## Lipschitz stability in inverse parabolic problems by the Carleman estimate.(English)Zbl 0992.35110

Summary: We consider a system $$y_t(t,x)=-\mathcal Ay(t,x)+g(t,x)$$ $$(0<t<T$$, $$x\in\Omega$$), $$y(\theta,x)=y_0(x)$$ $$(x\in \Omega)$$ with a suitable boundary condition, where $$\Omega\subset\mathbb{R}^n$$ is a bounded domain, $$-\mathcal A$$ is a uniformly elliptic operator of second order whose coefficients are suitably regular for $$(t,x)$$, $$\theta\in ]0,T[$$ is fixed, and the function $$g(t,x)$$ satisfies $$|g_t(t,x)|\leq C|g(\theta,x)|$$ for $$(t,x)\in [0,T]\times\overline{\Omega}$$. Our inverse problems are determinations of $$g$$ using overdetermining data $$y|_{]0,T[\times\omega}$$ or $$\{y|_{]0,T[\times\Gamma_0}$$, $$\nabla y|_{]0,T[\times\Gamma_0}\}$$, where $$\omega\subset\Omega$$ and $$\Gamma_0\subset\partial\Omega$$. Our main result is the Lipschitz stability in these stability problems. We also consider the determination of $$f=f(x)$$, $$x\in \Omega$$ in the case of $$g(t,x)=f(x)R(t,x)$$ with given $$R$$ satisfying $$R(\theta,\cdot)>0$$ on $$\overline\Omega$$. Finally, we discuss an upper estimation of our overdetermining data by means of $$f$$.

### MSC:

 35R30 Inverse problems for PDEs 35R35 Free boundary problems for PDEs 35K10 Second-order parabolic equations
Full Text: