A theorem of PDEs sais that the following Cauchy problem for the heat equation \begin{align*} & \partial_t u = \partial_{x}^2 u, \quad (t,x) \in \mathbb{R_+} \times \mathbb{R}, \\ & u|_{ t = 0 } = u_0 \in S'( \mathbb{R} ) \end{align*} admits a unique solution in $ C( \mathbb{R_+}, S'( \mathbb{R} ) ) $, where $ S'( \mathbb{R} ) $ is the class of tempered distributions.
There is an example which shows that the above problem is not uniquely solvable in $ D'( \mathbb{R_+} \times \mathbb{R} ) $. The construction goes like this. Put $ h( t ) = e^{ -1/t^2 } $, $ t >0 $, and $ h( t ) = 0 $ otherwise. Then it is possible to show that the series $ \sum_{k = 0}^{\infty} \partial_t^i \partial_x^j u_k( t, x ) $, with $ u_k( t, x ) = \frac{ 1 }{(2k)!} h^{(k)}(t) x^{2k} $, is normally convergent on $ [0, T] \times [-L, L] $ for any $ T > 0 $ and $ L > 0 $ and thus $ \sum_{k = 0}^{\infty} u_k( t, x ) $ defines a $ C^{ \infty }( \mathbb{R_+} \times \mathbb{R} ) $ function $ u $. This function, as well as zero function, is a solution of the heat equation satisfying the zero initial condition.
Now is there a direct way to prove that $ u $, constructed above, does not belong to $ C( \mathbb{R_+}, S'( \mathbb{R} ) ) $ to show that there is no contradiction with the theorem stated?
