How does $\partial_t u \notin L^2((0, T); H^{-1}(\mathbb{R}))$ look like?

Question

I have problems gaining intuition for the space $L^2\big((0, T); H^{-1}(\mathbb{R})\big)$. I always imagined these as being spatial distributional derivatives of a $L^2((0, T) \times \mathbb{R})$ function.

It gets more delicate when one looks into parabolic PDEs and encounters some $u \in L^2\big((0, T); H^1(\mathbb{R})\big)$. Many theorems require e.g. $\partial_t u \in L^2\big((0, T); H^{-1}(\mathbb{R})\big)$.

My problem is that I can't imagine an object $u \in L^2\big((0, T); H^1(\mathbb{R})\big)$ with $\partial_t u \notin L^2\big((0, T); H^{-1}(\mathbb{R})\big)$. Can someone give me some intuition on this and maybe even an example?

Answer 1

Take $v \in L^2(0,T) \setminus H^1(0,T)$ and $w \in H^1(\mathbb R)$. Now, define $u$ via $u(t,x) = v(t) w(x)$.