Stability estimate for an inverse problem of a hyperbolic heat equation from boundary measurement

Abstract

We are concerned with an inverse problem arising in thermal imaging in a bounded domain $\Omega \subset {\mathbb {R}}^n$, $n=2,3$. This inverse problem consists in the determination of the heat exchange coefficient q(x) appearing in the boundary of a hyperbolic heat equation with Robin boundary condition. A double-logarithmic stability estimate is developed.

1 Introduction

The classical model for heat propagation is given by the equations

$$\begin{aligned} u_t + \beta \text {div} q = 0, \end{aligned}$$

(1.1)

where uis the temperature (difference to a fixed constant reference temperature), $\beta$ is a positive physical constant and q is the heat flux vector satisfying the classical the well-known Fourier’s law

$$\begin{aligned} q + k \nabla \theta = 0 \end{aligned}$$

(1.2)

where k is the thermal conductivity depending on the properties of the material.

The model using classic Fourier’s law assumes the flux q to be proportional to the gradient of the temperature q at the same time $\theta$ as in (1.2) leads to the paradox of infinite heat propagation speed. That is, any thermal disturbance at a single point has an instantaneous effect everywhere in the medium. In other words, it is clear that Equation (1.2) together with the energy equation of the heat conduction (1.1) yields the classical heat transport equation (of parabolic type)

$$\begin{aligned} u_{t}(x,t)-\beta k \Delta u(x,t)=0 \end{aligned}$$

(1.3)

that allows an infinite speed for thermal signals.

The model using classic Fourier law exhibits the physical paradox of infinite propagation speed of signals. To eliminate this paradox a generalized thermoelasticity theory has been developed subsequently. The development of this theory was accelerated by the advent of the second sound effects observed experimentally in materials at a very low temperature. In heat transfer problems involving very short time intervals and/or very high heat fluxes, it has been revealed that the inclusion of the second sound effects to the original theory yields results which are realistic and very much different from those obtained with classic Fourier’s law. These models are based on hyperbolic-type equations for temperature and are closely connected with the theories of second sound, which view heat propagation as a wave-like phenomenon. In an idealized solid, for example, the thermal energy can be transported by quantized electronic excitations, which are called free electrons, and by the quanta of lattice vibrations, which are called phonons. These quanta undergo collisions of a dissipative nature, causing a thermal resistance in the medium. A relaxation time is associated with the average communication time between these collisions for the commencement of resistive flow. Although the convergence time for the solutions of the hyperbolic model to that of the parabolic model is small, it may become important when extremely short times are involved. It is under such situations that the assumption of a parabolic heat conduction model may lead to inaccurate modelling of the transient thermal behavior.

First relativistic heat conduction models with second sound were introduced by Cattaneo [6] by proposing a modification of Fourier’s law of heat conduction, i.e. the constitutive relation

$$\begin{aligned} \tau q_t + q + k \nabla u =0, \end{aligned}$$

(1.4)

where $\tau$ the relaxation constant that is, describes the time lag in the response of the heat flux to a gradient in the temperature. Cattaneo’s law is perhaps the most obvious, the most widely accepted and simplest generalization of Fourier’s law that gives rise to a finite speed of propagation of heat. Replacing the Fourier’s law (1.2) by the Cattaneo law (1.4) and combining with the law of balance of energy (1.1), we obtain a purely hyperbolic description of heat conduction (called also telegraph equation),

$$\begin{aligned} \tau u_{tt}+u_{t}-\beta k \Delta u=0 \end{aligned}$$

(1.5)

which predicts a finite signal speed equals to $1/\sqrt{\tau }$. Note that both laws, Fourier and Cattaneo, are dissipative with dissipations strong enough to produce exponential decay of the solutions for their corresponding heat equations.

In paper we investigate an inverse boundary coefficient problem in thermal imaging based on the purely hyperbolic heat equation (1.5). However, from a physical point of view, this equation is physically more realistic than (1.1) one based on Fourier’s law, since it propagates with finite speed. It is worth citing that several papers studying different inverse problems for the parabolic heat equation (1.1) and wave equation (e.g, [1,2,3,4,5, 12, 17, 20,21,22,23,24,25]). The inverse problem under consideration consists in recovering the unknown heat exchange (heat loss) coefficient q(x) appearing in the hyperbolic heat equation with Robin boundary condition. This recovery may be obtained from boundary temperature measurements. Let $\Omega$ be a bounded domain (that is, a non-empty open connected set) of class $C^3$ in ${\mathbb {R}}^{n}$; we denote by $\nu$ the outward unit normal vector to its boundary $\Gamma$. Let $\{\Gamma _0, \Gamma _1\}$ be a partition of $\Gamma$ and let $g : \Gamma _0\times (0,\infty )\rightarrow {\mathbb {R}}$ such that $supp(g)\subset \Gamma _0$ and $q : \Gamma _1\rightarrow {\mathbb {R}}$ be two given functions (q is nonnegative). We consider the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \tau u_{tt}(x,t)+u_{t}(x,t)= \Delta u(x,t),\quad (x,t)\in \Omega \times (0,\infty ),\\ \partial _\nu u(x,t)=g, \quad (x,t)\in \Gamma _0\times (0,\infty ),\\ \partial _\nu u(x,t)+q(x) u(x,t)=0,\quad (x,t)\in \Gamma _1\times (0,\infty ),\\ u(x,0)=u_0(x),\quad u_{t}(x,0)=u_1(x),\quad x\in \Omega , \end{array}\right. } \end{aligned}$$

(1.6)

In engineering applications, the stationary heat flux g corresponds to an uniform heating of the outer surface. Typically, this is the case when heat or flash lamps are used to provide the input flux g. In this paper, we study separately the two cases: stationary $(g:=g(x))$ and time-dependent heat flux $(g:=h(x,t))$. To be able to get some estimations of the solution, we assume that for a fixed positives constants $R_0$ and $R_1$, we have

$$\begin{aligned} \left\| u_0\right\| _{ L^2(\Omega )}\le & {} R_0, \nonumber \\ \left\| u_1\right\| _{L^2(\Omega )}\le & {} R_1. \end{aligned}$$

(1.7)

