Asymptotic behavior of solutions of mixed problem for linear thermo-elastic systems with microtemperatures

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1 Mathematica Aeterna, Vol. 8, 18, no. 4, 7-38 Aymptotic behavior of olution of mixed problem for linear thermo-elatic ytem with microtemperature Gulhan Kh. Shafiyeva Baku State Univerity Intitute of Mathematic and Mechanic of NAS of Azerbaijan Gunay R. Gadirova Intitute of Mathematic and Mechanic of NAS of Azerbaijan Abtract In thi paper we tudy the mixed problem with diipative boundary for linear thermo-elatic ytem with microtemperature. We invetigate the correctne of the mixed problem and etablih the exponential decreae in the energy norm of the olution. Mathematic Subject Claification: 35M1, 35B3, 8A17 Keyword: Thermo-elatic ytem with microtemperature, mixed problem, exponential decay energy function. 1 Introduction Thermo-elatic ytem decribe the elatic and thermal behavior of elatic heat conductive media, particularly the reciprocal action between elatic tree and temperature difference [1 5]. In recent year, the exitence, uniquene and aymptotic behavior of olution of the ytem of thermoelaticity ha been analyzed intenively [6,7,1,14,15] and the reference cited therein. Eringen [16] introduced a cla of micromorphic olid and called them microtretch olid. Microtretch olid of modeling porou media filled with ga or vicid fluid and compoite material with chopped elatic fiber. The material point i that of thee material can tretch and contract independently

2 8 Gulhan Kh. Shafiyeva and Gunay R. Gadirova of their tranlation and rotation. The exitence of olution the mixed problem and the Cauchy problem for different ytem of thi type tudied in the work [6 14]. The Cauchy problem for a emilinear thermo-elatic ytem with microtemperature in one pace variable are conidered in the follow work [8, 1, 13 15]. Statement problem and main reult In the domain [; ) [; 1] we conider the following thermo-elatic ytem with microtemperature: u tt µ u xx + bθ x = ϕ tt α ϕ xx + ωw x = θ t kθ xx + βu xt + gw x = w t γw xx + hϕ xt + mθ x =, (1) where u, ϕ, θ and w repreent the diplacement vector, microtretch, abolute temperature difference θ = T α T and microtemperature, repectively; µ, b, α, ω, k, β, g, γ, h and m are mooth function of (t, x) [, ) [, 1] with µ, α, k and γ being poitive. For ytem (1) we invetigate the mixed problem with boundary condition and the initial condition u(, x) = u (x), ϕ(, x) = ϕ (x), θ(, x) = θ (x), u(t, ) =, ϕ(t, ) =, () { ut (t, 1) + u x (t, 1) = ϕ t (t, 1) + ϕ x (t, 1) =, (3) θ(t, ) = θ(t, 1) =, w(t, ) = w(t, 1) = (4) u t (, x) = u 1 (x) ϕ t (, x) = ϕ 1 (x) w(, x) = w (x), x [, 1]. (5) The main purpoe of thi paper i to etablihing the behavior of olution of the problem (1)-(5) when µ, b, α, ω, k, β, g, γ, h and m ome contant and µ >, α >, k >, ω >. (6) Let there exit number λ i, i =, 1, uch that λ i >, i =, 1,, m = λ g,. (7) λ β = λ 1 b, λ ω = h

3 Aymptotic behavior of olution of mixed problem... 9 We introduce the following notation L = L (, 1), W 1 = {u : u W 1, u() = }, W 1 = W 1 (, 1) = {u : u W 1 (, 1), u() = u(1) = }, W = W (, 1). In the pace H = W 1 L W 1 L L L we define the calar product a follow: w, z H = λ 1 µ ν 1x z 1x dx + λ 1 ν x z x dx+ +λ α ν 3x z 3x dx + λ ν 4x z 4x dx + λ ν 5x z 5x dx + where w = (ν 1,..., ν 6 ), z = (z 1,..., z 6 ) H. We denote by H a following pace H = { w : w = (ν 1,..., ν 6 ) [ W W 1 W 1 ν 6x z 6x dx, [ ] ] W W 1, and by E(t) the energy function E (t) = ν 1x (1) + ν (1) =, ν 3x (1) + ν 4 (1) = } [ ut + ϕ t + u x + ϕ x + θ + w ] dx. In thi paper i obtained the following main reult: Theorem.1 Suppoe that condition (6),(7) are fulfilled. Then there exit number M 1 and d > uch that for any (u, u 1, ϕ, ϕ 1, θ, w ) H the inequality i true, where E () = E (t) Me dt E () [ ux + ϕ x + u 1 + ϕ 1 + θ + w ] dx.

