On a contact problem in thermoelasticity with second sound

Universität Konstanz

On a contact problem in thermoelasticity with second sound

Jan Sprenger

Konstanzer Schriften in Mathematik und Informatik Nr. 245, März 2008

ISSN 1430-3558

© Fachbereich Mathematik und Statistik

© Fachbereich Informatik und Informationswissenschaft Universität Konstanz

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ON A CONTACT PROBLEM IN THERMOELASTICITY WITH SECOND SOUND

JAN SPRENGER

Abstract. We investigate the existence and stability of a thermoelastic con- tact problem with second sound. Previous results established the existence and stability of a solution of the corresponding classical system in the case of radial symmetry. However, recent works have shown that sometimes stability can be lost when the classical Fourier heat conduction is substituted by Cattaneo’s Law. We show that also in this case this substitution does indeed lead to a loss in regularity that proves to be a major problem prohibiting the transfer of the existence proof for the classical problem to the problem with second sound, leaving the existence of a solution an open question. We then prove that, if a viscoelastic term is added to the equations providing additional regularity, existence and exponential stability - the second, as can be expected, only in the case of radial symmetry - follow.

1. Introduction

We consider a thermoelastic system that can come into contact with a rigid foun- dation. In particular, consider the equations of thermoelasticity with second sound on a bounded set Ω⊂Rⁿ with smooth boundary∂Ω = ΓC∪ΓN∪ΓD. On ΓD, the body is held fix, while on Γ_N tractions are zero. On Γ_C, the body is free, albeit its extension is limited by a rigid foundation. The temperature is held fixed at the en- tire boundary. If u=u(t, x),θ=θ(t, x) andq=q(t, x) describe the displacement, relative temperature and heat flow respectively, our equations take the form

∂_t²u_i−(C_ijklu_k,l)_,j−µ∂_tu_i,jj+m_ijθ_,j = 0 (1)

∂tθ+ div q+mij∂tui,j = 0 (2) τ0∂tqi+qi+Kijθ,j = 0 (3) On [0, T]×Ω, with initial values

u(0,·) =u0; ut(0,·) =u1; θ(0,·) =θ0; q(0,·) =q0 (4) satisfying

u0∈(H_Γ¹_D(Ω))ⁿ; u1∈(L²(Ω))ⁿ; θ0∈L²(Ω); q0∈(L²(Ω))ⁿ (5) and boundary conditions

θ|_∂Ω= 0 (6)

u|Γ_D = 0; σT|Γ_N = 0;

σ_ν≤0; u_ν≤0; σ_ν(u_ν−g) = 0; σ_T = 0 on Γ_C (7) Where

σij =Cijkluk,l+µui,j−mijθ

is the stress tensor and (withν being the exterior normal vector) σν=σijνiνj; σT =σν−σνν

The author would like to thank Prof. Dr. Racke for the opportunity to work on this interesting topic and helpful suggestions.

1

its normal and tangential components. We assume the elasticity module C = (Cijkl)i,j,k,l, the thermal expansion tensormand the heat conduction tensorK to satisfy

C_ijkl∈L^∞(Ω); ∃d_C>0∀η∈R^n×n:η_ijC_ijklη_kl ≥d_C|η|²; C_ijkl=C_jikl=C_klij

kij∈L^∞(Ω); ∃dk >0∀ξ∈Rⁿ:ξikijξj ≥dk|ξ|²; kij =kji

mij ∈L^∞(Ω); mij ≥0; mij=mji

wherek=K⁻¹in the sense of matrix inversion andµ≥0 is (for now) an arbitrary constant.

A few remarks on notation: We denote ∂ju=u,j, || · || :=|| · ||_(L2)^m, where mis either 1, n or n², which will be clear from the context. In addition, L^∞(H¹) :=

L^∞([0, T], H¹(Ω)) and likewise. H_Γ¹

C(Ω) denotes the space of weakly differentiable functions satisfying u|Γ_C = 0 in a weak sense. The technical problem in the han- dling of these equations lies in the boundary conditions for uon ΓC, which do not allow the well-known semi-group theoretic approach. Problems of this form arise naturally in the manufacturing of casts and pistons, cf. [8].

On the classical problem, i.eτ0=µ= 0, there are a number of papers available. In particular, Mun˜oz-Rivera and Racke [6] studied the corresponding classical prob- lem and derived existence and stability under the condition of radial symmetry. In the case of one space dimension, there are several results: Elliot and Tang [2] gave an existence result for more general boundary conditions; Mu˜noz Rivera and Jiang [5] gave an existence and stability result for a contact problem of two rods, and Gao and Mu˜noz Rivera [3] gave an existence and stability result for the semilinear case. Dropping the ∂²_tuterm in the first equation, one arrives at the quasi-statical case, where Shi and Shillor [8] proved the existence of a solution and Ames and Payne [1] gave a uniqueness result. Mun˜oz-Rivera and Racke [6] also prove the existence of a unique solution to the corresponding classical quasi-static problem and its exponential stability. One would - and, in fact, has for quite some time - expect these results to carry over to the fully hyperbolic problem, especially as the critical equation for the displacementuwhere the difficult boundary conditions arise remains unchanged. However, in a recent work, Racke and Fern´andez Sare [7]

showed that for a damped Timoshenko system, exponential stability is lost when substituting the Fourier Law of heat conduction by Cattaneo’s. In this light, the investigation of the behaviour of this particular system under a transition from clas- sical to hyperbolic heat conduction poses an interesting question. We shall indeed see that this transition leads to a loss in regularity that is not easily compensated, thus requiring the additional viscoelastic term (µ >0).