Furthermore, in order to avoid some difficulties, we shall assume that

$$\begin{aligned} \overline{{\Gamma _0}} \cap \overline{{\Gamma _1} }= \emptyset ,\qquad q\in C^1(\Gamma _1), \qquad {\Gamma _0}\not = \emptyset \ \text {or}\ q\not =0. \end{aligned}$$

(1.8)

In the case of time-dependent heat flux $(g:=h(x,t))$, we assume that

$$\begin{aligned} \int _{\Gamma _0}h(x,t)u(x,t)d\Gamma <0. \end{aligned}$$

(1.9)

Now, by using the same notations as in [9], we introduce the vector space

$$\begin{aligned} B_{s,r}({\mathbb {R}}^n)=\{y\in S'({\mathbb {R}}^n); {(1+\left| \xi \right| ^2)}^{s/2}{\hat{y}}\in L^r({\mathbb {R}}^n)\}, for\ s \in {\mathbb {R}}\ and\ 1 \le r\le \infty , \end{aligned}$$

equipped with the norm

$$\begin{aligned} \left\| y\right\| _{B_{s,r}({\mathbb {R}}^n)}=\left\| {\left( 1+\left| \xi \right| ^2\right) }^{s/2}{\hat{y}}\right\| _{L^r({\mathbb {R}}^n)}, \end{aligned}$$

where $S'({\mathbb {R}}^n)$ is the space of temperate distributions on ${\mathbb {R}}^n$, ${\hat{y}}$ denotes the Fourier transform of the function y and $B_{s,r}({\mathbb {R}}^n)$ is a Besov space.

Thereafter, we shall need that the solution to the problem (1.6) has some smoothness. In order to give sufficient conditions on data guaranteeing this smoothness, we may define the following sets of boundary coefficients:

$$\begin{aligned} D= & {} \{q\in B_{n-1/2,1}(\Gamma ); q\ge 0, q\ne 0 \}\\ D_M= & {} \{q\in D; \left\| q\right\| _{B_{n-1/2,1}}\le M\},\\ and D_M^0= & {} \{q\in D_M, supp(q)\subset \Gamma _1\}, \end{aligned}$$

where $M > 0$ is a given constant. We recall that the function q(x) (heat exchange coefficient) in (1.6) is known as the Robin coefficient with a support in $\Gamma _1$. So, the introduction of the space $D_M^0$ is suitable and it will be useful in the rest of the paper.

Now, we introduce $\gamma \times (0,\infty )$ as a subset of the accessible sub-boundary $\Gamma _0\times (0,\infty )$. We assume that $\gamma$ does not meet supp(g) ($g:=g(x)$ or $g:=h(x,t)$) and the following condition holds true:

$$\begin{aligned} \gamma \times (0,\infty )\subset \left( \Gamma _0\setminus supp(g)\right) \times (0,\infty ). \end{aligned}$$

(1.10)

The inverse problem associated to the problem (1.6) can be formulated as follows: Inverse problem. Determine q, supported on $\Gamma _1$, from the boundary measurements

$$\begin{aligned} u_{q|\gamma \times (0,\infty )}(x,t)=\psi ,(x,t)\in \gamma \times (0,\infty ), \end{aligned}$$

(1.11)

where $\Gamma _0$ is assumed to be a priori known and $u_q$ is the solution to the problem (1.6) with the coefficient q. In this paper, the error $q-{\tilde{q}}$ is obtained in the whole $\Gamma _1$ by a double-logarithmic stability. Following [10] and [9], we modulate the problem of detecting corrosion damage by electric measurements. For this end, we consider the following boundary value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta v(x)=0,\quad x\in \Omega ,\\ \partial _\nu v(x)=g(x),\quad x\in \Gamma _0,\\ \partial _\nu v(x)+q(x) v(x)=0,\quad x\in \Gamma _1. \end{array}\right. } \end{aligned}$$

(1.12)

Using Theorem 2.3 of [11] and the fact that $B_{n-1/2,1}(\Gamma )$ is continuously embedded in $B_{n-3/2,1}(\Gamma )$, we obtain that (for any $q\in D$) the problem (1.12) has a unique solution $v_q\in H^n(\Omega )$ $(n=2,3)$. Moreover, we have

$$\begin{aligned} \left\| v_q\right\| _{H^{n}(\Omega )}\le R_1,\quad \text{ for } \text{ all } q\in D_M^0, \end{aligned}$$

(1.13)

where $R_1=R_1(\Omega ,g,M)$ denotes a positive constant. In the next of this paper, we need the following Theorem in [ [8], Theorem 2] and Lemma in [ [7], Proposition 4.1].

Theorem 1.1

Let $\Omega \subset {\mathbb {R}}^n$ be a bounded domain with a ${C}^{1,1}$ (Hölder space) boundary $\partial \Omega$. Let $\gamma _0$ be an open domain of $\partial \Omega$ and P an elliptic operator. For all $\tau \in (0,1)$, there exist two positive constants c and $\epsilon _0$ such that, for all $\epsilon \in ]0,\epsilon _0[$ and for all $v \in H^2(\Omega )$, we have:

$$\begin{aligned} \left\| v\right\| _{H^1(\Omega )}\le e^{c/\epsilon }\left( \left\| Pv\right\| _{L^2(\Omega )}+\left\| v\right\| _{H^1(\gamma _0)}+\left\| \partial _\nu v \right\| _{L^2(\gamma _0)}\right) +\epsilon ^\tau \left\| v \right\| _{H^2(\Omega )}. \end{aligned}$$

Lemma 1.1

Let $0<\alpha <1$. Then, there exist three positive constants $\delta ^{*}$, $s^*:=\left| \ln \delta ^{*}\right|$ and $c=c(\Omega ,\Gamma )$ such that for any $0<\delta <\delta ^{*}$, $s>s^*$ and for a given $f\in {C}^\alpha (\Gamma )$ (Hölder space) we have:

$$\begin{aligned} \left\| f\right\| _{L^\infty (\Gamma )}\le \frac{c}{s^\alpha }+{e^s}\left\| fv_q\right\| _{L^\infty (\Gamma )}, \end{aligned}$$

(1.14)

where $v_q$ is the solution to (1.12).

In the following we distinguish the two cases: stationary $(g:=g(x))$ and time-dependent heat flux $(g:=h(x,t))$.

