So I have a partial differential equation:
$$P(D)u = \sum_{a \leq m}^{} c_{a} u^{a} = f$$
When I have the fundamental solution
$$P(D)T = \delta _{0} $$
I can solve the equation with (Malgrange-Ehrenpreis, Hörmander)
$$u =f\ast T\;\;\; for\;\;\; f \in C_{c} ^{\infty} $$
but what should I do if I have a partial differential equation with
$$ f = \chi _{ [0,1] } ? $$