This paper is organized as follows: In Section 2, we will give a proof for the existence of a weak solution. We will start following the approach of Mu˜noz Rivera and Racke [6] and then show why it can not be extended to this problem. To this end we will approximize the difficult boundary conditions on Γ_Cand obtain a penalized problem. We will then show that this penalized problem has a solution and give a sufficient condition for the convergence of this solution to a solution of our original problem - this is where the loss of regularity from the changed heat equation leaves its mark, as the conditions derived by Mu˜noz Rivera and Racke will no longer be sufficient. Finally, in section 3, we will prove a stability result in the radially symmetrical case, that is, the solutions to our problem decay to 0 exponentially. We will use a Lyapunov functional, similar to [6], although some changes are required to compensate for the different heat equation.

2

2. Existence

We will prove the existence of a solution in the following sense:

Definition 2.1. (u, θ, q)is a solution to (1)-(7) iff

u∈W^1,∞((L²)ⁿ)∩L^∞((H_Γ¹_D)ⁿ), θ∈L^∞(L²), q∈L^∞((L²)ⁿ) (8)

∂_tu(T,·), q(T,·)∈(L²(Ω))ⁿ; θ(T,·)∈L²(Ω); ∇u(T,·)∈L²(Ω) (9)

∀w∈W^1,∞((L²)ⁿ)∩L^∞((H_Γ¹_D)ⁿ), wν≤g onΓC:

−

T

Z

0

h∂tu, ∂twidt+hu(T,·), ∂tw(T,·)−∂tu(T,·)i − hu0, ∂tw(0,·)−u1i

+µ

T

Z

0

h∂_tu_i,j, w_i,jidt+

T

Z

0

hC_ijklu_k,l, w_i,jidt−

T

Z

0

hm_ijθ, w_i,jidt

+

T

Z

0

h∂tu, ∂tuidt−

T

Z

0

hCijkluk,l, ui,jidt−µ

2(||∇u(T,·)||²− ||∇u0||²)

+

T

Z

0

hmijθ, u_ijidt≥0

(10)

∀z∈W^1,∞(H₀¹) :

−

T

Z

0

hθ, zidt+hθ(T,·), z(T,·)i − hθ0, z(0,·)i −

T

Z

0

hqi, z,iidt

−

T

Z

0

hm_iju_i,j, ∂_tzidt+hm_iju_i,j(T,·), z(T,·)i − hm_iju_0i,j, z(0,·)i= 0

(11)

∀y∈W^1,∞((H¹)ⁿ) :

−τ₀

T

Z

0

hkijq_i, ∂_ty_jidt+τ₀hkijq_i(T,·), yj(T,·)i −τ₀hkijq_0i, y_j(0,·)i

+

T

Z

0

hkijqi, yjidt−

T

Z

0

hθ, yi,iidt= 0

(12)

u|ΓC ≤g a.e. (13)

We remark that all boundary conditions are represented in a weak sense in the above definiton. Also, we need the unusual condition (9) for condition (10) to make sense. This will be seen from the context in section 2.

To better handle the difficult boundary conditions in u, we consider the following penalized problem:

∂²_tu_i−(Cijklu_k,l),j−µu_i,jj+mijθ_,j = 0 (14)

∂tθ+ div q+mij∂tu_i,j = 0 (15) τ0∂tkijq_j+kijq_j+θ_,i = 0 (16)

3

with initial conditions

u(0,·) =u₀; u_t(0,·) =u₁; θ(0,·) =θ₀; q(0,·) =q₀ (17) and boundary conditions

θ|∂Ω = 0

u|_ΓD = 0; σ_T|_ΓN = 0 σ_ν =−1

(u_ν−g)⁺−∂tuν σT = 0 on ΓC (18) Note that only the boundary conditions on ΓChave been changed, everything else is identical to the original problem. We will see that σ_ν is bounded and therefore by (18) (u_ν−g)⁺ → 0 as → 0, satisfying (13). Next, we give a definition of a solution to the penalized problem. Letw^pp, y^p_p⊂H¹(Ω) be bases of (L²(Ω))ⁿand z^p_p⊂H₀¹(Ω) be a basis ofL²(Ω).