2 Stability of the determination of Robin coefficient in the stationary case

In this section, we establish a double logarithmic stability estimate for the determination of a boundary coefficient appearing in the problem (1.6) in the stationary case $(g:=g(x))$.

Let us denote by $u_q$ the solution to the problem (1.6) which can be decomposed into the sum

$$\begin{aligned} u_q= v_q+w_q, \end{aligned}$$

(2.1)

where $v_q$ is the solution to (1.12) and $w_q$ is the solution to the following problem

$$\begin{aligned}&\tau w_{tt}(x,t)+w_{t}(x,t)= \Delta w(x,t), \ \ (x,t)\in \Omega \times (0,\infty ), \end{aligned}$$

(2.2)

$$\begin{aligned}&\partial _\nu w(x,t)=0,\ (x,t)\in \Gamma _0\times (0,\infty ), \end{aligned}$$

(2.3)

$$\begin{aligned}&\partial _\nu w(x,t)+q(x)w(x,t)=0,\ (x,t)\in \Gamma _1\times (0,\infty ), \end{aligned}$$

(2.4)

$$\begin{aligned}&w(x,0)=u_{0}(x)-v(x), \ u_t(x,0)=u_1(x), \ \ x\in \Omega . \end{aligned}$$

(2.5)

2.1 Well-posedness and exponential stability of the problem (2.2)-(2.5)

We will begin by proving existence, uniqueness, and exponential stability results for the problem (2.2)-(2.5). We first define our solution spaces:

$$\begin{aligned} V=H^1_{\Gamma _0}( \Omega ) = \big \{ w\in H^1(\Omega )\ |\ \partial _\nu w=0\ \text {on} \ \Gamma _0\big \} \end{aligned}$$

with the norm

$$\begin{aligned} \Vert v\Vert ^2_V:=\int _{\Omega }|\nabla w|^2dx+\int _{\Gamma _1}qw^2d\Gamma , \end{aligned}$$

By hypothesis (1.8) the last expression defines a norm on V, which is equivalent to the norm induced by $H^1(\Omega )$; consequently, $V=H^1_{\Gamma _0}( \Omega )$ is a Hilbert space. Note that for function $w\in H^1_{\Gamma _0}( \Omega )$ the Poincaré’s inequality remains valid, that is

$$\begin{aligned} \Vert w\Vert \le C_p\Vert \nabla \,w\Vert , \end{aligned}$$

(2.6)

where $C_p > 0$ is the Poincaré’s constant. We define the following Hilbert space

$$\begin{aligned} {\mathscr {H}} = V \times L^2(\Omega )\equiv H^1_{\Gamma _0}( \Omega )\times L^2( \Omega ), \end{aligned}$$

endowed with the inner product

$$\begin{aligned} <W,V>_{\mathscr {H}}=\int _\Omega (\tau w v^*+\nabla v^*\nabla w)dx+\int _{\Gamma _1}w\,qv^* d\Gamma . \end{aligned}$$

(2.7)

In order to find a reasonable definition of the (weak) solution to the problem (2.2)-(2.5) we multiply the equation (2.2) by $v\in V$ and we integrate by parts. Using the boundary conditions (2.3), (2.4) we obtain that

$$\begin{aligned} \begin{aligned} \int _{\Omega } (\tau w_{tt} - \Delta w +w_t)v dx&= \int _{\Omega }(\tau w_{tt}v+ \nabla w \nabla v +w_tv)dx-\int _{\Gamma }v\, \partial _\nu wd\Gamma \\&=\int _{\Omega }(\tau w_{tt}v +\nabla w \nabla v +w_tv)dx+\int _{\Gamma _1}v\, qwd\Gamma \\&=0. \end{aligned} \end{aligned}$$

(2.8)

Introducing the new variable $v = w_t$, setting $W (t) = (w(t ), v(t ))^T$, the problem (2.2)-(2.5) can be written as a linear evolution equation in ${\mathscr {H}}$ of the form

$$\begin{aligned} \frac{d W(t)}{dt} = {\mathscr {A}}W(t),\qquad W(0) = W_0 \end{aligned}$$

(2.9)

where $W(0) = ( w_0, v_0)^{T} \in {\mathscr {H}}$ and ${\mathscr {A}}:{\mathscr {D}}({\mathscr {A}})\subset {\mathscr {H}}\rightarrow {\mathscr {H}}$ is the linear operator defined by

$$\begin{aligned} {\mathscr {A}} \left( { \begin{array}{*{20}c} w \\ v&{} \\ \end{array}}\right) = \left( {\begin{array}{*{20}c} v \\ \frac{1}{\tau }(\Delta w-v)&{}\\ \end{array}}\right) \end{aligned}$$

(2.10)

with domain

$$\begin{aligned} {\mathscr {D}}({\mathscr {A}})=\left\{ W\in V \times V\ :\ \Delta w-v\in {\mathscr {H}}\right\} . \end{aligned}$$

Note that $\mathscr {D(A)}$ is densely defined in ${\mathscr {H}}$. Now, we use the theory of semigroups of linear operators to prove the well-posedness of problem (2.2)-(2.5).

Lemma 2.1

Assume (1.8). The operator ${\mathscr {A}}$ generates a semigroup of contractions in ${\mathscr {H}}$.

Proof

Obviously $\mathscr {D(A)}$ is dense in ${\mathscr {H}}$ and the maximality of ${\mathscr {A}}$ is easily checked. We show now that ${\mathscr {A}}$ is dissipative. For any $W (t) = (w(t ), v(t ))^T$ be in $\mathscr {D(A)}$. From (2.10) and the inner product defined on ${\mathscr {H}}$, it is quite easy to check that

$$\begin{aligned} <{{\mathscr {A}}}W,W>_{\mathscr {H}}= & {} \int _{\Omega } ( \Delta w -v)v dx+ \int _{\Omega } \nabla v \nabla w dx+\int _{\Gamma } q w\,vd\Gamma \\= & {} -\int _{\Omega }|v|^2dx\le 0. \end{aligned}$$