4 3 Gulhan Kh. Shafiyeva and Gunay R. Gadirova 3 Exitence of olution of the problem (1)-(5) In pace H we define a linear operator A where Aw = ( ν, µ ν 1xx bν 5x, D (A) = H, α ν 3xx ων 6x, kν 5xx βν x gν 6x, γ ν 6xx hν 4x mν 5x ), w = (ν 1,..., ν 6 ) D (A). Lemma 3.1 A i diipative operator in H. Proof. Let w = (ν 1,..., ν 6 ) D (A). Then ( ) Aw, w H = λ 1 µ ν x ν 1x dx + λ 1 µ ν 1xx bν 5x νx dx+ ( ) +λ α ν 4x ν 3x dx + λ α ν 3xx ων 6x ν4 dx+ +λ (kν 5xx βν x gν 6x ) ν 5 dx + Integrating by part the obtained equality we have: γ Re Aw, w H = λ k Hence taking into account (6) that (γν 6xx hν 4x mν 5x ) ν 6 dx. ν 5x dx ν 6x dx λ 1 µ ν (1) λ α ν 4 (1). (8) Re Aw, w H. In thi way A i diipative operator. Lemma 3. A i invertible operator in H. Proof. Let h = (h 1,..., h 6 ) H. We conider the equation Aw = h, w D (A). (9) The equation (9) i equivalent to the boundary value problem ν = h 1 µ ν 1xx bν 5x = h ν 4 = h 3 α ν 3xx ων 6x = h 4 kν 5xx βν x gν 6x = h 5 γ ν 6xx hν 4x mν 5x = h 6 (1)

5 Aymptotic behavior of olution of mixed problem with the boundary condition: { ν1 () =, ν () =, ν 3 () =, ν 4 () =, ν 5 () = ν 5 (1) =, ν 6 () = ν 6 (1) =, (11) { ν1x (1) + ν (1) = ν 3x (1) + ν 4 (1) =. (1) Subtituting ν = h 1 and ν 4 = h 3 in other equation ytem (1) and boundary condition (11),(1) we obtain µ ν 1xx bν 5x = h α ν 3xx ων 6x = h 4 kν 5xx gν 6x = h 5 + βh 1x, (13) γν 6xx mν 5x = h 6 + hh 3x ν 1 () =, ν 3 () =, ν 5 () = ν 5 (1) =, ν 6 () = ν 6 (1) =, ν 1x (1) = h 1 (1), ν 3x (1) = h 3 (1). (14) Firt olve the ytem { kν5xx gν 6x = h 5 βh 1x γν 6xx mν 5x = h 6 hh 3x (15) with boundary condition { ν5 () = ν 5 (1) =, ν 6 () = ν 6 (1) =. (16) The problem (15),(16) ha a unique olution (ν 5, ν 6 ), where ν 5 W W 1, ν 6 W W 1. Subtituting ν 5 and ν 6 in the firt two equation of the ytem (13) we obtain the boundary value problem for the ytem with repect to the function ν 1 and ν 3 with inhomogeneou boundary condition. The obtained problem i alo olved by the tandard method. From Lemma 3.1 and Lemma 3. it follow that A maximally diipative operator, therefore A generate a trongly continuou emigroup U (t) = e ta. In thi cae w (t) = e ta w i a trong olution of the problem (1)-(5), if w H. If w H, then w (t) = e ta w i a weak olution of the problem (1)-(5). In thi intance if w H, w n H and w n w in H at n, then lim e ta w n = e ta w in C ([, T ] ; H) [17]. In thi ituation boundary n condition are undertood in the following ene: -for any ν, z W 1 take place the equalitie d dt u(t, 1), ν(1) u(t, 1), ν x(1) =, t [, T ], (17)

6 3 Gulhan Kh. Shafiyeva and Gunay R. Gadirova d dt ϕ(t, 1), z(1) ϕ(t, 1), z x(1) =, t [, T ]. (18) Thu the following theorem are valid: Theorem 3.1 Let the condition (6),(7) are fulfilled. Then for any (u, u 1, ϕ, ϕ 1, θ, w ) H problem (1)-(5) ha a unique olution (u, ϕ, θ, w), where u, ( ϕ C ([, ) ; W ) W 1 ) C 1 ([, ) ; W 1 ) C ([, ) ; L ), θ, w C [, ) ; W W 1 C 1 ([, ) ; L ) and the boundary condition (3),(4) are atified. Theorem 3. Let the condition (6),(7) are fulfilled. Then for any (u, u 1, ϕ, ϕ 1, θ, w ) H problem (1)-(5) ha a unique weak olution (u, ϕ, θ, w), where u, ϕ C ([, ) ; W 1 ) C 1 ([, ) ; L ), θ, w C ([, ) ; L ) and the boundary condition (3) are atified in the meaning of (17)-(18). 4 Proof of Theorem.1 For proof of Theorem.1 we ll ue the following lemma, which i proved in the paper V. Komornik [18]. Lemma 4.1 Let Z (t) : [, ) [, ) be a non-increaing function and aume that there exit contant c > uch that Z (t) dt MZ (),. Then ( Z (t) Z () exp 1 t ), t. c We define the functional + λ E (t) = λ 1 ϕ t dx + α λ u t dx + µ λ 1 ϕ xdx + λ u xdx+ θ dx + 1 It i eay to ee that the following lemma i true: w dx. Lemma 4. Let the condition (6),(7) are atified, then exit c 1 > and c > uch that c 1 E (t) E (t) c E (t).

