In evan's book, about the motivation for definition of weak solution to parabolic equation, it says (I am doubtful about the last sentence):
If we fix a function $v \in H_{0}^{1}(U)$, we can multiply the $\mathrm{PDE} \frac{\partial u}{\partial t}+L u=f$ by $v$ and integrate by parts, to find $(9) \quad\left(\mathbf{u}^{\prime}, v\right)+B[\mathbf{u}, v ; t]=(\mathbf{f}, v) \quad\left({ }^{\prime}=\frac{d}{d t}\right)$ for each $0 \leq t \leq T$, the pairing $(\cdot\ ,\cdot)$ denoting inner product in $L^{2}(U)$.
Next, observe that $$ u_{t}=g^{0}+\sum_{j=1}^{n} g_{x_{j}}^{j} \quad \text { in } U_{T} $$ for $g^{0}:=f-\sum_{i=1}^{n} b^{i} u_{x_{i}}-c u$ and $g^{j}:=\sum_{i=1}^{n} a^{i j} u_{x_{i}}(j=1, \ldots, n)$. Consequently (10) and the definitions from $\S 5.9 .1$ imply the right-hand side of (10) lies in the Sobolev space $H^{-1}(U)$, with $$ \left\|u_{t}\right\|_{H^{-1}(U)} \leq\left(\sum_{j=0}^{n}\left\|g^{j}\right\|_{L^{2}(U)}^{2}\right)^{1 / 2} \leq C\left(\|u\|_{H_{0}^{1}(U)}+\|f\|_{L^{2}(U)}\right) . $$ This estimate suggests it may be reasonable to look for a weak solution with $\mathbf{u}^{\prime} \in H^{-1}(U)$ for a.e. time $0 \leq t \leq T$
I'v got the point that $\left(\mathbf{u}^{\prime}, v\right)\in H^{-1}$, but what does the last sentence mean? How does the estimation work?
If we denote by $|\cdot|_1$ the $H^1_0$ norm $\|\nabla\cdot\|_{L^2}$, the $H^{-1}$ norm can be defined by: $$\|w\|_{H^{-1}}=\sup_{v\in H^1_0,|v|_1\le 1}\left<w,v\right>.$$ Then if $w$ is expressed by $w=g^0+\sum_{j=1}^ng^j_{x_j}$ where all the $g^i$ are in $L^2$, we have $$\left<w,v\right>=(g^0,v)-\sum_{j=1}^n(g^j,v_{x_j})$$ so that from Cauchy-Schwarz inequality, and then Poincaré inequality, $$|\left<w,v\right>|\le \|g^0\|_{L^2}\|v\|_{L^2}+\sum_{j=1}^n\|g^j\|_{L^2}|v|_1\le C|v|_1\sum_{j=0}^n\|g^j\|_{L^2}.$$ Thus $\|w\|_{H^{-1}}\le \sum_{j=0}^n\|g^j\|_{L^2}$ which applied to the expression for $u_t$ gives, assuming all coefficients are bounded, $$\|u_t\|_{H^{-1}}\le \|f\|_{L^2}+\max_i\|b^i\|_{L^\infty} |u|_1+\|c\|_{L^\infty}\|u\|_{L^2}+\max_{i,j}\|a^{ij}\|_{L^\infty}|u|_1$$ which gives the announced estimate using Poincaré inequality.