Solve the damped quasilinear wave equation $u_t+uu_x+u=0$ with $u(x,0)=f(x)$.

Question

Solve the damped quasilinear wave equation $u_t+uu_x+u=0$ with $u(x,0)=f(x)$. Determine if the solution breaks when $f$ satisfies the condition $f^\prime(x)>-1$ for all $x\in\mathbb{R}$.

I'm stumped at how to begin. Please help.

Answer 1

Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

$\dfrac{dt}{ds}=1$ , letting $t(0)=0$ , we have $t=s$

$\dfrac{du}{ds}=-u$ , letting $u(0)=u_0$ , we have $u=u_0e^{-s}=u_0e^{-t}$

$\dfrac{dx}{ds}=u=u_0e^{-s}$ , letting $x(0)=g(u_0)$ , we have $x=g(u_0)+u_0(1-e^{-s})=g(ue^t)+u(e^t-1)$ , i.e. $u=e^{-t}G(x+u(1-e^t))$

$u(x,0)=f(x)$ :

$G(x)=f(x)$

$\therefore u=e^{-t}f(x+u(1-e^t))$