7 Aymptotic behavior of olution of mixed problem Aume that (u, u 1, ϕ, ϕ 1, θ, w ) H, then E (t) i differentiable. Taking into account (1) we obtain that γ E (t) = kλ θxdx w xdx + λ 1 µ u x (1, t) u t (1, t) + λ α ϕ x (1, t) ϕ t (1, t). (19) By uing (3) from (19) we have Conequently, γ E (t) = kλ θxdx w xdx λ 1 µ u t (t, 1) λ α ϕ t (t, 1). () E (t) E (), t. (1) From the inequality (1) we directly obtain the following: Lemma 4.3 The following etimate are true: 1. θ x(t, x)dx 1 k E (t), t T ;. θ x(t, x)dxdt 1 k E (), T ; 3. w x(t, x)dx 1 γ E (t), t T ; 4. w x(t, x)dxdt 1 γ E (), T ; 5. u t (t, 1) dt 1 λ 1 µ E (), T ; 6. ϕ t (t, 1) dt 1 λ 1 α E (), T. Multiplying the firt equation of ytem (1) by u and integrating over the domain [, T ] [, 1] we get: = u ( u tt µ u xx + bθ x ) dxdt =

8 34 Gulhan Kh. Shafiyeva and Gunay R. Gadirova = u u t T t= dx µ u (t, 1) u x (t, 1) dt + u t dxdt + µ u x dxdt buθ x dxdt. () Similarly, multiplying the econd equation of ytem (1) by ϕ and integrating over the domain [, T ] [, 1] we have: = = ϕ ϕ t T t= dx ϕ ( ϕ tt α ϕ xx + ωw x ) dxdt = α ϕ (t, 1) ϕ x (t, 1) dt + From () and (3) follow that Hence we obtain + λ 1 λ 1 µ ϕ t dxdt + α u t dxdt + λ u x dxdt λ α ϕ x dxdt ωϕw x dxdt. (3) ϕ t dxdt ϕ x dxdt = = λ 1 u u t T t= dx λ 1µ u (t, 1) u x (t, 1) dt+ +λ ϕ ϕ t T t= dx λ α ϕ (t, 1) ϕ x (t, 1) dt+ +λ 1 b uθ x dxdt + λ ω ε (t) dt = λ 1 u u t T t= dx + λ ( λ θ + bλ 1 uθ x ) dxdt + +λ 1 µ u x dxdt + λ α ϕw x dxdt. ϕ ϕ t T t= dx+ ( w + λ ωw x ϕ ) dxdt+ ϕ xdxdt λ 1 µ u (t, 1) u x (t, 1) dt λ α ϕ (t, 1) ϕ x (t, 1) dt. (4)

9 Aymptotic behavior of olution of mixed problem Uing Holder inequality, embedding theorem [19] and Lemma 4.3 we get that ελ 1 µ C Similarly, we have λ 1 µ u (t, 1) u x (t, 1) dt ελ 1 µ C ελ α C u x (t, x) dxdt + 1 ε u x (t, 1) dt u x (t, x) dxdt + C (ε) E (). (5) λ α ϕ (t, 1) ϕ x (t, 1) dt ϕ x (t, x) dxdt + C (ε) E (). (6) Multiplying both ide of the firt equation of ytem (1) by xu x and integrating over the domain [, T ] [, 1] we obtain µ = xu x u xx dxdt + b xu tt u x dxdt xu x θ x dxdt. (7) Integrating (7) by part and taking into account the boundary condition (), (3) we get: µ u xdxdt + u t dxdt xu x µ u x (t, 1) dt + b T dx + xu t= x u t T t= dx Similarly, we have the following identity α xϕ x ϕ xdxdt + α ϕ x (t, 1) dt + ω xu x θ x dxdt =. (8) ϕ t dxdt T dx + xϕ t= x ϕ t T t= dx xϕ x w x dxdt =. (9)

