I've got this problem to solve:
Using Galerkin method, prove that there exists a weak solution of this differential equation:
$$-\Delta u = a(x) \circ \triangledown u - u_t +f(x)$$ on $\Omega$
$$u = 0$$ on $\partial \Omega$
$$u(x,0)=u_{0}(x)$$
where $f,u_0\in L^{1}(\Omega), a\in L^{\infty}(\Omega)$ are known.
I started with using definition of a weak solution. If $u\in W^{1,2}_{0}(\Omega)$ is a solution of the problem above (where $W^{1,2}_{0}(\Omega)$ is a Sobolev space with inner product defined $<f,g>=\int_{\Omega}\triangledown f\circ \triangledown g)$ then:
$$\forall \varphi \in W^{1,2}_{0}(\Omega) \int_{\Omega}\triangledown u \circ\triangledown \varphi -\int_{\Omega}(a(x)\circ\triangledown u) \varphi +\int_{\Omega}u_t\varphi = \int_{\Omega}f\varphi $$
Then I defined $W_m \subset W^{1,2}_{0}(\Omega)$ such that:
$$ W_m = lin\{\varphi_1,...,\varphi_m\} $$
where $\varphi_i$ are orthonormal. Let $u_m \in W_m$ be the solution of the problem above. We can write:
$$ u_m=\sum_{k=1}^{m}c_k^m\varphi_k$$
To finish the solution I need to know the coefficients $c^m_i$ and I have no idea how to do this. Thank you in advance for any help!