Definition 2.2. Let

u^ε₀, u^ε₁ ∈ (H^2,2(Ω)∩H₀¹(Ω))ⁿ q₀^ε ∈ (H¹(Ω))ⁿ

θ₀ ∈ H₀¹(Ω) Then(u, θ, q)is a solution to (14)-(18) iff

u^ε∈W^2,∞((L²)ⁿ)∩W^1,∞((H_Γ¹_D)ⁿ); θ^ε∈W^1,∞(L²)∩L^∞(H₀¹);

q^ε∈W^1,∞((L²)ⁿ)∩L^∞(D¹) (19)

u^ε(0,·) =u^ε₀; ∂tu^ε(0,·) =u^ε₁; θ^ε(0,·) =θ₀^ε; q^ε(0,·) =q₀^ε (20) and for allmost all t∈[0, T]

∀p∈N: h∂_t²u^ε(t,·), w^pi+µh∂_tu^ε_i,j(t,·), w_i,j^p i+hC_ijklu^ε_k,l(t,·), w_i,j^p i

− hmijθ^ε(t,·), w^p_i,ji=−1 ε

Z

Γ_C

(u^ε_ν(t,·)−g)⁺w^pdΓ−ε Z

Γ_C

∂tu^ε_ν(t,·)w^pdΓ (21)

∀p∈N: h∂_tθ^ε(t,·), z^pi+hdiv q^ε(t,·), z^pi+hm_ij∂_tu^ε_i,j(t,·), z^pi= 0 (22)

∀p∈N: τ₀hk_ij∂_tq_i^ε(t,·), y_j^pi+hk_ijq_i^ε(t,·), y_j^pi+h∇θ^ε(t,·), y_i^pi= 0 (23) To construct a solution to the penalized problem, we will use a Faedo-Galerkin- method. Note that, if (v, ψ, h) satisfy

v(0,·) =∂tv(0,·) =ψ(0,·) =h(0,·) = 0

h∂_t²v, w^pi+h(C_ijklv_k,l), w^p_i,ji+µh∂_tv_i,j, w^p_i,ji − hm_ijψ, w^p_i,ji

=hf, w^pi −1 ε

Z

Γ_C

(vν−g)⁺w^p_νdΓ−ε Z

Γ_C

(∂tvν)w^p_νdΓ (24) h∂tψ, z^pi+hdivh, z^pi+hmij∂tvi,j, z^pi=hb, z^pi (25) hτ₀k_ij∂_th_j, y_ii+hk_ijh_j, y_ii+h∇ψ, yi=he, y^pi (26) with

f_i := C_ijkl(u^ε_0k,l−tu^ε_1k,l))_,j+µu^ε_1i,jj−(m_ijθ^ε₀)_,j; (i= 1, . . . , n) b := −q^ε_0i,i+miju^ε_1i,j

e := −k_ijq_0j^ε −θ^ε_0,j

4

thenu:=v+u₀+tu₁,θ:=ψ+θ₀andq:=h+q₀ are a solution to the penalized problem. To find such (v, ψ, h), consider the following set of equations on [0, T]

h∂_t²v^m, w^pi_n+hC_ijklv_k,l^m, w^p_i,ji+µh∂_tv_i,j^m, w_i,j^p i − hm_ijψ, w^p_i,ji

=hf, w^pin−1 ε

Z

Γ_C

(v^m_ν −g)⁺w_ν^pdΓ−ε Z

Γ_C

∂tv_ν^mw^p_νdΓ (p= 1, ..., m) (27)

h∂tψ^m, z^pi+hdivh^m, z^pi+hmij∂_tv_i,j^m, z^pi=hb, z^pi(p= 1, ..., m) (28) τ0hkij∂th^m_i , y_j^pi+hkijh^m_i , y^p_ji+h∇ψ^m, y^pin =he, y^pin (p= 1, ..., m) (29) v(0,·) =∂tv(0,·) =ψ(0,·) =h(0,·) = 0 (30) where v^m(t, x) = a^m_p (t)w^p(x), ψ^m(t, x) = b^m_p(t)z^p(x) and h^m(t, x) = c^m_p(t)y^p(x) with unknown coefficents (a^m_p, b^m_p, c^m_p). Then (27)-(30) is a set of ordinary differ- ential equations for (a^m_p, b^m_p , c^m_p), thus possessing a solution with the regularity

v^m∈W^3,∞((H_Γ¹

D)ⁿ), ψ^m∈W^2,∞(H₀¹), h^m∈W^2,∞((H¹)ⁿ)

Note that the initial conditions are arbitrarily smooth and f, g, e are polynomial in t, allowing for a solution with the required smoothness.

Proposition 2.1. There exist(v, ψ, h)such that

(v^m)m

*∗ v in W^2,∞((L²)ⁿ)∩W^1,∞((H_Γ¹_C)ⁿ) (ψ^m)m

*∗ ψ inW^1,∞(L²) (h^m)_m *^∗ hin W^1,∞((L²)ⁿ)

Proof: Multiplying (27) by _dt^da^m_p , (28) by b^m_p and (29) by c^m_p respectively, we obtain after summarizing from 1 tom:

