I have just studied some elementary distribution theory. However, when attempting to apply them in solving partial differential equations I encounter the following confusion. Consider the heat equation:
$$ \frac{\partial u(x,t)}{\partial t} – k^2\Delta u(x,t) = 0,$$
it is certainly clear what this equality means in the sense of ordinary functions. However, when considering the solution $u(x,t)$ as a distribution my textbook makes the following remark:
"…we assume that $ u(x,t) \in C( [0,\infty),S'(\mathbb{R}^n) ) $, i.e., $u(x,t)$ is continuous in $t$, $t \geq 0$, with values in $ S'(\mathbb{R}^n) $…"
My question is what does this mean exactly? I mean, as far as I have encountered distributions has nothing to do with variables unless they are identified with measures or functions. Thus in general I must think of the distribution $u(x,t)$ as a functional acting on certain space of functions. I can try to make sense of this by perhaps looking at a map
$$ t \mapsto u(x,t) \in S'(\mathbb{R}^n) $$
which is continuous such that the argument $ x $ doesn't really play a role, hence I can consider $ u(x,.) $ as a distribution for every $t \in [0,1)$. Then again, if this is the case, how should I interpret derivative with respect to $t$?
Any insight would be grateful!
Thanks!
Best Answer
For $\phi \in S(\mathbb R^n)$ the differential equation should be interpreted as $$ \frac{\partial}{\partial t} \langle u(x,t), \phi(x) \rangle - k^2 \langle \Delta u(x,t), \phi(x) \rangle = 0. $$
I use $\langle u(x,t), \phi(x) \rangle$ as notation for application of a distribution on a smooth function. You might be used with some other notation.