Solution of partiell differential equation using the fundamental solution

by Chopin   Last Updated January 16, 2018 17:20 PM

So i have a partiell 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 partiell differential equation with

$$ f = \chi _{ \sqsubset0,1 \sqsupset } $$

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