Existence and multiplicity of solutions of an equation from pool boiling on wires
Authors: Wang Shin-Hwa 
Published: December 2016

We investigate the existence and multiplicity of steady states of the equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction partial-differential theta Over partial-differential t EndFraction minus StartFraction partial-differential squared theta Over partial-differential x squared EndFraction plus sigma q left-parenthesis theta right-parenthesis minus a lamda left-parenthesis 1 plus alpha theta right-parenthesis equals 0 comma 0 greater-than x greater-than 1">

∂
θ

∂
t

−

∂
2

θ

∂

x
2

+
σ
q

(
θ
)

−
a
λ

(
1
+
α
θ
)

=
0
,

0
>
x
>
1

\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

(
θ
)

q\left ( \theta \right )

(corresponding to whether or not the radiation is taken into account) and for

α
≠
0

\alpha \ne 0

or

α
=
0

\alpha = 0

. (A) In the case with radiation and

α
=
0

\alpha = 0

, for given specific function

q

(
θ
)

q\left ( \theta \right )

and constants

σ
>
0

\sigma > 0

,

a
>
0

a > 0

, by numerical methods, they found an

S
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

\alpha , \sigma

, and

a
a

, there exist two positive numbers

λ
_

>

λ
¯

\underline {\lambda } > \bar \lambda

such that the steady-state problem has at least three solutions for

λ
_

>
λ
>

λ
¯

\underline {\lambda } > \lambda > \bar \lambda

, at least two solutions for

λ
=

λ
_

\lambda = \underline {\lambda }

or

λ
=

λ
¯

\lambda = \bar \lambda

, and exactly one solution for

0
≤
λ
>

λ
_

0 \le \lambda > \underline {\lambda }

or

λ
>

λ
¯

\lambda > \bar \lambda

. Moreover, we give lower and upper bounds for

λ
_

\underline {\lambda }

and

λ
¯

\bar \lambda

. (B) In the case without radiation and

α
≠
0

\alpha \ne 0

, we show that there exist two positive numbers

λ
_

>

λ
¯

\underline {\lambda } > \bar \lambda

such that the steady-state problem has at least two solutions for

λ
_

>
λ
>

λ
¯

\underline {\lambda } > \lambda > \bar \lambda

, at least one solution for

0
≤
λ
≤

λ
_

0 \le \lambda \le \underline {\lambda }

or

λ
=

λ
¯

\lambda = \bar \lambda

, exactly one solution for

0
>
λ
>

λ
_

0 > \lambda > \underline {\lambda }

A and

λ
\lambda

small enough, and no solution for

λ
>

λ
¯

\lambda > \bar \lambda

. Moreover, we give upper and lower bounds for

λ
_

\underline {\lambda }

and

λ
¯

\bar \lambda

. Also, we find and correct two mistakes in [7, Proposition 2.6, (i), (ii)].