Inverse problem for a nonlinear parabolic equation with nonsmooth periodic coefficients

Abstract

We prove a result of stability for an inverse problem associated to a parabolic nonlinear equation by using a Carleman inequality. To do this, existence, uniqueness, regularity results and maximum principle for our problem are established. This stability inequality concerns the reconstruction of an $L^\infty ({\mathbb R}^n)$ coefficient, basing ourselves by taking an arbitrary open set of observation.

Appendix

Much of the demonstration of Theorem 2 is classic, we will give some indications.

We introduce the following weight functions:

$$\begin{aligned} \varphi (t, x) = \frac{ e^{\lambda \psi (x)}}{t(T-t)},\; \alpha (t, x) = \frac{e^{\lambda \psi } -e^{2\lambda ||\psi ||_{\infty }}}{t(T-t)}, \end{aligned}$$

(5.1)

where $\lambda >0$; remark that $\alpha <0$. We establish a Carleman estimate by taking into account the fact that the weight function $\psi$ is $L$-periodic. Set $\displaystyle w=e^{s\alpha }z.$ We have

$$\begin{aligned} L_1w+L_2w=f_s \; \text{ on }\; Q, \end{aligned}$$

with

$$\begin{aligned} L_1 w=-\Delta w-s^2\lambda ^2\varphi ^2|\nabla \psi |^2w-s\alpha _t w, \end{aligned}$$

(5.2)

$$\begin{aligned} L_2w= w_t+2s\lambda \varphi \nabla \psi .\nabla w+2s\lambda ^2\varphi w |\nabla \psi |^2, \end{aligned}$$

(5.3)

and

$$\begin{aligned} f_s=fe^{2s\alpha }-s\lambda \varphi w\Delta \psi +s\lambda ^2\varphi w |\nabla \psi |^2. \end{aligned}$$

(5.4)

So

$$\begin{aligned} \Vert f_s\Vert ^2_{L^2(Q)}= \Vert L_1w\Vert ^2_{L^2(Q)} + \Vert L_2w\Vert ^2_{L^2(Q)} +2(L_1w,L_2w)_{L_2(Q)}. \end{aligned}$$

(5.5)

We estimate the term $(L_1w,L_2w)_{L^2(Q)}$ which will be written in the following way

$$\begin{aligned} (L_1w,L_2w)_{L^2(Q)}=A_1+A_2+A_3, \end{aligned}$$

(5.6)

where

$$\begin{aligned} A_1&= -\int _Q\Big (\Delta w+s^2\lambda ^2\varphi ^2 |\nabla \psi |^2 w+s \alpha _t w\Big ) \Big ( w_t+2s\lambda ^2\varphi w |\nabla \psi |^2\Big )dx dt, \nonumber \\ A_2&= -\int _Q\Big (s^2\lambda ^2\varphi ^2|\nabla \psi |^2w+s\alpha _t w\Big )\Big (2s\lambda \varphi \nabla \psi .\nabla w\Big )dx dt, \nonumber \\ A_3&= -\int _Q\Big (\Delta w\Big ) \Big (2s\lambda \varphi \nabla \psi .\nabla w\Big )dx dt. \end{aligned}$$

(5.7)

Thanks to $L$-periodicity on $w, \varphi$ and $\psi$ and the fact that $w(0,.)=w(T,.)=0$, integration by parts gives