Then ${\mathscr {A}}$ is dissipative. To show that ${\mathscr {A}}$ is maximal we need to prove that $0\in \rho ({\mathscr {A}}).$ It suffices to find $W={(w, v)}^T\in \mathscr {D(A)}$ such that ${\mathscr {A}}{W}=F$ for any $F={(f_{1},f_{2})}^T\in {\mathscr {H}}.$ In terms of the components, we find

$$\begin{aligned} v= & {} f_{1}\in V, \end{aligned}$$

(2.11)

$$\begin{aligned} \Delta w-v= & {} \tau f_{2}\in L^{2}(\Omega ). \end{aligned}$$

(2.12)

By (2.11), we have $v\in V.$ We conclude that there exists a unique function w satisfying (2.12) such that $w\in {\mathscr {H}}\cap H^{2}(\Omega ).$ It is also clear from the regularity theory of elliptic systems that $\Vert W\Vert _{{\mathscr {H}}}\le C \Vert F\Vert _{{\mathscr {H}}},$ for a positive C,  and we conclude that 0 belongs to the resolvent of ${\mathscr {A}}.$

Then we conclude that operator ${\mathscr {A}}$ is m-dissipative in ${\mathscr {H}}$. Since $\mathscr {D(A)}$ is densely defined in ${\mathscr {H}}$, the lemma follows from the well-known Lumer-Phillips Theorem (e.g. see Pazy, [18]).

Now an application of the theory of semigroups (see Pazy, [18]) gives $\square$

Theorem 2.1

Assume (1.8). Given $(w_0, w_1) \in {\mathscr {H}}$ arbitrarily, the problem (2.2)-(2.5) has a unique solution satisfying

$$\begin{aligned} w \in C({\mathbb {R}}^+;H^1_{\Gamma _0}( \Omega )) \cap C({\mathbb {R}}^+;L^2(\Omega )). \end{aligned}$$

(2.13)

The energy $E : {\mathbb {R}}^+\rightarrow {\mathbb {R}}^+$ of the solution is non-increasing.

Recall the definition of the energy $E : {\mathbb {R}}^+\rightarrow {\mathbb {R}}^+$ of the solution, defined by

$$\begin{aligned} E = E(w) := \frac{1}{2}\int _{\Omega }(\tau |w_t|^2+|\nabla w|^2)dx+\frac{1}{2}\int _{\Gamma _1}qw^2d\Gamma . \end{aligned}$$

(2.14)

The following result shows in particular that the energy is non-increasing.

Lemma 2.2

Given $(w_0, w_1) \in \mathscr {D(A)}$ arbitrarily, the solution to problem (2.2)-(2.5) satisfies the energy equality

$$\begin{aligned} E(S) - E(T) = \int _S^T\int _{\Omega }|w_t|^2dxdt,\quad 0 \le S< T < \infty . \end{aligned}$$

(2.15)

Indeed, $E(S) \ge E(T)$.

Proof

We multiply equation (2.2) by $w_t$ and we integrate by parts in $\Omega \times (S, T)$. Using (2.3)-(2.4) we obtain that

$$\begin{aligned} \begin{aligned} \int _S^T\int _{\Omega } (\tau w_{tt} - \Delta w +w_t)w_t dx dt&=\frac{1}{2}\Big [\int _{\Omega }(\tau w_{t}^2+ |\nabla w |^2)dx\Big ]^T_S+\int _S^T\int _{\Omega }|v|^2dxdt-\int _S^T\int _{\Gamma }w_t \partial _\nu d\Gamma dt \\&= \frac{1}{2}\Big [\int _{\Omega }(\tau w_{t}^2+ |\nabla w |^2)dx+\int _{\Gamma _1}qw^2d\Gamma dt \Big ]^T_S+\int _S^T\int _{\Omega }|v|^2dxdt \\&=0. \end{aligned} \end{aligned}$$

and (2.15) follows from the definition of the energy. $\square$

Now we shall prove the exponential stabilization of the problem (2.2)-(2.5) which is based on the following inequality used in Haraux [13] and Lagnese [14].

Theorem 2.2

Let $E: {\mathbb {R}}_+\rightarrow {\mathbb {R}}^+$ be a non-increasing function and assume that there exists a constant $T > 0$ such that

$$\begin{aligned} \displaystyle \int _t^{\infty }E(s)ds\le TE(t),\quad \forall t\in {\mathbb {R}}_+. \end{aligned}$$

Then,

$$\begin{aligned} E(t)\le E(0)e^{^{1-\frac{t}{T}}},\quad \forall t\ge T. \end{aligned}$$

Now, we can cite the following result:

Theorem 2.3

Assume (1.8). Given $(w_0, w_1) \in \mathscr {D(A)}$ arbitrarily, then there exist two positive constant $C_1$ and $\eta$ such that the energy of the solution to problem (2.2)-(2.5) satisfies

$$\begin{aligned} \frac{1}{2}\Vert W(t)\Vert ^2_{\mathscr {H}}= {E}(t)\le C_1{E}(0)e^{-\eta t}\quad \forall t\ge 0. \end{aligned}$$

(2.16)

Proof

Replace v into (2.8) by w and integrate over $[S, T] \times \Omega$, we obtain

$$\begin{aligned} \begin{aligned} \int _S^T \int _{\Omega } (\tau u_{tt} - \Delta w +w_t)u dx dt&= \int _S^T\int _{\Omega }(\tau w_{tt}u+ \nabla w \nabla w +w_tw)dxdt+\int _S^T\int _{\Gamma _1} qw\,w d\Gamma dt\\&=\tau \Big [\int _{\Omega } w_{t}udx\Big ]^T_S-\tau \int _S^T\int _{\Omega } w_t^2dxdt+\int _S^T\int _{\Omega }|\nabla w|^2 dxdt \\&\quad +\frac{1}{2}\Big [\int _{\Omega } w^2dx\Big ]^T_S+\int _S^T\int _{\Gamma _1} qw\,w d\Gamma dt \\&=0. \end{aligned} \end{aligned}$$

(2.17)

Thus, from the definition of E, we deduce

$$\begin{aligned} 2\int _S^T {E}(t)dt=-\tau \Big [\int _{\Omega } w_{t}udx\Big ]^T_S+2\tau \int _S^T\int _{\Omega } w_t^2dxdt-\frac{1}{2}\Big [\int _{\Omega } w^2dx\Big ]^T_S. \end{aligned}$$

(2.18)

