Let $u : (0, \infty) \times \mathbb{R} \rightarrow \mathbb{R} $ such that $u(t,\cdot) \in L^1(\mathbb{R}) \cap L^\infty(\mathbb{R})$ for all $t \in (0,\infty)$ and $u$ is continuously differentiable in $t$.
I.e. $u \in C^1((0,\infty); L^1 \cap L^\infty)$
I would like to know how we can show, explicitly, that the integral $\int_{\mathbb{R}} \partial_t |u(t)| \text{d}x$ is well defined. In particular, I am off course worried about the points at which $u$ hits 0.
If $u$ is constantly 0 along any open time interval, then of course the time derivative of $u$ is well defined as 0 along that interval. Thus, we are only concerned with the points at which $u$ initially hits 0 or passes through it.
In what way is $\partial_t |u(t)| $ expanded to include these points, where the derivative is normally not defined?
I understand that of course, since $u$ is continuously differentiable, such points will make up a set of 0 measure. But that on its own should not be enough to say the integral is finite, right? A set of 0 measure can still cause problems if the function blows up to $\infty$ on that set.
How do we show explicitly that this problem does not occur?
We can say the absolute value of a $W^{1,k}$ function is also $W^{1,k}$ thanks to the Chain Rule for Sobolev Spaces, as per L. Evans's "Partial Differential Equations", Chapter 5, problem 17.