A very odd partial differential equation.

1. Feb 27, 2008

mhill

Studying a problem of difussion i came across the PDE (first order)

$$\partial _{t} U(x,t) = U(x,t)^{2}$$

with initial condition $$\partial _{t} U(x,0) = \delta (x)$$

i have found the solution $$U(x,t)= \frac{ \delta (x) }{ t \delta (x) -1}$$

however does this make sense ? from the definition of delta function the only values that U(x,t) can have for different positions 'x' are just 0 or infinity , and if i use a Taylor expansion i get

$$U(x,t)= - \delta (x) ( t\delta (x) +t^{2} (\delta (x) )^{2}+......)$$

which makes no sense for me