1 2

d dt



||∂tv^m||²+hCijklv_k,l^m, v^m_i,ji+1 ε

Z

Γ_C

|(u^m_ν −g)⁺|²dΓ +||ψ^m||²+τ0||h^m||²





+µ||∂t∇v^m||²+ε Z

Γ_C

|∂tv^m_ν |²dΓ +hkijh^m_i , h^m_j i+hdivh^m, ψ^mi+h∇ψ^m, h^mi

=hf, ∂tv^mi+hb, ψ^mi+he, h^mi

(31) where we used that

Z

Γ_C

(v^m_ν −g)⁺∂tv^mdΓ = d dt

Z

Γ_C

|(v^m_ν −g)⁺|²dΓ

As one easily checks by partial integration,

hdivh^m, ψ^mi+h∇ψ^m, h^mi= 0

5

and therefore, after integrating (31) on (0, t), we obtain by Gronwall’s inequality

||∂tv^m(t,·)||n ≤ C hCijklv_k,l^m(t,·), v^m_i,j(t,·)i ≤ C

1 ε

Z

Γ_C

|(v_ν^m(t,·)−g)⁺|²dΓ ≤ C

||h^m(t,·)||_n ≤ C ε

t

Z

0

Z

ΓC

|∂_tv_ν^m(t,·)|²dΓdt ≤ C (32)

t

Z

0

hkijh^m_i (t,·), h^m_j (t,·)idt ≤ C (33)

Using the smoothness of the functions (v^m, ψ^m, h^m), we see that they satisfy the time-derivated system

h∂_t³v^m, w^pi_n+hC_ijkl∂_tv_k,l^m, w_i,j^p i+µh∂_t²v_i,j^m, w_i,j^p i − hm_ij∂_tψ, w^p_i,ji

= −1 ε

Z

Γ_C

∂t(v^m_ν −g)⁺w^p_νdΓ−ε Z

Γ_C

∂_t²v_ν^mw^p_νdΓ +h∂tf, w^pin (34)

h∂_t²ψ^m, z^pi+h∂tdiv h^m, z^pi+hmij∂_t²v_i,j^m, z^pi= 0 (35)

τ0hkij∂_t²h^m_i , y^p_ji+hkij∂th^m_i , y^p_ji+h∂t∇ψ^m, y^pin = 0 (36) Multiplying (34) by _dt^d2₂a^m_p , (35) by _dt^db^m_p and (36) by _dt^dc^m_p respectively, we obtain similar to (31)

1 2

d dt

||∂_t²v^m||²_n+hCijkl∂tv_k,l^m, ∂tv_i,j^mi+||∂tψ^m||²+τ0||∂th^m||²_n +µ||∂_t²∇v^m||²_n×n+hkijh^m_i , h^m_j i

= h∂_tf, ∂_t²v^mi −1 ε

Z

ΓC

∂_t(v_ν^m−g)⁺∂_t²v^mdΓ−ε Z

ΓC

∂_t²v_ν^m∂_t²v^m_ν dΓ

Observe that in general it is not

∂t |∂t(v^m_ν −g)⁺|²= 2∂t(v_ν^m−g)⁺∂_t²v^ma.e.

since the distributional second derivative of (v_ν^m−g)⁺need not be regular. However, using (32),

1 2

d dt

||∂_t²v^m||²_n+hCijkl∂tv_k,l^m, ∂tv_i,j^mi+||∂tψ^m||²+τ0||∂th^m||²_n +µ||∂_t²∇v^m||²_n×n+hkijh^m_i , h^m_j i

≤ h∂_tf, ∂_t²v^mi_n+ 1 2ε³

Z

ΓC

|∂_tv^m_ν|²dΓ−ε 2

Z

ΓC

|∂_t²v_ν^m|²dΓ

≤ h∂tf, ∂_t²v^min+Cε

(37)

6

where C→ ∞ as→0. For constant we conclude, using Gronwall’s inequality again, that

(v^m)m is bounded in W^2,∞((L²)ⁿ)∩W^1,∞((H_Γ¹_D)ⁿ) (ψ^m)m is bounded in W^1,∞(L²)

(h^m)_m is bounded in W^1,∞((L²)ⁿ)

from which the claimed convergence follows.

We can now show

Theorem 2.1. There is a solution to the penalized problem.

Proof: Take (v, ψ, h) as in Proposition 2.1. Define u:=v+u0+tu1

θ:=ψ+θ0

q:=h+q₀

Then it is clear that (u, θ, q) have the desired regularity (19) and fulfill the initial conditions (20). Using Lemma 1.4 from [4], we obtain the convergence

u^ε→uinC¹([0, T],(L²(Γ_C))ⁿ)

It then follows from the convergence proved in Theorem 2.1 that (u, θ, q) satisfy

(21)-(23).

Now we will prove the convergence of solutions to the penalized problem. As we can see in the proof of Proposition (2.1), we can not use the second energy level to gain estimates on the convergence of (u, θ, q), asis now no longer constant.

Therefore, we lose one level of regularity in time. This loss is grave, since we will no longer have convergence of some terms in the equations, i.e. it is generally unknown if the limits (u, ψ, q) are solutions to the original problem. However, ifµ > 0, the viscoelastic term will provide us with the missing regularity and an existence proof is possible. This will be shown in detail in the proof of Theorem 2.2.

