This question arises from my studies concerning first order partial differential equations, in particular from the definition of weak solution for a conservation law.
Let $q \in C^1(\mathbb{R})$ and $u=u(t,x)$ be a bounded function on $[0,+\infty)\times \mathbb{R}$. Let $v=v(t,x)\in C^1([0,+\infty)\times\mathbb{R})$ with compact support.
What are sufficient conditions for the function $[uv_t+ (q(u))v_x]$ to be integrable on $(0,+\infty)\times \mathbb{R}$, ie the (improper) integral $$\int_{0}^{+\infty}dt\int_{-\infty}^{+\infty}[uv_t+ (q(u))v_x] \,dx$$ exists?
Note
With $v_t,v_x$ I mean partial derivative of $v$ with respect to $t,x$.
Thanks a lot in Advance.
The integral exists in Lebsgue sense without any further hypothesis. Since $\int_0^{N}\int_{-N}^{N}$ in the Lebesgue sense coincides with the Riemann integral it follows that the improper Riemann integral also exists.