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Stability result for two coefficients in a coupled hyperbolic-parabolic system

  • Patricia Gaitan EMAIL logo and Hadjer Ouzzane

Abstract

This work is concerned with the study of the inverse problem of determining two coefficients in a hyperbolic-parabolic system using the following observation data: an interior measurement of only one component and data of two components at a fixed time over the whole spatial domain. A Lipschitz stability result is proved using Carleman estimates.

MSC 2010: 35R30; 35M30; 35Q92

A Appendix

Proof of Lemma 2.

Let us start with the estimation of 2. We have

dE1dt=sΩtφ|Y|2e2sφdx+Ω|Y|tYe2sφdx=sΩtφ|Y|2e2sφdx+Ω(LY-A(x)Y)Ye2sφ𝑑x.

Then, we obtain

dE1dt-sΩtφ|Y|2e2sφdx+12Ωe2sφA(x)(|Y|2)dx=ΩYLYe2sφ𝑑x.

After an integration by parts, we get

(A.1)dE1dt-sΩ(tφ+φA(x))|Y|2e2sφ𝑑x+12Γ+A(x)ν|Y|2e2sφ𝑑σ=ΩYLYe2sφ𝑑x+12ΩA(x)|Y|2e2sφ𝑑x.

Moreover, for all s>0 large enough, from Assumption 1 (iv) we obtain

(A.2)dE1dt+sCΩ|Y|2e2sφ𝑑xΩYLYe2sφ𝑑x.

Using the formula 2abεa2+b2ε with ε=sC, we estimate the right-hand side as follows:

|ΩYLYe2sφ𝑑x|12sCΩ|Y|2e2sφ𝑑x+12sCΩ|LY|2e2sφ𝑑x.

Substituting this estimate into (A.2), we have

dE1dt+sCE1(t)12sCΩ|LY|2e2sφ𝑑x.

On the other hand, for t(T-η,T), using the Gronwall Lemma, we obtain

E1(t)e-sC(t-(T-η))E1(T-η)+esC(T-t-η)2scT-ηtΩe2sφ(τ)|LY(τ)|2𝑑x𝑑τe-sC(t-(T-η))E1(T-η)+12sCT-ηTΩe2sφ(τ)|LY(τ)|2𝑑x𝑑τ.

Integrating this inequality for t between T-η et T, we get

T-ηTE1(t)𝑑tE1(T-η)T-ηTe-sC(t-(T-η))𝑑t+T-ηT12sCT-ηTΩe2sφ(τ)|LY(τ)|2𝑑x𝑑τ𝑑tE1(T-η)TT-ηe-sc(t-(T-η))𝑑t+η2sc0TΩe2sφ|LY|2𝑑x𝑑t.

Finally, we have

(A.3)T-ηTE1(t)𝑑tCsE1(T-η)+Cs0TΩe2sφ|LY|2𝑑x𝑑t.

Now, we want to estimate E1(T-η) by E1(τ) for τ(η,T-η). We use (A.1) and we integrate between τ and T-η to obtain

τT-ηdE1dt𝑑t+12τT-ηΓ+A(x)ν|Y|2e2sφ𝑑σ𝑑t=τT-ηsΩ(tφ+φA(x))|Y|2e2sφ𝑑x𝑑t+12ΩA(x)|Y|2e2sφ𝑑x𝑑t+τT-ηΩYLYe2sφ𝑑x𝑑t.

Thus we have

τT-ηdE1dt𝑑tCsτT-ηΩ|Y|2e2sφ𝑑x𝑑t+CsτT-ηΩ|LY|2e2sφ𝑑x𝑑t,

which gives

E1(T-η)-E1(τ)CsηT-ηΩ|Y|2e2sφ𝑑x𝑑t+Cs0TΩ|LY|2e2sφ𝑑x𝑑t.

Integrating between η and T-η, we obtain, for s>0 sufficiently large,

(A.4)E1(T-η)CsηT-ηE1(t)𝑑t+Cs0TΩ|LY|2e2sφ𝑑x𝑑t.

Finally, thanks to (A.4) and (A.3), we obtain

T-ηTE1(t)𝑑tCηT-ηE1(t)𝑑t+Cs0TΩe2sφ|LY|2𝑑x𝑑t.

That is

2CηT-ηΩ|Y|2𝑑x𝑑t+Cs0TΩe2sφ|LY|2𝑑x𝑑t.

Next, we will estimate 4. We will need the following auxiliary lemma (we refer to [3, 10, 26] for the proof):

Lemma 1.

Let φC2(Ω) such that 1>|φ|δ>0. There exist s0>0 and C>0 such that, for all ss0 and all ZH01(Ω),

s2Ωe2sφ|Z|2𝑑xCΩe2sφ|Z|2𝑑x.

We have

dE2dt=sΩtφ|Z|2e2sφdx+Ω|Z|tZe2sφdx=sΩtφ|Z|2e2sφdx+Ω(PZ+ΔZ)Ze2sφ𝑑x.

Then

dE2dt-sΩtφ|Z|2e2sφdx-ΩZΔZe2sφ𝑑x=ΩZPZe2sφ𝑑x.

An integration by parts leads to

dE2dt-sΩtφ|Z|2e2sφdx+Ω|Z|2e2sφ𝑑x+2sΩe2sφZZφdx=ΩZPZe2sφ𝑑x.

On the other hand, we have

2sΩe2sφZZφdx=sΩe2sφφ|Z|2dx=-sΩe2sφ(2s|φ|2+Δφ)|Z|2𝑑x.

Thus

dE2dt-Cs2Ω|φ|2|Z|2e2sφ𝑑x+Ω|Z|2e2sφ𝑑xCsΩ(|tφ|+|Δφ|)|Z|2e2sφ𝑑x+ΩZPZe2sφ𝑑x.

Applying Lemma 1 to the last inequality, we obtain

dE2dt-Cs2Ω|φ|2e2sφ|Z|2𝑑x+Cs2Ω|Z|2e2sφ𝑑xΩZPZe2sφ𝑑x+CsΩ|Z|2e2sφ𝑑x.

For s large enough, the last term of the right hand side is absorbed by the last term of the left hand side, so we have

dE2dt-Cs2Ω|φ|2e2sφ|Z|2𝑑x+Cs2Ω|Z|2e2sφ𝑑xΩZPZe2sφ𝑑x.

This leads, for s large, to the following inequality:

dE2dt+Cs2Ω(1-|φ|2))e2sφ|Z|2dxΩZPZe2sφdx.

According to the assumption of Lemma 2 we obtain

dE2dt+CsΩe2sφ|Z|2𝑑xΩZPZe2sφ𝑑x.

In the same way as for E1, we obtain

4CηT-ηΩ|Z|2e2sφ𝑑x𝑑t+Cs0TΩ|PZ|2e2sφ𝑑x𝑑t.

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Received: 2015-2-3
Revised: 2016-1-27
Accepted: 2016-3-30
Published Online: 2016-5-27
Published in Print: 2017-6-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

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