Proposition 2.2. There exist(u, θ, q)such that

u^ε* u^∗ in W^1,∞((L²)ⁿ)∩L^∞(H_Γ¹_C) θ^ε* θ^∗ in L^∞(L²)

q^ε* q^∗ in L^∞((L²)⁽) If µ >0, then

u^ε* uinW^1,2((H_Γ¹_C))

Proof: By the regularity of (u^ε, θ, q), we can subsitute them for (w^p, z^p, y^p) in (21), (22) and (23) respectively and obtain

d dt



||∂tu^ε||²+hCijklu^ε_k,l, u^ε_i,ji+||θ^ε||²+hkijq_i^ε, q_j^εi+1 ε

Z

Γ_C

|(u^ε_ν−g)⁺|²dΓ





+µ||∂t∇u^ε||²+hkijq_i^ε, q^ε_ji+ε Z

Γ_C

|∂tu^ε_ν|²dΓ = 0

(38) where we again used that

<divq, θ >+<∇θ, q >= 0

7

Integrating from 0 tot and using Gronwall’s inequality, we conclude the existence of a constantC=C(||u^ε₀||,||θ₀^ε||,||q₀^ε||) such that for allt >0

||∂tu^ε(t,·)||n ≤ C hC_ijklu^ε_k,l(t,·), u^ε_i,j(t,·)i ≤ C

||θ^ε(t,·)|| ≤ C hkijq^ε_i(t,·), q^ε_j(t,·)i ≤ C 1

ε Z

Γ_C

|(u^ε_ν(t,·)−g(·))⁺|²dΓ ≤ C

µ

t

Z

0

||∂_t∇u^ε(s,·)||²_nds ≤ C

t

Z

0

hkijq_i^ε(s,·), q^ε_j(s,·)ids ≤ C

ε

t

Z

0

Z

ΓC

|∂tu^ε_ν(s,·)|²dΓds ≤ C

This implies the desired convergence.

Proposition 2.3. Let(u, θ, q)be the functions from Proposition 2.2. Then u^ε(T,·)* u(T,^∗ ·) in L^∞(H_Γ¹

C) u^ε_t(T,·)* u^∗ _t(T,·) in L^∞(L²)

θ^ε(T,·)* θ(T,^∗ ·) in L^∞(L²) q^ε(T,·)* q(T,^∗ ·) in L^∞(L²)

Proof: Note that due to the regularity of the solutions to the penalized problem, u, u_t, θandqare continuous in time by Sobolev’s Imbedding Theorem. Therefore, the asserted convergence holds by the estimates gained in the proof for Theorem

2.2.

Theorem 2.2. Let µ >0. Let (u0, u1, θ0, q0)∈(H_Γ¹_D(Ω))ⁿ×(L²(Ω))²ⁿ⁺¹. Then there exists a solution to (1)-(7).

Proof: Let

(u^ε₀)_ε, (u^ε₁)_ε ⊂ (H₀¹(Ω)∩H^2,2(Ω))ⁿ (q₀^ε)ε ⊂ (H¹(Ω))ⁿ

(θ₀^ε)ε ⊂ H¹(Ω) with

u^ε₀−→u0 in (H_Γ¹_D)ⁿ (39)

u^ε₁−→u1 in (L²(Ω))ⁿ (40)

θ₀^ε−→θ0 in L²(Ω) (41)

q^ε₀−→q₀ in (L²(Ω))ⁿ (42) Let (u, θ, q) be the solutions to the penalized problem for each >0 and (u, θ, q) be the limits from Proposition 2.2. Then (u, θ, q) will satisfy (8).

8

We can substitute z^p in (22) for anyz∈W^1,∞(H₀¹) and obtain h∂_tθ^ε, zi+hdivq^ε, zi+hm_ij∂_tu^ε_i,j, zi= 0

Integrating from 0 to T we arrive at

hθ(T,·)^ε, z(T,·)i − hθ^ε₀, z(0,·i −

T

Z

0

hθ^ε, ∂tzidt−

T

Z

0

hq^ε_i, z,iidt

+hmiju^ε_i,j(T,·), z(T,·)i − hmiju^ε_0i,j, z(0,·)i −

T

Z

0

hmiju^ε_i,j, zidt= 0

(43)

Using Propositions 2.2 and 2.3, we conclude by taking the limit→0 that (u, θ, q) fulfill (11).

Similarly, substitutingy^p in (23) for anyy∈W^1,∞(H¹) and integrating yields

hkijq_i^ε(T,·), y(T,·)i − hkijq0i^ε, y(0,·)i −

T

Z

0

hkijq^ε_i(t,·), ∂tyj(t,·)idt

+

T

Z

0

hk_ijq^ε_i(t,·), y_j(t,·)i+

T

Z

0

hθ^ε(t,·), y_i,i(t,·)idt= 0

(44)

Again, taking the limit→0 and using Propositions 2.2 and 2.3 we conclude that (u, θ, q) fulfill (12).

