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I'm trying to check the well-posedness of the following equation:

$\pmatrix{u\\v}_t$ = $\pmatrix{4/3 & 0 \\ 1 & 0}$\pmatrix{u\\v}_{xx}$+$\pmatrix{0 & -2/3 \\ 1 & 0}$\pmatrix{u\\v}_{xy}$

As far I understand, in order to show well-posedness, I have to prove that the energy of the equation is bounded, that is:$\int a{\lVert{v}\rVert}^2+b{\lVert{v}\rVert}^2 < M$ where $a$, $b$ and $M$ are constants. Does this need to be proven to hold for all $a$ and $b$ or I can choose their values to my convenience?

Is there a book (or online resource) that contains discussion of this type of problems?

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    Why can you not? Observe that if $\int a\|u\|^2 + b\|v\|^2 \leq M$, then for any c,d > 0 you have $\int c\|u\|^2 + d\|v\|^2 \leq \max(\frac{c}{a},\frac{d}{b}) \int a\|u\|^2 + b\|v\|^2 \leq \max(\frac{c}{a},\frac{d}{b}) M$ The problem that I was alluding to in my comment above is that for certain $a,b$ the inequality is more easily proven: it is$a$lot easier to prove the conservation of energy for the linear wave equation when you write the energy as $ E(u) = \int c^2|u_t|^2 + |u_x|^2 $ then to directly show that $\int 5 c^2|u_t|^2 + 2.3 |u_x|^2 $ is bounded.2012-06-19

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