How to understand partial differential equations in the sense of distribution?

Question

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!

Answer 1

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.

Answer 2

Just to add something to md2perpe's answer.

In your question you mentioned the space $u(x,t) \in C( [0,\infty),S'(\mathbb{R}^n) )$.

Sometimes, especially if you work with the evolution PDEs, it is convenient to understand variable $t$ diferent than the $x$ (it is very useful to have solution that is continuous mapping from $[0,\infty)$ to some Banach space X). Based on the textbook you are using you maybe want to find more about those type of spaces.

More about them you could find in: (there are other books but I prefer this two)

Weak and Measure-Valued Solutions to Evolutionary PDEs - J. Necas, J. Malek, M. Rokyta, M. Ruzicka, 1996 - subsection 1.2.6

Linear and Quasi-linear Evolution Equations in Hilbert Spaces - P. Cherrier, A. Milani, 2012 - subsection 1.7

Also a bunch of titles of some amazing books that deal with distribution theory you could find on this mathoverflow question (I prefer A Guide to Distribution Theory and Fourier Transforms - R. Strichartz, 1994 - first chapter for intuition about distributions).

Hope this helps you somehow.