If $u:[0,\infty)\times\mathbb R\to\mathbb R$ is a $C^\infty$ solution of $u_t(t,x)-u_{xx}(t,x)=0$ then $\frac{d}{dt}\int_\mathbb R u(t,x)\,dx\equiv0$

Question

Problem: Suppose $u:[0,\infty)\times\mathbb R\to\mathbb R$ is a smooth solution to $u_t(t,x)-u_{xx}(t,x)=0$ such that $u$ and its derivatives are integrable. Show that $$\frac{d}{dt}\left[\int_\mathbb R u(t,x)\,dx\right]\equiv0.$$

My Attempt: Differentiating under the integral and applying the hypothesis we get $$\frac{d}{dt}\left[\int_\mathbb R u(t,x)\,dx\right]=\int_\mathbb{R}u_t(t,x)\,dx=\int_\mathbb{R}u_{xx}(t,x)\,dx=u_x(t,x)\bigg\vert_{x\to-\infty}^{x\to\infty}.$$ But I cannot see how the above limits converge to zero. I also thought about applying some energy estimate, using the energy $$E(t)=\int_{\mathbb R}u(t,x)^2\,dx,$$ but I could not see a way to go with this either.

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Could anyone point me in the right direction, with a hint only.
Thank you for your time and appreciate any help.

Answer 1

Hint: if $f$ and $f'$ are both integrable then $$\lim_{x\rightarrow\pm\infty} f(x)=0$$ (use the mean value theorem)