Solution of partiell differential equation using the fundamental solution

Question

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

$$ f = \chi _{ [0,1] } ? $$

Answer 1

Exactly the same thing. Observe that $$ \chi_{[0,1]}\ast\phi\in C_c^\infty\quad\forall\phi\in C_c^\infty. $$ Then you can define $T\ast\chi_{[0,1]}$ by $$ \langle T\ast\chi_{[0,1]},\phi\rangle=\langle T,\phi\ast\chi_{[0,1]}\rangle\quad\forall\phi\in C_c^\infty. $$