n) L = 1. ( μ(τ) n t τ (τ) ) 1 dτ F(u n )dx. (3.4) μ (τ) t n (τ) dτ Hence, by integrating (3.3) with respectto t from to t,weget E n (t)+ω We now claim that θ n dτ E n (). (3.5) ( un (t), θ n (t), n t ) W, (3.6) for all t [, T]andsufficientlylargen. Indeed, if (u, θ, )=(,,),then(u, θ, ) W.IfI(u, θ, )>,then,frome() < d, i.e., 1 u u 1 + J(u, θ, )<d, it follows that J(u, θ, )<d. Hence(u, θ, ) W. Thus(u n (), θ n (), n ) W for sufficiently large n due to (3.). As a result, assertion (3.6) follows as desired. If it was not the case, there would exist a < t < T such that (u n (t ), θ n (t ), t n ) W, i.e.,

7 LiuBoundary Value Problems (18) 18:161 Page 7 of 11 I(u n (t ), θ n (t ), t n )=and(u n (t ), θ n (t ), t n ) (,, ), or J(u n (t ), θ n (t ), t n )=d. Due to (3.4), (3.5)and(3.), we get 1 u nt + 1 u nt + J ( u n, θ n, n t ) < d, (3.7) for all t [, T]andsufficientlylargen.ThistellsusthatJ(u n (t ), θ n (t ), t n )=d is impossible. On the other hand, if I(u n (t ), θ n (t ), t n )=and(u n (t ), θ n (t ), t n ) (,, ), then, by the definition of d,wegetj(u n (t ), θ n (t ), t n ) d, which contradicts (3.7). We deduce from (A 1 )and(a )that J ( u n, θ n, n t ) γ α ( un + g ( u n ) u n + θ n + t ) αγ n L + 1 γ I( u n, θ n, t n). Combining this with (3.4) (3.6)and(3.), we arrive at γ α ( un + g ( u n ) u n + θ n + t ) αγ n L + 1 u nt + 1 u nt + ω for all t [, T]andsufficientlylargen.Moreover, f (u n ) ( ) p q αγ q βq u n p p βq C p u n p < β q C p γ α d, θ n dτ < d, (3.8) where q = p p 1,andC is the constant for the Sobolev embedding H () H 1 () L p (). Hence there exist (u, θ, t ) and subsequences of {u n }, {θ n }, {n t},stillrepresentedbythe same notations (and we shall not repeat this), such that, as n, u n u weakly star in L (, T; H () H 1 ()), (3.9) u n u strongly in L p ()anda.e.in [, T], (3.1) u nt u t weakly star in L (, T; H 1 ()), (3.11) θ n θ weakly star in L (, T; L () ) ( and weakly in L (, T; H 1 ()) if ω > ), (3.1) t n t weakly star in L (, T; L), (3.13) f (u n ) χ weakly star in L (, T; L q () ), for any T >.Inviewof[18, Lemma 1.3], we have χ = f (u). According to the Aubin Lions lemma, we have u n u strongly in L (, T; H 1 ()),

8 LiuBoundary Value Problems (18) 18:161 Page 8 of 11 which, together with (3.8), (3.9)and(3.11), gives u n u strongly in C (, T; H 1 ()). (3.14) Therefore, by the arguments similar to the proof given in [14, p ], we can pass to the limit in the approximate problem (3.1) (3.). Thus (u, θ, t ) W is a global solution to problem (.1) (.3). Next, we prove (.4). Indeed, note that ( G un ) G ( u ) 1 = g(ϑ 1 )dσ un u C 1 u n u, where ϑ 1 = σ u n +(1 σ ) u,<σ < 1. Hence it follows from (3.14)that lim G( u n ) = G ( u ). (3.15) n Furthermore, F(u n )dx F(u)dx f (ϑ ) un u dx f (ϑ ) q u n u p C u n u p, where ϑ = σ u n +(1 σ )u. This, together with (3.1), yields lim F(u n )dx = F(u)dx. (3.16) n Consequently, by (3.9), (3.11) (3.13), (3.5), (3.15), (3.16)and(3.), we obtain 1 u t + 1 u t + 1 u + 1 θ + 1 t L + ω θ dτ ( 1 lim inf n u nt + 1 u nt + 1 u n + 1 θ n + 1 n t L ) + ω θ n dτ lim inf n ( E n () 1 G( u n ) + = E() 1 G( u ) + F(u)dx. Thus the proof of Theorem.1 is complete. ) F(u n )dx 4 Proof of Theorem. We divide the proof of this theorem into two cases.

