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.
On applications of maximal regularity to inverse problems for integrodifferential equations of parabolic type / F. Colombo, D. Guidetti, A. Lorenzi - In: Evolution equations / [a cura di] G. Ruiz Goldstein, R. Nagel, S. Romanelli. - New York : Dekker, 2003. - ISBN 0824709756. - pp. 77-89
On applications of maximal regularity to inverse problems for integrodifferential equations of parabolic type
A. LorenziUltimo
2003
Abstract
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.Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.