We shall estimate the terms of the right member of the above expression. By using the Young’s and Poincaré’s inequalities, we deduce

$$\begin{aligned} \begin{aligned} -\tau \Big [\int _{\Omega } w_{t}udx\Big ]^T_S&\le \tau \int _{\Omega } w_{t}udx\le C(\int _{\Omega } w_{t}^2dx+\int _{\Omega } |\nabla \,w|^2dx)\le C E(t),\quad \forall \,t> 0\\&\le CE(S),\quad \forall \,S>0 \\ -\frac{1}{2}\Big [\int _{\Omega } w^2dx\Big ]^T_S&\le \int _{\Omega } w^{2}dx\le C\int _{\Omega } |\nabla \,w|^2dx\le C E(t),\quad \forall \,t> 0\\&\le CE(S),\quad \forall \,S> 0 \\ 2\tau \int _S^T\int _{\Omega } w_t^2dxdt&= 2\tau (E(S)-E(T))\le CE(S),\quad \forall \,S > 0. \end{aligned} \end{aligned}$$

(2.19)

Substituting (2.19) into (2.18) yields

$$\begin{aligned} 2 \int _S^T E(t)dt \le cE(S),\quad \forall \,S > 0. \end{aligned}$$

Applying Theorem 2.2, hence (2.16) follows for every $(w_0, w_1) \in \mathscr {D(A)}$. $\square$

2.2 Double logarithmic stability estimate

In this subsection, we establish a double logarithmic stability estimate for the inverse problem in thermal imaging described above. Here, we require the heat flux on a part of the accessible boundary $\Gamma _0$ be the same at every time. Moreover, we assume that the boundary function g introduced in the problem (1.6) depends only on the space variable x and $supp(g)\subset \Gamma _0$. Now, let us consider the following assumption:

where $k \ge 0$ is an integer. Note that g is not identically equal to zero.

Now, we assume that we have (1.8), (1.11), ($A_1$) and (1.10). By applying stationary heat flux, we get the following result.

Theorem 2.4

Let the functions $u_q$ and $u_{{\tilde{q}}}$ be solutions to (1.6) with coefficients q and ${\tilde{q}}$, respectively. Let the constants $\alpha \in (0,1)$ and $\sigma \in (0,1)$. Then, there exist two positive constants $A=A(\sigma ,\Omega ,\Gamma _i,\gamma )$ and $B=B(\Omega ,\gamma )$ such that for any $q,{\tilde{q}}\in D_M^0\cap {C}^\alpha (\Gamma )\cap C^1(\Gamma )$, we have:

$$\begin{aligned} \left\| q-{\tilde{q}}\right\| _{L^\infty (\Gamma _1)}\le \frac{A}{\left| \ln \left| \ln \left( B\left\| u_q-u_{{\tilde{q}}}\right\| _{L^\infty ((0,\infty );L^2(\gamma ))}\right) \right| \right| ^\sigma }, \end{aligned}$$

where ${C}^\alpha (\Gamma )$ is the Hölder space, A and B are independent of q and ${\tilde{q}}$.

Proof

Let $\gamma \subset \Gamma _0\subset \Gamma$ be defined as in (1.10). Let $u_q$ (resp. $u_{{\tilde{q}}}$) be the solution to the problem (1.6) with the coefficient q (resp. with the coefficient ${\tilde{q}}$). Analogously, we can define the functions $v_q$ and $v_{{\tilde{q}}}$ . Using relation (2.1), and the continuity of the trace operator, we can find a constant $\lambda >0$ such that for all $t>0$

$$\begin{aligned} \left\| v_q-v_{{\tilde{q}}}\right\| _{L^2(\gamma )}\le & {} \left\| v_q-u_q(\cdot ,t)\right\| _{L^2(\gamma )}+\left\| v_{{\tilde{q}}}-u_{{\tilde{q}}}(\cdot ,t)\right\| _{L^2(\gamma )}+\left\| u_{{q}}(\cdot ,t)-u_{{\tilde{q}}}(\cdot ,t)\right\| _{L^2(\gamma )},\\\le & {} \left\| v_q-u_q(\cdot ,t)\right\| _{L^2(\Gamma )}+\left\| v_{{\tilde{q}}}-u_{{\tilde{q}}}(\cdot ,t)\right\| _{L^2(\Gamma )}+\left\| u_{{q}}(\cdot ,t)-u_{{\tilde{q}}}(\cdot ,t)\right\| _{L^2(\gamma )},\\\le & {} \lambda \left\| v_q-u_q(\cdot ,t)\right\| _{L^2(\Omega )}+\lambda \left\| v_{{\tilde{q}}}-u_{{\tilde{q}}}(\cdot ,t)\right\| _{L^2(\Omega )}+\left\| u_{{q}}(\cdot ,t)-u_{{\tilde{q}}}(\cdot ,t)\right\| _{L^2(\gamma )},\\\le & {} \lambda \left\| w_q(\cdot ,t)\right\| _{L^2(\Omega )}+\lambda \left\| w_{{\tilde{q}}}(\cdot ,t)\right\| _{L^2(\Omega )}+\left\| u_{{q}}-u_{{\tilde{q}}}\right\| _{L^\infty ((0,\infty );L^2(\gamma ))}, \end{aligned}$$

By Theorem2.3, we get

$$\begin{aligned} \left\| w_q(\cdot ,t)\right\| _{L^2(\Omega )}\rightarrow 0, \left\| w_{{\tilde{q}}}(\cdot ,t)\right\| _{L^2(\Omega )}\rightarrow 0, ast\rightarrow +\infty . \end{aligned}$$

Then, we have

$$\begin{aligned} \left\| v_q-v_{{\tilde{q}}}\right\| _{L^2(\gamma )}\le \left\| u_{{q}}-u_{{\tilde{q}}}\right\| _{L^\infty ((0,\infty );L^2(\gamma ))}. \end{aligned}$$

(2.20)

Let us define ${\tilde{v}}=v_q-v_{{\tilde{q}}}$ satisfying $\Delta {\tilde{v}} =0$. Then, we use the interpolation inequality (e.g. [16]), to obtain

$$\begin{aligned} \left\| {\tilde{v}} \right\| _{H^1(\gamma )}\le & {} C_1\left\| {\tilde{v}} \right\| ^{2/3}_{H^2(\gamma )} \left\| {\tilde{v}} \right\| ^{1/3}_{L^{2}(\gamma )}. \end{aligned}$$