9 LiuBoundary Value Problems (18) 18:161 Page 9 of 11 Case 1. (u, θ, ) (,, ). Let δ m =1 1 m, u m = δ m u, θ m = δ m θ and m = δ m, m =,3,....Weconsiderproblem (.1) (.) with the following initial conditions: u(x,)=u m (x), u t (x,)=u 1 (x), θ(x,)=θ m (x), (x, τ)= m (x, τ). (4.1) From I(u, θ, ), and J ( δu, δθ, δ t) = 1 δ u + 1 G( δ u ) + 1 δ θ + 1 δ t L F(δu)dx, I ( δu, δθ, δ t) = δ d dδ J( δu, δθ, δ t), it is easy to verify that there exists a unique δ = δ (u ) 1suchthatJ(δu, δθ, δ t )isstrictly increasing for δ [, δ ] and assumes the maximum at δ = δ.hencej(u m, θ m, m )< J(u, θ, )andi(u m, θ m, m )>.Moreover, J(u m, θ m, m ) γ α ( um + g ( u m ) u m + θ m αγ + m t ) 1 L + γ I(u m, θ m, m ) >. We further obtain E m () = 1 u u 1 + J(u m, θ m, m )>, and E m () < 1 u u 1 + J(u, θ, )=E() = d. Hence, we conclude from Theorem.1 that problem (.1) (.) and(4.1) admits a global solution (u m (t), θ m (t), m t ) W satisfying (u mtt, ϕ 1 )+( u m, ϕ 1 )+(g( u m ) u m, ϕ 1 ) +( u mtt, ϕ 1 ) ν( θ m, ϕ 1 )=(f(u m ), ϕ 1 ), (θ mt, ϕ )+ω( θ m, ϕ )+(m t, ϕ ) L + ν( u mt, ϕ )=, (mt t, ϕ 3) L =(θ m, ϕ 3 ) L (mτ t, ϕ 3) L,

10 LiuBoundary Value Problems (18) 18:161 Page 1 of 11 with u m (x,)=u (x), θ m (x,)=θ (x), m (x, τ)= (x, τ), u mt (x,)=u 1 (x), and E m (t)+ω Consequently, θ m dτ E m (). γ α ( um + g ( u m ) u m + θ m + t αγ m ) L + 1 u mt + 1 u mt + ω θ m dτ < d, BytheargumentssimilartotheproofofTheorem.1, weseethatproblem(.1) (.3) admits a global solution (u, θ, t ) W. Case. (u, θ, )=(,,). In this case, it is clear that J(u, θ, )=.Thus E() = 1 u u 1. Let δ m =1 1 m and u m1 = δ m u 1 (x), m > 1 and consider problem (.1) (.) with the following initial conditions: u(x,)=u (x), u t (x,)=u m1 (x), θ(x,)=θ (x), (x, τ)= (x, τ). (4.) Note that <E m () = 1 u m1 + 1 u m1 < E(). We conclude from Theorem.1 that problem (.1) (.)and(4.) admits a global solution (u m, θ m, m t ) W. The remainder of the proof is the same as in Case 1. Acknowledgements The author would like to thank the reviewers for the valuable suggestions. Funding This work is supported by the Science and Technology Plan Project of Gansu Province in China (Grant No. 17JR5RA79). List of abbreviations u p, u L p () ; u, u ; L, L μ (R+ ; H 1 ()); H, H () H 1 () L () L. Availability of data and materials Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

11 LiuBoundary Value Problems (18) 18:161 Page 11 of 11 Competing interests The author declares that he has no competing interests. Authors contributions The author carried out all the work in this manuscript, and read and approved the final manuscript. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: July 18 Accepted: 17 October 18 References 1. Lagnese, J.E.: Boundary Stabilization of Thin Plates. SIAM, Philadelphia (1989). Lagnese, J.E., Lions, J.L.: Modelling Analysis and Control of Thin Plates. Masson, Paris (1988) 3. Gurtin, M.E., Pipkin, A.C.: A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. Anal. 31(), (1968) 4. Coleman, B.D., Gurtin, M.E.: Equipresence and constitutive equations for rigid heat conductors. Z. Angew. Math. Phys. 18(), (1967) 5. Ball, J.M.: Stability theory for an extensible beam. J. Differ. Equ. 14(3), (1973) 6. Berger, H.M.: A new approach to the analysis of large deflections of plates. J. Appl. Mech., (1955) 7. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.: Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation. Commun. Contemp. Math. 6(5), (4) 8. Dickey, R.W.: Free vibrations and dynamic buckling of the extensible beam. J. Math. Anal. Appl. 9(), (197) 9. Eden, A., Milani, A.J.: Exponential attractors for extensible beam equations. Nonlinearity 6(3), (1993) 1. Ma, T.F., Narciso, V.: Global attractor for a model of extensible beam with nonlinear damping and source terms. Nonlinear Anal. 73(1), (1) 11. Woinowsky-Krieger, S.: The effect of an axial force on the vibration of hinged bars. J. Appl. Mech. 17, (195) 1. Wu, H.: Long-time behavior for a nonlinear plate equation with thermal memory. J. Math. Anal. Appl. 348(),65 67 (8) 13. Barbosa, A.R.A., Ma, T.F.: Long-time dynamics of an extensible plate equation with thermal memory. J. Math. Anal. Appl. 416(1), (14) 14. Giorgi, C., Pata, V., Marzocchi, A.: Asymptotic behavior of a semilinear problem in heat conduction with memory. Nonlinear Differ. Equ. Appl. 5(3), (1998) 15. Grasselli, M., Pata, V.: Uniform attractors of nonautonomous dynamical systems with memory. In: Lorenzi, A., Ruf, B. (eds.) Evolution Equations, Semigroups and Functional Analysis, 1st edn. vol. 5, pp Birkhäuser, Basel () 16. Payne, L.E., Sattinger, D.H.: Sadle points and instability of nonlinear hyperbolic equations. Isr. J. Math. (3 4), (1975) 17. Sattinger, D.H.: On global solution of nonlinear hyperbolic equations. Arch. Ration. Mech. Anal. 3(), (1968) 18. Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris (1969)