Equation C(x,t) is an Oscillatory integral that is obtained by solving a partial differential equation with Fourier transform. The partial differential equation is ill-posed in inverse time.
$$C(x,t)= \int_0^{\infty}\int_{-\infty}^{\infty}B(\omega,y)e^{i\omega x}dy d\omega $$ $$B(\omega,y)= f(y)*e^{-i\omega y+iv\omega t+tD\omega^2+tE\omega^4} $$ $$\partial^4 C/\partial x^4= ? $$
