Existence of solution for this parabolic PDE

Question

The parabolic PDE $$\langle u', v \rangle + a(u,v) = \langle f, v \rangle$$ has a unique solution $u \in L^2(0,T; H^1)$ with $u' \in L^2(0,T;H^{-1})$ if $a$ is a bounded and coercive bilinear form (assuming $f$ is nice).

I want to know if the PDE $$\langle gu', v \rangle + a(u,v) = \langle f, v \rangle$$ or equivalently, $$\langle u', gv \rangle + a(u,v) = \langle f, v \rangle?$$ has a unique solution, where $g$ is a smooth function (depending on time and space) which is bounded above and below by positive numbers? I do not want to "divide by $g$" because that messes up my bilinear form and I can't show coercivity anymore. I think the Galerkin approach is not amenable here.

Thanks. (Posted here also: https://math.stackexchange.com/questions/265312/existence-of-solution-for-this-parabolic-pde)

Edit: In the Galerkin method I can't seem to extract a convergent subsequence. I end up with the same problems (needing a constrain to be satisfied) so I guess this may not be solvable...

Answer 1

You can use a weighted inner product instead of the standard $L^2$-inner product. In any case, the Galerkin method ought to work. There is also a direct method, that sometimes goes under the name of Lions' extension of Lax-Milgram lemma.