From Proposition 2.3, it is immediately clear that (u, θ, q) satisfy (9). Using Lemma 1.4 from [4] again, it follows from Proposition 2.2 that

u^ε−→u inC⁰([0, T],(L²(ΓC))ⁿ) therefore, since

1 ε

Z

ΓC

|(u^ε_ν(t,·)−g(·))⁺|²dΓ ≤C,

we conclude that

Z

Γ_C

|(u_ν(t,·)−g(·))⁺|²dΓ = 0

and therefore (13) is satisfied.

Note that we did not useµ >0 yet, therefore everything we proved so far will also hold if µ= 0. The critical part is in fact the convergence of quadratic terms that appear in (10), as we will see in the following calculations.

For any w∈L^∞(H_Γ¹

D)∩W^1,∞(L²) we substitutew_p in (23) byw−uand obtain h∂_t²u^ε, w−u^εi+hCijklu^ε_k,l, w_i,j−u^ε_i,ji

+µh∂tu^ε_i,j, wi,j−u^ε_i,ji+hmijθ^ε, wi,j−u^ε_i,ji

= −1 ε

Z

ΓC

(u^ε_ν−g)⁺(w_ν−u^ε_ν)dΓ−ε Z

ΓC

∂_tu^ε_ν(w_ν−u^ε_ν)dΓ

9

Integrating from 0 to T we arrive at

h∂tu^ε(T,·), w(T,·)i − h∂tu^ε(T,·), u^ε(T,·)i − hu^ε₁, w(0,·)−u^ε₀i

−

T

Z

0

h∂_tu^ε(t,·), ∂_tw(t,·)idt+

T

Z

0

hC_ijklu^ε_k,l(t,·), w_i,j(t,·)idt

+

T

Z

0

h∂tu^ε(t,·), ∂tu^ε(t,·)idt−

T

Z

0

hCijklu^ε_k,l(t,·), u^ε_i,j(t,·)idt

+µ

T

Z

0

h∂_tu^ε_i,j(t,·), w_i,j(t,·)idt−µ

2 ||∇u^ε(T,·)||²− ||∇u^ε₀||²

+

T

Z

0

hmijθ^ε(t,·), wi,j(t,·)idt−

T

Z

0

hmijθ^ε(t,·), u^ε_i,j(t,·)idt

= 1 ε

T

Z

0

Z

Γ_C

(u^ε_ν(t,·)−g)⁺(u^ε_ν(t,·)−g)−(u^ε_ν(t,·)−g)⁺(w_ν(t,·)−g)dΓdt

−ε

T

Z

0

Z

Γ_C

∂tu^ε_ν(t,·)wν(t,·)dΓdt+ε 2

Z

Γ_C

|u^ε_ν(T,·)|²− |u^ε_0ν|²dΓ

≥ − ε





T

Z

0

Z

ΓC

∂_tu^ε_ν(t,·)w_ν(t,·)dΓdt+1 2

Z

ΓC

|u^ε_ν(T,·)|²− |u^ε_0ν|²dΓ





(45)

Using Propositions 2.2 and 2.3, we see that the right hand side of (45) will converge to 0 as→0, since weak-* convergent series are bounded in norm. For the left hand side we can again conlude the convergence of all terms that are linear in (u, θ, q).

However, the convergence of the quadratic terms, namely the L²(0, T)-Norms of

||u_t||,hCijklui,j, uk,li andhmijθ, ui,ji remains an issue. While we know the terms will be bounded, we can not conclude their convergence to the respective terms for u, as weak-* convergence does not imply norm convergence.

Note that it is not possible to circumvent this problem by simply taking estimates for the second order energy and giving a strong solution, since the second order energy is not (trivially) bounded in . We remark that Mun˜oz Rivera and Racke [6] encountered a similar problem, which could be circumvented by reducing the problem to the radially symmetrical case and using an estimate obtained via com- pensated compactness. However, it is not possible to utilize this for our problem, since we do not have a bound on∇θ, which is a necessary component of the proof in [6].

Therefore, we shall useµ >0, which will yield u^ε* uin W^1,2((H_Γ¹_C))

by Proposition 2.2. From this we can conclude the uniform convergence of u_t as well as ∇u. It is then possible to take the limit→ 0 in (45) and conclude that

(u, θ, q) will satisfy (10).

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3. Stability

In general, one can not expect the exponential stability of a thermoelastic problem that is not radially symmetric. Therefore, we shall restrict our problem to the ra- dially symmetric, isotropic and homogenous case, i.e. we assume that the following conditions hold:

The domain Ω is radially symmetric, in this case annular:

Ω =B(0,1)\B(0, r0), 1> r0>0; ΓD=∂B(0, r0); ΓC=∂B(0,1); ΓN =∅ The coefficients satisfy the following symmetry conditions:

C_ijkl=λδ_ijδ_kl+ν(δ_ikδ_jl+δ_jkδ_il)

m_ij =mδ_ij, K_ij =κδ_ij, g(x) =g≥0 f.a. x∈Γ_C

Additionally, we shall assume that the solution to the problem as derived in the previous section is unique in this case, which implies that with radially symmetric initial data and the above assumptions on the coefficients, the solution itself will be radially symmetric. We shall first investigate the stability of the penalized problem, which will transfer to the original problem by a simple continuity argument.

