Evans' Partial Differential Equations p.136 3.4.1 Shocks, entropy condition

In Evans' PDE textbook p. 136, 3.4.1 Shocks, entropy condition:

More precisely, assume \left\lbrace \begin{aligned} &v : \Bbb R\times [0, \infty) \to \Bbb R \text{ is smooth, with}\\ &\text{compact support} \end{aligned} \right. \tag{2}

We call $$v$$ a test function. Now multiply the PDE $$u_t + F(u)_x = 0$$ by $$v$$ and integrate by parts: \begin{aligned} 0&=\int_{0}^{\infty}\int_{-\infty}^{\infty}(u_t+F(u)_x)\, v\, dx dt& \\ &=-\int_{0}^{\infty}\int_{-\infty}^{\infty}uv_t\, dxdt - \int_{-\infty}^{\infty}uv\, dx|_{t=0} \\ &\qquad - \int_{0}^{\infty}\int_{-\infty}^{\infty}F(u)v_x\, dxdt . \end{aligned} \tag{3}

But my computation is $$\int_{0}^{\infty}\int_{-\infty}^{\infty}(u_t+F(u)_x)v dxdt=\\-\int_{0}^{\infty}\int_{-\infty}^{\infty}uv_t dxdt+\int_{-\infty}^{\infty}uv dx|_{t=0}^{\infty}\\+\int_{0}^{\infty}F(u)v dtdx|_{-\infty}^{\infty}-\int_{0}^{\infty}\int_{-\infty}^{\infty}F(u)v_x dxdt$$

How to get the answer?

Since $$v$$ is smooth with compact support, its values at $$x = \pm\infty$$ and $$t = +\infty$$ are zero. This is also true for $$uv$$, $$F(u)v$$, etc. Hence, the additional terms obtained in your (correct) computation vanish, and the formula $$(3)$$ of Evans' book is obtained.