An inverse problem in the theory of combustion of materials with memory
The authors study the following inverse problem in a Banach space \(X\): Find the functions \(h,u\) and \(\rho\) from the relations \[ u'(t)=Au(t)+\int_0^t h(t-s)Au(s) ds+f(u(t),\rho(t)),\quad t\in [0,T], \] \[ \rho'(t)=B\rho(t)+g(u(t),\rho(t)),\quad t\in[0,T], \] \[ u(0)=u_0,\quad \rho(0)=\rho_0, \qquad \Phi(u(t))=\ell(t), \quad t\in[0,T], \] where \(A\) and \(B\) are linear closed operators with dense domains, and \(\Phi\) is a linear bounded functional on \(X\). Using a fixed point theorem the authors prove the local (with respect to \(t\)) solvability of the inverse problem. Then, this result is applied to prove local solvability of the inverse problem with integral overdetermination for a system of combustion equations having a convolution term which describes the thermal memory of the material.
scientific article; zbMATH DE number 1449011