PDEs Given the divergence form of a Parabolic PDE with Bounded coefficients: initial/boundary data: , a weak solution with (see Sobolev Space Valued Function) satisfies the following variational formulation: Misplaced &&\left\langle \frac{d}{dt} u(t), v \right\rangle +B_t(u,v) := \left\langle \frac{d}{dt} u(t), v \right\rangle +\int_U \sum_{i,j=1}^n A^{ij}(t, x) \partial_{x_i} u \space \partial_{x_j} v +\sum_{i=1}^n b^i(t, x) \partial_{x_i} u \space v + c(t, x) uv \space dx = \left\langle f, v \right\rangle_{L^2(U)} \\ &u(0)=g \end{aligned}$$ if the above holds for all $v \in H_0^1(U)$ and a.e. $t \in [0, T]$. A weak solution automatically satisfies the regularity result of [[Sobolev Space Valued Function]]s that $u \in C([0,T]; L^2(U))$; this is consistent with the physical interpretation that the solution should evolve continuously in time from the initial condition.