Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\), and consider a parabolic problem with nonhomogeneous term: \[ \begin{cases} u_t(x,t)=s\Delta u(x,t)+ q(x)u(x,t)+ \alpha(x,t)f(x), \quad x\in\Omega,\;t\in (0,T)\\ u(x,0)=0, \quad x \in\Omega\\ u(x,t)=0, \quad x\in\partial \Omega,\;t\in(0,T) \end{cases} \tag{1} \] where \(s>0\) is the diffusion parameter and, \(q\) and \(\alpha\) are given functions defined respectively on \(\overline\Omega\) and \(\overline\Omega \times[0,T]\). We discuss the following inverse problem:
(IP) Determine \(u(x,t)\) and \(f(x)\), \(x\in\Omega\), \(t\in(0,T)\) satisfying (1) and \(u(x,T)= h(x)\), \(x\in\Omega\) for a given \(h\).
In this paper, using the analyticity of \(u\) with respect to \(s\), we prove the well-posedness of (IP) in Hölder spaces except for a finite set of \(s\).