With our assumptions, the equations take the form

∂_t²u^ε−µ∂t4u^ε−λ14u^ε−(λ1+λ2)∇divu^ε+m∇θ^ε= 0 (46)

∂tθ^ε+ divq^ε+mdiv∂tu^ε= 0 (47) τ₀∂_tq^ε+q^ε+κθ^ε= 0 (48) with Lame-Moduliλ₁,λ₂satisfying 2λ₁+nλ₂>0 and constantsκ >0 andm6= 0.

The boundary conditions to the penalized problem then read θ^ε|∂Ω= 0, u^ε|Γ_D = 0 µ∂_t∂u^ε

∂ν ·ν+λ₁∂u^ε

∂ν ·ν+ (λ₁+λ₂)divu^ε=−1

ε(u^ε_ν−g)⁺−ε∂_tu^ε_ν on Γ_C (49) As mentioned above, solutions to this problem will also be radially symmetric, so we can write

u^ε(t, x) =xw(t,|x|), θ^ε(t, x) =ψ(t,|x|), q^ε(t, x) =xh(t,|x|) Writingr:=|x|, (w, ψ, h) will then satisfy the equations

∂_t²w−µ∂twrr−µ∂t

1

rwr−ν1wrr−ν2

r wr+m

rθr= 0 (50)

∂tψ+nh+rhr+mn∂tw+mr∂twr= 0 (51) τ0∂th+h+κ

rψr= 0 (52)

We will now show that the energy of the penalized problem, defined by E^ε(t) :=||∂_tu^ε||²+λ₁||∇u^ε||²+κ||θ^ε||²+||q^ε||²+1

ε Z

ΓC

|(u^ε_ν−g)⁺|²dΓ

11

decays exponentially as time goes to infinity, i.e.

E^ε(t)≤αE₀^εe^−βt

We will use the technique of a Lyapunov functional, constructing the negative terms of the energy and combining the respective functionals in a final estimate. First, one easily sees by multiplying (46) with ∂tu, (47) with κθ and (48) with q and integrating over Ω, that the energy satisfies

d

dtE^ε(t)≤ −C2 (µ||∂tu^ε||²_n+||q^ε||²_n) (53) Proposition 3.1. Let

F1(t) :=h∂tu^ε, u^εin+ε Z

ΓC

|u^ε_ν|²dΓ−µ||∇u^ε||²_n×n then for any δ1>0

d

dtF1(t)≤ −(C3−δ1)||∇u^ε||²_n×n−1 ε

Z

ΓC

|(u^ε_ν−g)⁺|²dΓ +||∂tu^ε||²_n+C4

δ₁||θ^ε||² (54) Proof:

d

dth∂tu^ε, u^εi

=||∂tu^ε||²+h∂_t²u^ε, u^εi

=||∂tu^ε||²−λ1||∇u^ε||²−µh∂t∇u^ε,∇u^εi − hmθ^ε,div u^εi

−(λ1+λ2)||divu^ε||²−1 ε

Z

ΓC

(u^ε_ν−g)⁺uνdΓ−ε Z

ΓC

∂tu^ε_νu^ε_νdΓ

(55)

Estimating

|hmθ^ε,divu^εi| ≤C4

δ1||∇u^ε||²+1 δ1

||θ^ε||²

for any δ1>0 and

−1 ε

Z

ΓC

(u^ε_ν−g)⁺u^ε_νdΓ

≤ −1 ε

Z

Γ_C

|(u^ε_ν−g)⁺|²dΓ

we obtain the desired result.

Proposition 3.2. Let

Ψ(t, r) :=

r

Z

r0

ψ(t, s)ds and

F2(t) :=−τ0 1

Z

r0

Ψ(t, r)h(t, r)dr Then for any δ2, δ3>0

d

dtF2(t)≤ C δ2+δ3

1

Z

r0

|h(t, r)|²dr−κ−δ2

r0 1

Z

r0

|ψ(t, r)|²dr+δ3 1

Z

r0

|∂tw(t, r)|²dr (56)

12

Proof: By (51), Ψ satisfies

∂tΨ +

r

Z

r0

nhds+

r

Z

r0

shds+

r

Z

r0

mn∂twds+

r

Z

r0

ms∂twsds= 0 (57)

Multiplying (57) withhr and integrating, we obtain for anyδ₃>0

−

1

Z

r₀

(∂_tΨ)hrdr=n

1

Z

r₀

hr

r

Z

r₀

hdsdr+

1

Z

r₀

hr

r

Z

r₀

sh_sdsdr

+mn

1

Z

r0

hr

r

Z

r0

∂tw(t, s)dsdr+m

1

Z

r0

h

r

Z

r0

s∂twsdsdr

≤ C δ₃

1

Z

r0

|h(t, r)|²dr+δ3 1

Z

r0

|w(t, r)|²dr

(58)

Multiplying (52) by Ψrand integrating, we obtain

τ₀

1

Z

r₀

Ψ(∂_th)rdr+

1

Z

r₀

Ψhrdr+

1

Z

r₀

κΨ_rrΨdr= 0

We have, by definition of Ψ,

1

Z

r₀

κΨrr(t, r)Ψ(t, r)dr=−κ

1

Z

r₀

|Ψr(t, r)|²dr=−κ

1

Z

r₀

|ψ(t, r)|²dr

Using Poincar´e’s Inequalitiy for Ψ, this implies for any δ₂>0

−

1

Z

r₀

Ψ(∂_th)rdr≤ C δ2

1

Z

r₀

|h|²dr−(κ−δ₂)

1

Z

r₀

|ψ|²dr (59)

Combining (58) and (59), we obtain the desired result.

