Hi I am supposed to show using the following example (which is given in the problem) the existence of non analytic solutions to the heat equation.
$ u_{t} -u_{xx}=0,\hspace{10pt} u(0,x)=e^{-x^4}$
Where $-\infty<x<\infty$.
The problem I have is I don't really know what I should do. I am using Evans's book and using the theorems I can write the solution as
$u(t,x)=\frac{1}{\sqrt{4\pi t}}$ $\int_{-\infty}^{\infty} e^{(\frac{{-(x-y)^2})}{4t}-y^4}$ $dy $
But I don't know what to do after this. Is there a way to evaluate the integral or test analyticity without doing so? Any hints or help is appreciated greatly.
Edit: Corrected the wrong formula