By relations (1.13) and (2.20), we obtain

$$\begin{aligned} \left\| {\tilde{v}} \right\| _{H^1(\gamma )}\le C_2\left\| u_{{q}}-u_{{\tilde{q}}}\right\| ^{1/3}_{L^\infty ((0,\infty );L^2(\gamma ))}. \end{aligned}$$

(2.21)

Consequently, we have $\partial _\nu {\tilde{v}} =\partial _\nu v_q-\partial _\nu v_{{\tilde{q}}}=0$ on $\gamma$. Using the inequality (2.21) and Theorem1.1, we get

$$\begin{aligned} \left\| {\tilde{v}} \right\| _{H^1(\Omega )}\le C_6e^{c/\epsilon }\left\| u_{{q}}-u_{{\tilde{q}}}\right\| ^{1/3}_{L^\infty ((0,\infty );L^2(\gamma ))}+\epsilon ^\tau \left\| {\tilde{v}} \right\| _{H^2(\Omega )}. \end{aligned}$$

(2.22)

Minimizing (2.22) on $\epsilon$, we obtain the existence of three positive constants $A_1$, $B_1$ and $\tau$ satisfying

$$\begin{aligned} \left\| {\tilde{v}} \right\| _{H^{1}(\Omega )} \le \frac{A_1}{\left| \ln \left( B_1{\left\| u_{{q}}-u_{{\tilde{q}}}\right\| _{L^\infty ((0,\infty );L^2(\gamma ))}}\right) \right| ^{\tau }}. \end{aligned}$$

(2.23)

From the interpolation inequality (e.g. [16]), by (1.13), we get:

$$\begin{aligned} \left\| \partial _\nu {\tilde{v}} \right\| _{L^2(\Gamma )}\le & {} C_4 \left\| \partial _\nu {\tilde{v}} \right\| ^{1/2}_{H^{-1/2}(\Gamma )}\left\| \partial _\nu {\tilde{v}} \right\| ^{1/2}_{H^{1/2}(\Gamma )},\\\le & {} C_5 \left\| {\tilde{v}} \right\| ^{1/2}_{H^{1}(\Omega )}\left\| {\tilde{v}} \right\| ^{1/2}_{H^{2}(\Omega )},\\\le & {} C_6 \left\| {\tilde{v}} \right\| ^{1/2}_{H^{1}(\Omega )}, \end{aligned}$$

where $C_4,C_5,C_6$ are positive constants. By using trace theorem, we get

$$\begin{aligned} \left\| {\tilde{v}} \right\| _{L^{2}(\Gamma )}+\left\| \partial _\nu {\tilde{v}} \right\| _{L^{2}(\Gamma )}\le C_7\left\| {\tilde{v}} \right\| ^{1/2}_{H^{1}(\Omega )}, \end{aligned}$$

(2.24)

where $C_7$ is a positive constant. By (2.23) and (2.24), we get the existence of two positive constants $A_2$ and $B_2$ such that

$$\begin{aligned} \left\| {\tilde{v}} \right\| _{L^{2}(\Gamma )}+\left\| \partial _\nu {\tilde{v}} \right\| _{L^{2}(\Gamma )}\le \frac{A_2}{\left| \ln \left( B_1{\left\| u_{{q}}-u_{{\tilde{q}}}\right\| _{L^\infty ((0,\infty );L^2(\gamma ))}}\right) \right| ^{\tau /2}}. \end{aligned}$$

(2.25)

Returning to the definition of ${\tilde{v}}=v_q-v_{{\tilde{q}}}$ and the problem (1.12), we get

$$\begin{aligned} \partial _\nu v_q-\partial _\nu v_{{\tilde{q}}}= & {} q v_q-{\tilde{q}}v_{{\tilde{q}}},\\ \partial _\nu {\tilde{v}}= & {} (q-{\tilde{q}})v_q+{\tilde{q}}v_q-{\tilde{q}}v_{{\tilde{q}}},\\ \partial _\nu {\tilde{v}}= & {} (q-{\tilde{q}})v_q+{\tilde{q}}{\tilde{v}}. \end{aligned}$$

Then, we have

$$\begin{aligned} (q-{\tilde{q}})v_q=-\partial _\nu {\tilde{v}} -{\tilde{q}}{\tilde{v}}. \end{aligned}$$

(2.26)

Combining (2.25) and (2.26) we obtain the existence of two positive constants $A_3$ and $B_3$ such that

$$\begin{aligned} \left\| (q-{\tilde{q}})v_q\right\| _{L^2(\Gamma _1)}\le \left\| (q-{\tilde{q}})v_q\right\| _{L^2(\Gamma )}\le \frac{A_3}{\left| \ln \left( B_1{\left\| u_{{q}}-u_{{\tilde{q}}}\right\| _{L^\infty ((0,\infty );L^2(\gamma ))}}\right) \right| ^{\tau /2}}. \end{aligned}$$

(2.27)

By using Lemma1.1 where $f=q-{\tilde{q}}$, relation (2.27) and minimizing (1.14) on s ensure the existence of three positive constants A, B and $\sigma$ satisfying

$$\begin{aligned} \left\| q-{\tilde{q}}\right\| _{L^2(\Gamma _1)}\le \frac{A}{\left| \ln \left| \ln \left( B\left\| u_{q}-u_{{\tilde{q}}}\right\| _{L^\infty ((0,\infty );L^2(\gamma ))}\right) \right| \right| ^\sigma }. \end{aligned}$$

(2.28)

Using the interpolation inequality, there exist two poisitive constants $C_8$ and $\theta <1$ such that:

$$\begin{aligned} \left\| q-{\tilde{q}}\right\| _{L^\infty (\Gamma _1)}\le C_8 \left\| q-{\tilde{q}}\right\| ^{1-\theta }_{L^2(\Gamma _1)} \end{aligned}$$

(2.29)

Using (2.28) and (2.29), we obtain the existence of three positive constants A, B and ${\sigma }$ satisfying

$$\begin{aligned} \left\| q-{\tilde{q}}\right\| _{L^\infty (\Gamma _1)}\le \frac{A}{\left| \ln \left| \ln \left( B\left\| u_{q}-u_{{\tilde{q}}}\right\| _{L^\infty ((0,\infty );L^2(\gamma ))}\right) \right| \right| ^\sigma }. \end{aligned}$$

$\square$

3 Stability of the determination of Robin coefficient in the time-dependent heat flux

In this section, we establish a double logarithmic stability estimate for the determination of a boundary coefficient appearing in the problem (1.6) for the time-dependent heat flux case $(g:=h(x,t))$.

