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I need to solve this:

$$ \begin{cases} U_{tt}= \Delta U + |x|^2 \sin t \\ U(x,0) = |x|^4 + |x|^2 \\ U_t(x,0) = |x|^4 - |x|^2 \end{cases} $$

in 3 dimensions $x = \{x_1, x_2, x_3\}$

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Are you looking for solutions in $\mathbb{R}^3\times [0,\infty[$? In this case, perhaps you could use superposition principle, Kirckhhoff formula and Duhamel principle (see Evan's PDEs book, §§2.4.1.c & 2.4.2) to build a solution... But this way to the solution really requires a lot of work. – Pacciu Dec 27 '11 at 1:36
How to rewrite the PDE in terms of the dependent variable $U$ and the independent variables $x_1$ , $x_2$ and $x_3$ ? – doraemonpaul Oct 24 '12 at 2:35
How to factorize $x_1^2+x_2^2+x_3^2−t^2$ in $\mathcal{C}$ ? – doraemonpaul Dec 21 '12 at 18:35
Does $|x|=\sqrt{x_1^2+x_2^2+x_3^2}$ ? – doraemonpaul Dec 21 '12 at 18:36

Try solutions of the form $U(x,t) = (6-|x|^2) \sin(t) + \dfrac{F(|x|+t)+ G(|x|-t)}{|x|}$

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Hmm, didn't notice how old this question was. @night-owl: why did you edit it now? Are you interested in a solution? – Robert Israel Jul 5 '12 at 7:08

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