An inverse problem for an abstract nonlinear parabolic integrodifferential equation
type: Item
instance of: scholarly article
title: An inverse problem for an abstract nonlinear parabolic integrodifferential equation
zbMATH Open document ID: 0786.45014
author: Maurizio Grasselli
author: Alfredo Lorenzi
published in: Differential and Integral Equations
publication date: 1993-06-29T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 74A15
Mathematics Subject Classification ID: 80A20
zbMATH DE Number: 219695
zbMATH Keywords: Banach spaces
zbMATH Keywords: analytic semigroup
zbMATH Keywords: nonlinear Volterra integral equations
zbMATH Keywords: heat conduction
zbMATH Keywords: materials with memory
reviewed by: Eugenio Sinestrari
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 219695