**Step 1: Testing with $u$.** Set $v = u(t) \in H^1_0(U)$ in the weak formulation: \begin{align*} u_t(t) \circ u(t) + B[u(t), u(t); t] = f(t) \circ u(t). \end{align*} The first term satisfies the identity: \begin{align*} u_t \circ u = \frac{1}{2}\frac{d}{dt}\|u(t)\|_{L^2(U)}^2. \end{align*} This identity requires justification (it is not immediate in infinite dimensions since $u_t \in H^{-1}$, not $L^2$); it follows from a [mollification](/page/Standard%20Mollifier)-in-time argument or from the properties of the Gelfand triple. **Step 2: Elliptic coercivity.** By uniform ellipticity and the structure of $B$, there exist constants $\beta > 0$ and $\gamma \ge 0$ (from Gårding's inequality for the elliptic part $L$) such that: \begin{align*} B[u, u; t] \ge \beta\|u\|_{H^1_0(U)}^2 - \gamma\|u\|_{L^2(U)}^2. \end{align*} **Step 3: Absorbing the right-hand side.** By Young's inequality: \begin{align*} |f \circ u| \le \|f\|_{H^{-1}} \|u\|_{H^1_0} \le \frac{\beta}{2}\|u\|_{H^1_0}^2 + \frac{1}{2\beta}\|f\|_{H^{-1}}^2. \end{align*} **Step 4: Differential inequality.** Combining Steps 1–3: \begin{align*} \frac{1}{2}\frac{d}{dt}\|u\|_{L^2}^2 + \frac{\beta}{2}\|u\|_{H^1_0}^2 \le \gamma\|u\|_{L^2}^2 + \frac{1}{2\beta}\|f\|_{H^{-1}}^2. \end{align*} Gronwall's inequality applied to $\varphi(t) := \|u(t)\|_{L^2(U)}^2$ yields the estimate.