Suppose $F: \mathbb{R} \rightarrow \mathbb{R}$, strictly convex function and consider the Riemann problem $u_{t}+(F(u))_{x}=0$, and $u_{L}, u_{R}$ are constants and satisfying $u_{L}< u_{R}$. Find a weak solution by using rarefaction wave.
I proved this result and my answer is
$$u(x,t)=\left\{\begin{array}{lr} u_{L}, & \text{for } x\leq F'(u_{L})t\\ (F')^{-1}(\frac{x}{t}), & \text{for } F'(u_{L})t\leq x\leq F'(u_{R})t\\ u_{R}, & \text{for } x\geq F'(u_{R})t \end{array}\right\} $$
Is there any mistake in this solution? I want to prove that the given solution is a weak solution. I applied the definition, but my integral is not zero. Anyone, please refer me the proof that shows this solution is a weak solution but not classic.
This is my attempt:
$\begin{aligned} \int_{0}^{\infty}\!\int_{-\infty}^{\infty} \left[ \phi_t u + \phi_x F(u)\right] \mathrm{d}x\, \mathrm{d}t &= \int_{0}^{\infty}\!\int_{-\infty}^{F'(u_{L})t} \left[ \phi_t u_{L} + \phi_x F(u_{L})\right] \mathrm{d}x\, \mathrm{d}t \\ &+ \int_{0}^{\infty}\!\int_{F'(u_{L})t}^{F'(u_{R})t} \left[ \phi_t ((F')^{-1}\frac{x}{t}) + \phi_x F((F')^{-1}(\frac{x}{t}))\right] \mathrm{d}x\, \mathrm{d}t \\ &+\int_{0}^{\infty}\!\int_{F'(u_{R})t}^{\infty} \left[ \phi_t u_{R} + \phi_x F(u_{R})\right] \mathrm{d}x\, \mathrm{d}t \end{aligned}$
$\begin{aligned} &=& u_{L}(\int_{-\infty}^{0} dx \!\int_{0}^{\infty} \phi_t dt + \int_{0}^{\infty} dx \!\int_\frac{x}{F'(u_{L})}^{\infty} \phi_t dt)+ F(u_{L}) \int_{0}^{\infty} dt\!\int_{-\infty}^{F'(u_{L}t)} \phi_x dx +u_{R} \int_{0}^{\infty} dx\! \int_{0}^{\frac{x}{F'(u_{R})}} \phi_t dt + F(u_{R}) \int_{0}^{\infty} dt\!\int_{F'(u_{R})t}^{\infty} \phi_x dx \end{aligned}$
$\begin{aligned} &=& -\int_{-\infty}^{\infty} \phi(x,0)u(x,0)dx-u_{L} \int_{0}^{\infty} \phi(x,(F'(u_{L})^{-1}x)dx +u_{R} \int_{0}^{\infty} \phi(x,(F'(u_{R})^{-1}x)dx+ F(u_{L}) \int_{0}^{\infty} \phi(F'(u_{L})t,t)dt - F(u_{R}) \int_{0}^{\infty} \phi(F'(u_{R})t,t)dt\end{aligned}$
I am confused, how to solve this part $\int_{0}^{\infty}\!\int_{F'(u_{L})t}^{F'(u_{R})t} \left[ \phi_t ((F')^{-1}\frac{x}{t}) + \phi_x F((F')^{-1}(\frac{x}{t}))\right] \mathrm{d}x\, \mathrm{d}t$.
