On applications of maximal regularity to inverse problems for integrodifferential equations of parabolic type

The authors study the nonlinear inverse problem of identifying the couple $(u,m)$ in the integrodifferential equation of parabolic type (1) $D_tu(t,x)=Au(t,x)+\int_0^tm(t-s)Au(s,x)ds+f(t,x)$ in $[0,T]\times\Omega\subsetR^n$ with an initial condition (2) $u(0,t)=u_0(x)$, a boundary condition (3) $Bu(t,x)=g(t,x)$ on $[0,T] \times\partial\Omega$ and an overspecification condition (4) $\Psi[u(t,·)]=g(t)$ for $t\in[0,T]$. Here, $\Psi$ is a given bounded functional, $A$ is a second order strongly elliptic operator and $B$ is a differential operator of order not exceeding one. Using analytic semigroup theory, fixed point arguments and maximal regularity results for abstract Cauchy problems, the authors prove existence and uniqueness theorems for the inverse problem (1)--(4) in properly chosen function spaces. Extensions to more general linear state equations (1), to time- and space-dependent memory kernels $m$ and to special nonlinear state equations (1) arising in population dynamics and in the theory of spread of infections are given.