Let us decompose $u_q$ the solution to problem (1.6) into the sum

$$\begin{aligned} u_q= v_q+w_q+u_q^0, \end{aligned}$$

(3.1)

where $v_q$ is the solution to (1.12), $w_q$ is the solution to (2.2)-(2.5) and $u^0_q$ is the solution to the following problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \tau u^0_{tt}(x,t)+u^0_{t}(x,t)= \Delta u^0(x,t),\quad (x,t)\in \Omega \times (0,\infty ),\\ \partial _\nu u^0(x,t)=h(x,t)-g(x), \quad (x,t)\in \Gamma _0\times (0,\infty ),\\ \partial _\nu u^0(x,t)+q(x) u^0(x,t)=0,\quad (x,t)\in \Gamma _1\times (0,\infty ),\\ u^0(x,0)=0,\quad u^0_{t}(x,0)=0,\quad x\in \Omega , \end{array}\right. } \end{aligned}$$

(3.2)

In the following we need the following result.

3.1 Asymptotic stability of the problem (1.6)

We define the following Hilbert space

$$\begin{aligned} {\mathscr {H}} = H^1(\Omega ) \times L^2(\Omega , \end{aligned}$$

endowed with the inner product

$$\begin{aligned} <U,V>_{\mathscr {H}}=\int _\Omega (\tau v v^*+\nabla u\nabla u^*)dx+\int _{\Gamma _0}h(x,t)\, u d\Gamma +\int _{\Gamma _1}u\,qu^* d\Gamma . \end{aligned}$$

(3.3)

By using the same previous argument, it is easy to prove that the operator ${\mathscr {A}}$ generates a semigroup of contractions in ${\mathscr {H}}$ and the problem (1.6) is well posed under the condition (1.9). In the next, we will show an asymptotic behavior result for the unique solution to (1.6) in the state space ${\mathscr {H}}$.

Lemma 3.1

The resolvent operator $(\lambda I-{\mathscr {A}} )^{-1}\ :\ {\mathscr {H}}\rightarrow {\mathscr {H}}$ is compact for any $\lambda > 0$ and hence the canonical embedding $\Im :\mathscr {D(A)}\rightarrow {\mathscr {H}}$ is compact, where is equipped with the graph norm.

Proof

We seek $U=(u, v)\in \mathscr {D(A)}$ such that $( I-{\mathscr {A}} ){U}=F$ for any $F=(f_{1},f_{2})\in {\mathscr {H}}.$ In terms of the components, we find

$$\begin{aligned} v= & {} u-f_{1}\nonumber \\ \Delta u-v= & {} -\tau (v-f_{2})\in L^{2}(\Omega )\nonumber \\ \partial _\nu u(x,t)= & {} h(x,t)\nonumber \\ \partial _\nu u(x,t)= & {} -q(x)u(x,t) \end{aligned}$$

(3.4)

Using Lax-Milgram Theorem (see Brezis (1992)), one can readily show that the above system has a unique solution and thus the operator $(\lambda I-{\mathscr {A}} )^{-1}$ exists and maps ${\mathscr {H}}$ into $\mathscr {D(A)}$. Finally, by using Sobolev embedding, we deduce that $(\lambda I-{\mathscr {A}} )^{-1}$ is compact. The proof follows then from the well-known result of Kato (1976).$\square$

The main result of this subsection is:

Theorem 3.1

Assume (1.8) and (1.9) hold. Given $(u_0, u_1) \in {\mathscr {H}}$ arbitrarily, the solution $U (t) = (u(t ), v(t ))$ to problem (1.6) tends in ${\mathscr {H}}$ to $(\wp ,0)$ as $t\rightarrow \infty$, where

$$\begin{aligned} \wp =\frac{1}{\text {mes}( \Omega )}\int _{\Omega }(u_0+ u_1)dx. \end{aligned}$$

Proof

we shall first recall that the set

$$\begin{aligned} \omega (u_0,u_1) = \{(y,z) \in {\mathscr {H}} \big | \exists t_n \rightarrow \infty ;\ \Vert (u(t_n),u_t(t_n))-(y,z)\Vert _{\mathscr {H}}\rightarrow 0 \text {as}\ n\rightarrow \infty \} \end{aligned}$$

is called the $\omega -$limit set z. Applying LaSalle’s principle [15], we have:

• i) and it is compact set.

• ii) $\omega (u_0,u_1)$ is invariant under the semi-group S(t) of contraction in ${\mathscr {H}}$.

• iii) Let $(u(t), v(t)) = S(t)(u_0, u_1)$ be a solution to (1.6), then $\lim _{t\rightarrow \infty }(u(t), v(t))\in \omega (u_0,u_1)$.

• iv) $\omega (u_0,u_1)\subset \mathscr {D(A)}$.

• v) $t\rightarrow \Vert S(t)\omega \Vert ^2_{\mathscr {H}}$ is a constant function for any $(y,z)\in \omega (u_0,u_1)$.

From (iii), it is sufficient to prove that $\omega (u_0,u_1)$ contains only element of the form$(\wp ,0)$. Let $Y=(y,z)\in \omega (u_0,u_1)$ and $U(t)=(u(t), v(t))$ the solution to (1.6) corresponding to Y, then we have

$$\begin{aligned} \frac{d}{dt}\Vert S(t)Y\Vert ^2_{\mathscr {H}}=0\Rightarrow<\frac{d}{dt}(S(t)Y),S(t)Y>_{\mathscr {H}}=0\Rightarrow <\frac{d}{dt}U(t),U(t)>_{\mathscr {H}}=0. \end{aligned}$$

Therefore

$$\begin{aligned} <{{\mathscr {A}}}U,U>_{\mathscr {H}}=-\int _{\Omega }|v|^2dx=0. \end{aligned}$$

Thus $v=u_t=0$ on $\Omega$, which implies that u is a constant with respect to t. Since $\Delta u=\tau \partial _{tt}u+\partial _t u$, we deduce that $\Delta u=0$. Therefore, by Green’s formula and (1.6), we have

$$\begin{aligned} -\int _\Omega \Delta u\, udx=\int _\Omega |\nabla u|^2dx-\int _{\Gamma _0}h(x,t)\, u d\Gamma +\int _{\Gamma _1}qu^2 d\Gamma =0. \end{aligned}$$

Since the three integrals are positive, then we conclude that u is a constant with respect to x on $\Omega$ and vanishes on $\Gamma _0$ and $\Gamma _1$. Finally, $u=\wp$ on $\Omega$ where $\wp$ is a constant and $u_{|_{\Gamma _0}}=u_{|_{\Gamma _1}}=0$.

