Abstract
We investigate the existence and multiplicity of steady states of the equation $\\\\\\\\\\\\\\\\frac{{\\\\\\\\\\\\\\\\partial \\\\\\\\\\\\\\\\theta }}{{\\\\\\\\\\\\\\\\partial t}} - \\\\\\\\\\\\\\\\frac{{{\\\\\\\\\\\\\\\\partial ^2}\\\\\\\\\\\\\\\\theta }}{{\\\\\\\\\\\\\\\\partial {x^2}}} + \\\\\\\\\\\\\\\\sigma \\\\\\\\\\\\\\\\lambda \\\\\\\\\\\\\\\\left( {1 + \\\\\\\\\\\\\\\\alpha \\\\\\\\\\\\\\\\theta } \\\\\\\\\\\\\\\\right) = 0,0 < x < 1,$ 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(θ) (corresponding to whether or not the radiation is taken into account) and for α ≠ 0 or α = 0. (A) In the case with radiation and α = 0, for given specific function q(θ) and constants σ > 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 α, σ, and a, there exist two positive numbers λ̲ < λ̅ such that the steady-state problem has at least three solutions for λ̲ < λ < λ̅, at least two solutions for λ = λ̅ or λ = λ̅, and exactly one solution for 0 ≤ λ < λ̲ or λ > λ̅. Moreover, we give lower and upper bounds for λ̲ and λ̅. (B) In the case without radiation and α ≠ 0, we show that there exist two positive numbers λ̲ < λ̅ such that the steady-state problem has at least two solutions for λ̲ < λ < λ̅, at least one solution for 0 ≤ λ ≤ λ̲ or λ = λ̅, exactly one solution for 0 < λ < λ̲ and λ small enough, and no solution for λ > λ̅. Moreover, we give upper and lower bounds for λ̲ and λ̅. Also, we find and correct two mistakes in [7, Proposition 2.6, (i), (ii)].