An inverse problem for an abstract nonlinear parabolic integrodifferential equation (Q1802863)

The following equation is considered \((*\) is the convolution) \[ u'(t)+(h*u')(t)+A(u(t))+(K*B(u))(t)=f(t),\;t\in]0,T[\text{ a.e.} \tag{1} \] where \(A,B:Y\to X\), \(u,f:]0,T[\to X\) and \(h,K:]0,T[\to\mathbb{R}\). Here \(X\) and \(Y\) are Banach spaces and \(u,h,K\) are unknown. Equation (1) is supplemented by the conditions (2) \(u(0)=u_ 0\); \(Lu(t)=\psi(t)\), \(t\in]0,T[\) a.e.; \(M(u(t))+(K*N(u))(t)=\varphi(t)\), \(t\in]0,T[\) a.e., where \(L\) is a linear functional on \(X\), \(M,N\) are nonlinear continuous functionals on \(Y\), \(u_ 0\in Y\), and \(\psi,\varphi\) are real functions. The main assumption is that the Fréchet derivative of \(A\) in \(u_ 0\) is the generator of an analytic semigroup in \(X\) with domain \(Y\): with additional conditions on the data it is proved that \(u\), \(h\) and \(K\) are uniquely determined if \(T>0\) is small enough. The method of proof is based on a transformation of the problem to a system of three nonlinear Volterra integral equations of the second kind. The problem has applications to the heat conduction theory in materials with memory with unknown (or scarcely known) relaxation or memory functions \(h\) and \(K\). The linear case was treated by analogous methods by the first author [Math. Methods. Appl. Sci. 15, No. 3, 167-186 (1992; Zbl 0753.45010)].