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As we know we can not multiply distribution

does it mean that there are no weak distributional solution to non linear differntial equation ??

for example $ xy'(x) ==$ has the solution $ y(x)= CH(x)$ for the heaviside step function

however if there is a term of the form $ y(x)y'(x) $ and you try for your nonllinear ode the solution distribution $ y(X)=h(x) $

you should handel with $ H(x)\delta (x) $ or this could be enven worse for example $ \delta(x) \delta ' (x) $

sop what happens with weaks solutions for nonlinear ODE ?

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  • $\begingroup$ Let's take $f\in L^1_{loc}$ such that $ff'\in L^1_{loc}$ and consider an equation for $g$: $gg'=ff'$. It's nonlinear, yet it has a solution in the sense of distributions. $\endgroup$ – TZakrevskiy Jan 17 '14 at 23:42

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