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Let $u(x,t)$ solves $u_{tt}-u_{xx}=0$ with initial conditions $u(x,0)=\phi(x)$ and $u_t(x,0)=0$. Here $\phi(x)$ is a smooth function that vanishes outside a bounded interval, say $[a,b]$. Show that $$ \lim_{t \rightarrow \infty} \int\limits_{\Bbb R} u_x^2 dx\quad\text{ and }\quad\lim_{t \rightarrow \infty} \int\limits_{\Bbb R} u_t^2 dx $$ both exist and they are equal. Represent their values in terms of $\phi$.

I don't have a constructive idea in mind. Would appreciate if someone can direct me to specific topics or give me a general idea. Also should I involve epsilon-delta language?

Added: bounty is awarded to the answer to this problem.

Ѕᴀᴀᴅ
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Beacon
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2 Answers2

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Use the solution to the IVP $$ u(x,t) = \frac{1}{2}(\phi(x+t) + \phi(x-t)). $$ Hint: the limit is a definite integral with integrand containing $\phi'$.

J. Heller
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Hint.

Using the Laplace transform

$$ s^2U(s,x)-U_{xx}(x,x)=s \phi(x) $$

and solving for $x$ giving

$$ U(s,x)=e^{s x} \left(c_1(s)-\int^x \frac{1}{2} \phi (\eta ) e^{-\eta s} \, d\eta \right)+e^{-s x} \left(c_2(s)+\int^x \frac{1}{2} \phi (\xi ) e^{s \xi } \, d\xi \right) $$

and now use the final value theorem in

$$ \int_{\mathbb{R}}U_x(s,x)^2 dx,\ \ \ \int_{\mathbb{R}}s^2U(s,x)^2 dx $$

Cesareo
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