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Title Inverse problem and Lipschitz stability for the heat equation with a discontinuous diffusion coefficient

Name Julie Valein
Country France
Email julie.valein@univ-lorraine.fr
Co-Author(s) Emmanuelle Cr\'epeau and Lionel Rosier
Submit Time 2014-02-27 10:49:29
Special Session 75: Differential and difference equations on graphs and their applications
We consider a star-shaped network $\mathcal{R}$ of $n+1$ edges $e_j$, of length $l_j>0$, $j\in\{0,..,n\}$, connected at one vertex that we assume to be the origin $0$ of all the edges. We consider on this plane 1-D network a heat equation with a different diffusion coefficient on each string, given by the following system $$\left\{ \begin{array}{lll} u_{j,t}(x,t)-(c_j(x)u_{j,x}(x,t))_x=g_j(x,t),&\quad\forall j\in\left\{ 0,\cdots,n \right\}, (x,t)\in (0,l_j)\times (0,T),\\ u_j(l_j,t)=h_j(t),&\quad \,\forall j\in\left\{ 0,\cdots,n \right\}, t\in(0,T),\\ u(x,0) = u^0(x), &\quad x\in\mathcal{R}, \end{array}\right.\qquad (1) $$ under the assumptions of continuity and of Kirschoff law at the vertex $0$, given by $$u_j(0,t)=u_i(0,t)=:u(0,t),\quad\forall i,j\in\left\{0,...,n\right\},\,0