$$\begin{aligned} A_1&= \int _Q\Big \{\frac{s^2 \lambda ^2}{2}( \varphi ^2|\nabla \psi |^2)_tw^2 -\frac{s}{2}\alpha _{tt}w^2+2s \lambda ^2\varphi |\nabla \psi |^2|\nabla w|^2\\&\quad +\,2s\lambda ^2 w \nabla w.\nabla (\varphi |\nabla \psi |^2) -2s^3\lambda ^4\varphi ^3 w^2|\nabla \psi |^4-2s^2\lambda ^2\varphi \alpha _t w^2|\nabla \psi |^2\Big \}dx dt.\\ A_2&= \int _Q\Big \{3s^3\lambda ^4\varphi ^3|\nabla \psi |^4w^2+s^3\lambda ^3\varphi ^3w^2\nabla . \Big (|\nabla \psi |^2 \nabla \psi \Big ) \nonumber \\&\quad \quad +\,s^2\lambda w^2 \nabla . \Big (\varphi \alpha _t (\nabla \psi )\Big ) \Big \}dx dt.\\ A_3&= \int _Q\Big \{2s\lambda ^2\varphi \Big (\Big (\nabla \psi .\nabla w\Big )^2+2s\lambda \varphi \Big (\nabla w. (\nabla \nabla \psi ).\nabla w\Big ) \Big ) \nonumber \\&\quad \quad -\,s\lambda ^2\varphi |\nabla \psi |^2|\nabla w|^2-s\lambda \varphi |\nabla w|^2\Delta \psi \Big \}dx dt. \end{aligned}$$

we get

$$\begin{aligned} (L_1w,L_2w)_{L^2(Q)}&= \int _Q \left\{ s^3\lambda ^4\varphi ^3|\nabla \psi |^4 w^2+s\lambda ^2\varphi |\nabla \psi |^2|\nabla w|^2\right. \nonumber \\&\quad \quad \left. +2s\lambda ^2\varphi (\nabla \psi .\nabla w)^2\right\} dx dt+X_1, \end{aligned}$$

(5.8)

where $X_1$ is a term which can be absorbed.

Now inserting (5.8) in (5.5), we obtain

$$\begin{aligned} ||f_s||_{L^2(Q)}^2&= ||L_1w||_{L^2(Q)}^2+||L_2w||_{L^2(Q)}^2\nonumber \\&\quad +2\int _Q \{s^3\lambda ^4\varphi ^3|\nabla \psi |^4 w^2+\,s\lambda ^2\varphi |\nabla \psi |^2|\nabla w|^2\nonumber \\&\quad +2s\lambda ^2\varphi (\nabla \psi .\nabla w)^2\} dx dt+2X_1. \end{aligned}$$

(5.9)

Taking into account (5.4), we see that

$$\begin{aligned} ||f_s||_{L^2(Q)}^2\le 3\left( ||fe^{s\alpha }||_{L^2(Q)}^2+||s\lambda \varphi w\Delta \psi ||_{L^2(Q)}^2+ ||s\lambda ^2\varphi w |\nabla \psi |^2||_{L^2(Q)}^2\right) . \end{aligned}$$

(5.10)

From the Proposition 2, there exists $\delta$ such that $0<\delta \le |\nabla \psi |^2, x\in \overline{\Omega \backslash \omega }$. We get

$$\begin{aligned}&||L_1w||_{L^2(Q)}^2+||L_2w||_{L^2(Q)}^2+\int _Q\left( s^3\lambda ^4\varphi ^3w^2+s\lambda ^2\varphi |\nabla w|^2\right) dx dt \nonumber \\&\quad \le C\left( \int _{(0,T)\times \omega }\left( s^3\lambda ^4\varphi ^3w^2+s\lambda ^2\varphi |\nabla w|^2\right) dx dt +||{f}e^{s\alpha }||_{L^2(Q)}^2\right) . \end{aligned}$$

(5.11)

It remains to estimate the terms $w_t$ and $\Delta w$. We use the following relations:

$$\begin{aligned} \Delta w&= -L_1 w+s^2\lambda ^2\varphi ^2|\nabla \psi |^2w+s\alpha _t w, \\ w_t&= L_2w-2s\lambda \varphi \nabla \psi .\nabla w-2s\lambda ^2\varphi w |\nabla \psi |^2. \end{aligned}$$

To obtain the Carleman inequality (1.2) in $z$, we use the fact that $w=e^{s\alpha }z$ in $Q$. $\square$