Defining

L(t) :=N E^ε(t) +F1(t) +δ4F2(t)

whereδ4 will be chosen later, we easily see that for large enoughN there existC1, C₂>0 such that

C1E(t)≤L(t)≤C2E(t) (60)

Now, we can prove the essential theorem of this section.

Theorem 3.1. Let µ >0. Then the system is exponentially stable, i.e. there is a β >0such that

E^ε(t)≤αE₀^εe^−βt

13

Proof: Using (53), (54) and (56), we conclude that d

dtL(t)≤ −N C2(µ||∂tu^ε||²_n+||q^ε||²_n)−(C3−δ1)||∇u^ε||²_n×n

−1 ε

Z

Γ_C

|(u^ε_ν−g)⁺|²dΓ +||∂tu^ε||²_n+C₄ δ1

||θ^ε||²

+δ₄



 C₅ δ2+δ3

1

Z

r₀

|h(t, r)|²dr−κ−δ₂ r0

1

Z

r₀

|ψ|²dr+δ₃

1

Z

r₀

|∂tw|²dr





≤(1 +C6δ3δ4−N C2µ)||∂tu^ε||²_n+ ( δ4C7

δ2+δ3

−N C2)||q^ε||²_n+ (δ1−C3)||∇u^ε||²_n×n

−1 ε

Z

Γ_C

|(u^ε_ν−g)⁺|²dΓ + (C4

δ1

−δ4

κ−δ2

r0

)||θ^ε||²

Choosing δ1 < C3 and δ2 < κ, then δ4 > _δ ^r0C4

1(κ−δ2) and (arbitrarily) δ3 = 1 we conclude that, for sufficiently large N, there is aC >0 such that

d

dtL(t)≤ −CE^ε(t)

Using (60), this proves our theorem.

Note that δ3 is not really needed for the construction of the Lyapunov functional and could have been left as 1. However, we want to point out that the positive u_t term arising from F₂is not a problem; the problem requiring µ >0 is the positive u_tterm arising fromF₁, which can not be made arbitrarily small without losing the negative terms for the derivatives of u. For the classical problem, Mun˜oz-Rivera and Racke [6] showed that this term can be handled by adding additional functions to the Lyapunov functional; however, this gives rise to a positive ∇θterm. While

∇θis given as a negative term from the energy itself in the classical case, this does not hold forτ0 >0; in fact we do not know anything about derivatives ofθ. It is therefore necessary to gain the negativeutterm by other means, one of them being the viscoelastic term. If we define the energy of the original problem as

E(t) :=||∂tu||²+||∇u||²+||θ||²+||q||²

we see by the lower semicontinuity of the norms of weak*-convergent series, using Proposition 2.2, that

lim inf

ε→0 E^ε(t)≥E(t) Using the strong convergence of initial data, we obtain

E(t)≤E^ε(t)≤αE^ε(0) exp(−βt)→αE(0) exp(−βt) This proves our final theorem:

Theorem 3.2. Letµ >0. Then there areα,β >0such that E(t) ≤ αE(0) exp(−βt)

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References

[1] K.A. Ames and L.E. Payne. Uniqueness and continuous dependence of solutions to a multidi- mensional thermoelastic contact problem.J. Elasticity, 34:139–148, 1994.

[2] C.M. Elliot and Q. Tang. A dynamic contact problem in thermoelasticity.Nonlinear Analysis, 23:884–898, 1994.

[3] H. Gao and J. Mu˜noz Rivera. Global existence and decay for the semilinear thermoelastic contact problem.J. Differential Equations, 186:52–68, 2002.

[4] J.U. Kim. A boundary thin obstacle problem for a wave equation.Comm. Partial Differential Equations, 14:1011–1026, 1989.

[5] J. Mu˜noz Rivera and S. Jiang. The thermoelastic and viscoelastic contact of two rods.J. Math.

Anal. Appl., 217:423–458, 1998.

[6] J. Mu˜noz Rivera and R. Racke. Multidimensional contact problems in thermoelasticity.SIAM J. Appl. Math, 58:1307–1337, 1998.

[7] H. Fern´andez Sare and R. Racke. On the stability of damped timoshenko systems - cattaneo versus fourier law.Archive Rational Mech. Anal. (to appear), 2007.

[8] P. Shi and M. Shillor. Existence of a solution to the n dimensional problem of thermoelastic contact.Comm. Partial Differential Equations, 17:1597–1618, 1992.

Jan Sprenger, Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Wien, Austria

E-Mail: jsprenger@asc.tuwien.ac.at

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