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

Prove that the equation $\Large u_t = p(u)u_x ; t0$ 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) $\Large u_t = u \sin (u) u_x$

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