Existence and multiplicity of solutions of an equation from pool boiling on wires

We investigate the existence and multiplicity of steady states of the equation \[ \frac {{\partial \theta }}{{\partial t}} - \frac {{{\partial ^2}\theta }}{{\partial {x^2}}} + \sigma q\left ( \theta \right ) - a\lambda \left ( 1 + \alpha \theta \right ) = 0, \qquad 0 < x < 1\] with Dirichlet boundary conditions and initial conditions. This equation was derived and studied by Joly, Kernevez, and Llory [7] and Joly [8] in studying thermal effects from pool boiling, in which wires are heated by the Joule effect and are cooled in a bath of boiling water at constant pressure. They studied the steady-state problem for two kinds of heat flux density $q\left ( \theta \right )$ (corresponding to whether or not the radiation is taken into account) and for $\alpha \ne 0$ or $\alpha = 0$. (A) In the case with radiation and $\alpha = 0$, for given specific function $q\left ( \theta \right )$ and constants $\sigma > 0$, $a > 0$, by numerical methods, they found an $S$-shaped bifurcation diagram and three solutions for some parameter values. We prove this rigorously, for a specific range of parameters of physical interest. Specifically, we show that, for specific values of $\alpha , \sigma$, and $a$, there exist two positive numbers $\underline{lambda} < \bar \lambda$ such that the steady-state problem has at least three solutions for $\underline{\lambda} < \lambda < \bar \lambda$, at least two solutions for $\lambda = \underline{\lambda}$ or $\lambda = \bar \lambda$, and exactly one solution for $0 \le \lambda < \underline{\lambda}$ or $\lambda > \bar \lambda$. Moreover, we give lower and upper bounds for $\underline{\lambda}$ and $\bar \lambda$. (B) In the case without radiation and $\alpha \ne 0$, we show that there exist two positive numbers $\underline{\lambda} < \bar \lambda$ such that the steady-state problem has at least two solutions for $\underline{\lambda} < \lambda < \bar \lambda$, at least one solution for $0 \le \lambda \le \underline{\lambda}$ or $\lambda = \bar \lambda$, exactly one solution for $0 < \lambda < \underline{\lambda}$ A and $\lambda$ small enough, and no solution for $\lambda > \bar \lambda$. Moreover, we give upper and lower bounds for $\underline{\lambda}$ and $\bar \lambda$. Also, we find and correct two mistakes in [7, Proposition 2.6, (i), (ii)].