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Suppose u is a smooth solution of the following problem:

$u_{xxt}+u_{xx}−u^3 =0 $ in $[0,1]×(0,∞)$,

$u(0,t)=u(1,t)=0 $ $\forall t \geq 0$

with initial data $u(x, 0) = x(x − 1)$.Show that $u(x,t)$ uniformly tends to zero as $t → \infty$.

Actually I think I have to use energy method with $E(t) = \int_{0}^{1}(u_x)^{2}(x,t)dt$, but don't know how. Could you give me some hint for it?

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Multiply the equation by $u$ and integrate by parts in space: $$ \int u_xu_{xt}=\frac{1}{2}\frac{d}{dt}\|u\|_{H^1}^2=-\int u_x^2-u^4\leq -\|u\|_{H^1}^2 $$ Consequently, using Sobolev embedding, $$ \|u(t)\|_{L^\infty}\leq c\|u(t)\|_{H^1}\leq \|u_0\|_{H^1}e^{-t} $$ Right?

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Oh.. Integrate by parts.. Thank you! Actually I don't know what the sobolev embedding is, but I can get it with the energy method. – user124697 Feb 13 '14 at 5:42
By Sobolev embedding I mean the inequality $\|u\|_{L^\infty}\leq c\|u\|_{H^1}$ – guacho Feb 13 '14 at 6:30

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