# Can a non-linear ODE or PDE have a distributional solution ??

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 ?

• 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. – TZakrevskiy Jan 17 '14 at 23:42