A nonlinear parameter dependent boundary value problem in one space dimension (Q5952242)

The authors establish continuous dependence on \(\varepsilon\) of the solution of \[ \begin{aligned} &u_t^\varepsilon=u^\varepsilon_{xx}\\ &(a_0 u^\varepsilon_{xx}-b_0 u^\varepsilon_x)|_{x=0}=c_0 \beta_0(u^\varepsilon(t,0),\varepsilon)\\ &(a_1 u^\varepsilon_{xx}-b_1 u^\varepsilon_x)|_{x=1}=c_1 \beta_1(u^\varepsilon(t,1),\varepsilon)\\ & u^\varepsilon(0,\cdot)=u_0\end{aligned} \] assuming that \(\beta_j:{\mathbb R}\times {\mathbb R}^M\to{\mathbb R}\) is differentiable and \(\beta_j(\cdot,\varepsilon)\) is strictly increasing for \(j=0,\;1\), and a set of further hypotheses is satisfied. An application is given to a thermal radiation problem. The work is closely related to papers by \textit{J. A. Goldstein} and \textit{C. Y. Lin} [J. Differ. Equ. 68, 429-443 (1987; Zbl 0626.35048)], and \textit{G. R. Goldstein} [Math. Methods Appl. Sci. 16, 779-798 (1993; Zbl 0796.35084)].