Identification of a nonlinear parameter in a parabolic equation from a linear equation
The nonlinear parabolic system of the form: \[ \begin{cases} {\partial u \over \partial t} - \nabla \bigl( a(u) \nabla u \bigr) = f(x,t) \\ a(u) {\partial u \over \partial n} = g(x,t) \\ u(x, 0) = u_0 (x) \end{cases} \tag{1} \] is considered. The problem consists in finding the nonlinear parameter \(a(u)\) when an information about the state solution \(u\) of (1) is known. Results on existence are first given. Then, the identification problem and its error estimate is studied. \(\Lambda\) penalized functional is introduced and the identification of the nonlinear parameter is simplified by replacing this parameter with a linear one. Numerical approximation and test results are finally given.
scientific article; zbMATH DE number 863608