Let
$$Tu=\bigg(t\Psi +x \partial_x\bigg)u=0$$
where $\Psi=\partial_{tt}.$
Define a kernel which solves this equation by
$$ u(t,x)=\exp\bigg(\frac{t}{\log x} \bigg) $$
for all $t>0$ and $u,x\in(0,1)$. Consider a square domain with boundary $\Omega \in \Bbb R^2$ s.t. $\Omega=[0,1]^2.$ Define the initial condition
$$ \lim_{t \to 0} u(t,x):=\psi(x)$$
where the limit is taken in the sense of distributions. Also define the condition
$$ u(t,x)=0 \space\space\space x \in \partial \Omega $$
I believe I've set up the conditions correctly.
Is $u(t,x)$ the fundamental solution for $Tu=0?$