The uniqueness of solution of an inverse problem for a heat conduction quasilinear equation (Q5947807)

The initial-boundary value problem \[ \frac{\partial u(x,t)}{\partial t} = \frac{{\partial}^2 u(x,t)}{\partial x^2} + f(u(x,t)), \quad (x,t) \in Q_T, \tag{1} \] \[ u\mid_{t=0} = \varphi (x), \quad 0\leq x < \infty, \tag{2} \] \[ \frac{\partial u}{\partial x}\mid_{x=0} = \omega(t), \tag{3} \] where \(Q_T = \{0<x< \infty, 0<t<T\}\) and \(T>0\), is considered. The uniqueness and existence results were presented in [\textit{I. K. Glad, N. L. Njort} and \textit{N. G. Ushakov}, Upper bounds for the MISE of kernel density estimators, Preprint Dept. of Statistics, Univ. of Oslo (1998)]. Here, an inverse problem in relation to the problem (1)--(3) is stated as follows. Let the function \(f(\eta)\) be known only in the domain of values \(G(u)\) of the function \(\varphi\), i.e. \[ f(\eta) = \alpha(\eta), \quad \eta \in G(\varphi), \tag{4} \] where \(\alpha(\eta)\) is a given function. Find the function \(f(\eta)\) in the domain of values \(G(u)\), \(G(\varphi)\neq G(u)\) from the following additional information about a solution \(u(x,t)\): \[ u \mid_{x=0}= \psi (t), \quad 0 \leq t \leq T, \tag{5} \] where \(\psi(t)\) is a known function. The problem (1)--(5) has been discussed in several papers, where the uniqueness was proved for \(f(u)\) uniformly differentiable and \(f(\eta)\eta \leq 0\). Now, the uniqueness is stated for a more general class of functions.