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Consider the problem $$\begin{cases}u_{tt}=u_{xx}-2au_{x}+a^{2}u, (x,t)\in\mathbb{R}^{2}\\[8pt] u(x,0)=f(x), x\in\mathbb{R}\\[8pt] u_{t}(x,0)=g(x), x\in\mathbb{R} \end{cases}$$ where $a>0$.

(a)Study the existence and uniqueness of local solutions analytical of the problem, assuming $f$ and $g$ analytic in a neighborhood of $(0,0)$.

(b)Assuming $f\in C^{2}(\mathbb{R}), g\in C^{1}(\mathbb{R})$, show that there exists a unique solution of the problem in $C^{2}(\mathbb{R})$ and display such solution.

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I see this is your seventh question here and yet you've accepted only one answer so far. If you get a satisfying answer, please accept your favorite one. –  Git Gud Feb 5 '13 at 20:02
    
What's wrong with my question? Why did I receive $-1$? –  Manoel Feb 5 '13 at 20:53
    
I didn't downvote you, it was someone else. –  Git Gud Feb 5 '13 at 21:01
    
I did not downvote either but I can see NO personal thought or comment in this question. This is bad practice since (1) it makes difficult to adjust an answer suited to the level of your understanding, and (2) one could imagine that you are simply dumping your homework on the site and waiting for a full proof to appear to put your name on it and hand it back. –  Did Feb 5 '13 at 21:17
    
Who gives downvote in question could say why, can help the questioner! Simply give downvote, can take on all questions. –  Manoel Feb 6 '13 at 14:28

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