Limit of the derivative of the solution to Burgers Equation

Question

Let $u$ be the solution to Burgers equation $$u_t + u u_x = \epsilon u_{xx}$$ I want to prove that $$\lim_ {x \to \pm \infty}\partial_x u(x,t) \, \, \,\text{exists}.$$ But I don't even know if it's true

Answer 1

It isn't true in general. Consider for instance $\epsilon=0$ and the initial-value problem $u(x,0) = x^2$. Then the solution $u(x,t)$ at $t>0$ isn't defined over the whole real line (cf. various related posts on this site), so that there is little chance to be able to compute $u_x$ as $x$ approaches $-\infty$.