Hence the $\omega -$limit set contains only elements of the form $(\wp ,0)$, where $\wp$ is a constant, and we find

$$\begin{aligned} \lim _{t\rightarrow \infty }(u(t), v(t))=(\wp ,0). \end{aligned}$$

(3.5)

To find the expression of $\wp$, we use $\Delta u=\tau \partial _{tt}u+\partial _t u$ and we have:

$$\begin{aligned} \frac{d}{dt}\int _\Omega (\tau u_t+u)dx=\int _\Omega \Delta \, udx=0. \end{aligned}$$

Therefore $\int _\Omega (\tau u_t+u)udx$ is a constant function. Thus,

$$\begin{aligned} \int _\Omega (\tau u_t(x,t)+u(x,t))dx=\int _\Omega (\tau u_1(x)+u_0(x))dx,\quad \forall t>0. \end{aligned}$$

Using (3.5) and passing to the limit, we get

$$\begin{aligned} \wp =\frac{1}{\text {mes}(\Omega )}\int _\Omega (\tau u_1(x)+u_0(x))dx. \end{aligned}$$

$\square$

Remark 1

Furthermore, if the initial data $(u_0,u_1)$ satisfies the additional condition

$$\begin{aligned} \int _\Omega (\tau u_1(x)+u_0(x))dx =0, \end{aligned}$$

(3.6)

then the constant $\wp$ of Theorem 3.1 is zero, i.e., $U(t)=(u(t), v(t))\rightarrow (0,0)$ in ${\mathscr {H}}$ as $t\rightarrow \infty$.

3.2 Double logarithmic stability estimate

Here, we assume that the boundary function g introduced in the problem (1.6) depends on the space variable x and the time variable t $(g:=h(x,t))$. In this subsection, we suppose that we have the following assumptions

($A_2$):

${\tilde{h}}=\chi _{\Gamma _a}h\in C((0,\infty ),H^{3/2}(\partial \Omega ))$, and $\partial _t{\tilde{h}}\in C((0,\infty ),H^{1/2}(\partial \Omega ))$.

Now, we assume that we have (1.8),(1.9), (1.10), ($A_2$) and (1.11). By applying time-dependent heat flux, we get the following result.

Theorem 3.2

Let the functions $u_q$ and $u_{{\tilde{q}}}$ be solutions to (1.6) with coefficients q and ${\tilde{q}}$, respectively. Let the constants $\alpha \in (0,1)$ and $\sigma \in (0,1)$. Then, there exist two positive constants $A=A(\sigma ,\Omega ,\Gamma _i,\gamma )$ and $B=B(\Omega ,\gamma )$ such that for any $q,{\tilde{q}}\in D_M^0\cap {C}^\alpha (\Gamma )\cap C^1(\Gamma )$, we have:

$$\begin{aligned} \left\| q-{\tilde{q}}\right\| _{L^\infty (\Gamma _1)}\le \frac{A}{\left| \ln \left| \ln \left( B\left\| u_q-u_{{\tilde{q}}}\right\| _{L^\infty ((0,\infty );L^2(\gamma ))}\right) \right| \right| ^\sigma }, \end{aligned}$$

where ${C}^\alpha (\Gamma )$ is the Hölder space, A and B are independent of q and ${\tilde{q}}$.

Proof

Let $\gamma \subset \Gamma _0\subset \Gamma$ be defined as in (1.10). Let $u_q$ (resp. $u_{{\tilde{q}}}$) be the solution to the problem (1.6) with the coefficient q (resp. with the coefficient ${\tilde{q}}$). Analogously, we can define the functions $v_q$ and $v_{{\tilde{q}}}$ are solutions to (1.12). Using relation (3.1) and trace theorem, we can find two constant $\lambda >0$ and ${\tilde{\lambda }}>0$ such that for all $t>0$

$$\begin{aligned} \left\| v_q-v_{{\tilde{q}}}\right\| _{L^2(\gamma )}\le & {} \lambda \left\| w_q(\cdot ,t)\right\| _{L^2(\Omega )}+\lambda \left\| w_{{\tilde{q}}}(\cdot ,t)\right\| _{L^2(\Omega )}\nonumber \\&\quad +\left\| u_{{q}}-u_{{\tilde{q}}}\right\| _{L^\infty ((0,\infty );L^2(\gamma ))}\nonumber \\&\quad +{\tilde{\lambda }}\left\| u^0_q(\cdot ,t)\right\| _{L^2(\Omega )}+{\tilde{\lambda }}\left\| u^0_{{\tilde{q}}}(\cdot ,t)\right\| _{L^2(\Omega )}. \end{aligned}$$

where $w_q$ and $w_{{\tilde{q}}}$ are solutions to (2.2)-(2.5), $u^0_q$ and $u^0_{{\tilde{q}}}$ are solutions to (3.2). By Theorem2.3 and letting t tends to infinity, we get

$$\begin{aligned} \left\| w_q(\cdot ,t)\right\| _{L^2(\Omega )}\rightarrow 0, \left\| w_{{\tilde{q}}}(\cdot ,t)\right\| _{L^2(\Omega )}\rightarrow 0. \end{aligned}$$

Then, by remark 1 and letting t tends to infinity, we get

$$\begin{aligned} \left\| u^0_q(\cdot ,t)\right\| _{L^2(\Omega )}\rightarrow 0, \left\| u^0_{{\tilde{q}}}(\cdot ,t)\right\| _{L^2(\Omega )}\rightarrow 0. \end{aligned}$$

One can proceed as done in the proof of Theorem 2.4 to get the announced result. $\square$