Question

Let \(\Large p: \mathbb R \to \mathbb R \) be a differentiable function.

Prove that the equation \(\Large u_t = p(u)u_x ; t>0\) has a solution satisfying the functional relation \(\Large u=f(x+p(u)t)\) , where \(\Large f\) is a differentiable function. In particular, find such solution for the following equations:

(a) \(\Large u_t = ku_x\)

(b) \(\Large u_t= uu_x\)

(c) \(\Large u_t = u \sin (u) u_x\)