Continuous dependence results for an inverse problem in the theory of combustion of materials with memory (Q2764721)

Let \(\Omega \subset {\mathbb R}^n\), \(n=1,2,3\), be a bounded domain with a smooth boundary \(\partial \Omega\). The author is concerned with the continuous dependence on the data of the solution \((u,\rho,h)\) to the following integro-differential identification problem related to the theory of combustion of materials with memory:NEWLINENEWLINEfind three functions \(u:[0,T]\times\Omega \to {\mathbb R}\), \(\rho:[0,T]\times\Omega \to {\mathbb R}\) and \(h:[0,T]\to {\mathbb R}\) satifying the equations NEWLINE\[NEWLINE D_tu(t,x)= \text{div}\,(d_1(x)\nabla u(t,x)) NEWLINE\]NEWLINE NEWLINE\[NEWLINE +\int_0^t h(t-s) \text{ div}\,(d_1(x)\nabla u(s,x))\,ds + f(u(t,x),\rho (t,x)),\quad (t,x)\in [0,T]\times\Omega tag1NEWLINE\]NEWLINE NEWLINE\[NEWLINE D_t\rho (t,x) = \text{ div}\,(d_2(x)\nabla \rho (t,x)) + g(u(t,x),\rho (t,x)),\quad (t,x)\in [0,T]\times\Omega \tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(0,x)=u_0(x),\quad \rho (0,x)=\rho_0(x),\quad x\in \Omega \tag{3}NEWLINE\]NEWLINE NEWLINE\[NEWLINE D_{\nu}u(t,x) = D_{\nu}\rho (t,x)=0, \quad (t,x)\in [0,T]\times\partial\Omega \tag{4}NEWLINE\]NEWLINE NEWLINE\[NEWLINE \int_{\Omega} \varphi(x)u(t,x)\,dx = \ell(t),\quad t\in [0,T] \tag{5}NEWLINE\]NEWLINE NEWLINEHere \(d_1,d_2:{\mathbb R}\times {\mathbb R}\to {\mathbb R}_+\), \(f,g:{\mathbb R}\times {\mathbb R}\to {\mathbb R}\), \(u_0:\Omega \to {\mathbb R}\), \(\rho_0: \Omega \to {\mathbb R}\), \(\ell:[0,T]\to {\mathbb R}\), \(\varphi:\Omega \to \Omega\) are given (smooth) functions, while \(D_{\nu}\) denotes the outward normal derivative on \(\partial\Omega\). NEWLINENEWLINETo prove the continuous dependence the author makes use of an abstract version of problem (1)--(5) related to a general Banach space \(X\) and takes advantage of the analytic theory of semigroups of linear bounded operators. NEWLINENEWLINEThe present paper completes the results obtained in [\textit{F. Colombo} and \textit{A. Lorenzi}, Adv. Differ. Equ. 3, No, 1, 133--154 (1998; Zbl 0958.35145)], where only existence and uniqueness of \((u,\rho,h)\) were proved.NEWLINENEWLINEFinally, in the appendix a deduction is given of the evolution equations governing the combustion of materials with memory.