10 36 Gulhan Kh. Shafiyeva and Gunay R. Gadirova Uing Holder and Young inequalitie, Lemma 4.3, and inequality (1) from (8) we obtain that ( µ ε ) Similarly, from (9) we get ( α ε ) u xdxdt C ε E (). ϕ xdxdt C ε E (). (3) Further, uing Poincare inequality [] and Lemma 4.3, we have that w dxdt C wxdxdt CE (). Uing Holder and Young inequalitie we alo obtain bλ 1 Similarly, we get that ωλ uθ x dxdt ε ε w x ϕdxdt ε u xdxdt + C b λ 1 ε θ xdxdt u xdxdt + C ε E (). (31) ϕ xdxdt + C ε E (). (3) Chooing ε > mall enough from (4)-(3), we have that E (t) dt CE (). (33) Let (u, u 1, ϕ, ϕ 1, θ, w ) H. Then there exit (u n, u 1n, ϕ n, ϕ 1n, θ n, w n ) H uch that (u n, u 1n, ϕ n, ϕ 1n, θ n, w n ) (u, u 1, ϕ, ϕ 1, θ, w ) in H. Hence, paing to the limit we obtain that inequality (1) i alo valid for initial data (u, u 1, ϕ, ϕ 1, θ, w ) H, but inequality (33) i valid for weak olution too. Thu, by Lemma 4.1 we get E (t) Me d t E(), t, (34) where M > and d > not depend on t >. Note that the tatement of the Theorem.1 follow from (34) and Lemma 3..

11 Aymptotic behavior of olution of mixed problem Reference [1] S. Jiang, R. Racke, Evolution equation in thermoelaticity, Monograph and Survey in Pure and Applied Mathematic, 11(CRC/Chapman and Hall, ). [] S. Chen, Y.G. Wang, Propagation of ingularitie of olution to hyperbolic parabolic coupled ytem, Math. Nachr., 4 (), [3] M. E. Gurtin, A. C. Pipkin, A general theory of heat conduction with finite wave peed, Arch. Ration. Mech. Analyi, 31 (1968), [4] D.D. Joeph, L. Prezioi, Heat wave, Rev. Mod. Phy., 61 (1989), [5] R. Racke, Lecture on nonlinear evolution equation: initial value problem, (Braunchweig:Vieweg, 199). [6] R. Racke, Thermoelaticity with econd ound: exponential tability in linear and nonlinear 1-d, Math. Meth. Appl. Sci., 5 (), [7] R. Racke, Aymptotic behavior of olution in linear - or 3-d thermoelaticity with econd ound, Quart. Appl. Math., 61 (3), [8] R. Racke, Y.G. Wang, Propagation of ingularitie in one-dimenional thermoelaticity, J. Math. Analyi Applic., 3 (1998), [9] M. Reiig, Y.G. Wang, Linear thermoelatic ytem of type III in 1-D, Preprint. [1] M.A. Tarabak, On exitence of mooth olution in one-dimenional nonlinear thermoelaticity with econd ound, Quart. Appl. Math., 5 (199), [11] Y.G. Wang, Microlocal analyi in nonlinear thermoelaticity, Nonlin. Analyi, 54 (3), [1] Y.G. Wang, M. Reiig, Parabolic type decay rate for 1-D-thermoelatic ytem with time-dependent coefficient, Monath, Math., 138 (3), [13] A.M. Abd El-Latief, S.E. Khader, Exact Solution of Thermoelatic Problem for a One-Dimenional Bar without Energy Diipation, Hindawi Publihing Corporation, ISRN Mechanical Engineering, 14(14), Article ID 69459, 6 page.

12 38 Gulhan Kh. Shafiyeva and Gunay R. Gadirova [14] E. Jaime, Mucoz Rivera, Maria Grazia Nao, Ramon Quintanilla, Decay of olution for a mixture of thermoelatic one dimenional olid, Computer and Mathematic with Application, 66 (13), [15] M. Slemrod,Global exitence, uniquene and aymptotic tability of claical mooth olution in one dimenional nonlinear thermoelaticity, Arch. Rational Mech. Anal., 76 (1981), [16] A. C. Eringen, Linear theory of micropolar elaticity, J. Math. Mech., 15 (1996), [17] T. Kato, Quai - linear of evolution equation, with application to partial differential equation, Springer, Lecture Note in Mathematic, 448( 1975), 5 7. [18] V. Komornik, Decay etimate for the wave equation with internal damping, International Serie Num. Math. Birkhauer Verlag Bael, 118 (1994), [19] R.A. Adam, Sobolev Space,Academic Pre, New York, (1975), 68pp. [] L.C. Evan, Partial differential equation, nd ed., 19(1), 749pp. Received: June 9, 18

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