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Consider a positive solution to

$(*)\qquad du_{xx}=u(\alpha-u)$ in $(a,b)$, $u(a)=0=u(b)$,

where $d$ and $\alpha$ are positive constants. Prove $u(x) \leqslant \alpha$ on $[a,b]$.

i) Suppose to the contrary that $u(x_0) > \alpha$ for some $x_0 \in (a,b)$. show there is (a',b') \subseteq (a,b) so that $u(x) > \alpha$ on (a',b') with u(a')=u(b')=\alpha\,.

ii) let w(x) =u(x) - \alpha, x \in [a',b'], and write $(*)$ in terms of $w$.

iii) Multiply the resulting equation by $w$, integrate from a' to b' and find a contradiction to the original solution.

I am having a little bit of trouble with part ii of this question, any help is greatly appreciated. Thank you.

1 Answers 1

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Question ii) is rather easy. Being $\alpha$ a constant its derivative is zero and your equation just becomes

$dw_{xx}=-(w+\alpha)w.$

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    Question iii) seems to make sense only when you multiply by $w'$ and not $w$. In this way you are able to integrate this equation exactly and get the contradiction as the homework asks. The same you can do with equation (*) just multiplying by $u'$.2011-12-13