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?