I am looking for some help to determine the large-time behavior of the unique solution for the equation in $\mathbb R^+ \times \mathbb R$ $$u_t+\vert\nabla u\vert^\frac{2}{3}=0,\ \ \ \ u(0,x)=-\cos x$$ More specifically, I am thinking about how to determine the behavior of u at $(2015,0)$ or $(n,0)$ for any large integer $n$, but I have no idea that where to start.
Many thanks to the help in advance.
Long comment: The Hamilton-Jacobi PDE is derived from a canonical transformation involving a type-2 generating function $u(t,x)$ which makes vanish the new Hamiltonian $K=H(u_x) + u_t$. Here, $H(p)=|p|^{2/3}$ is the original Hamiltonian.
For convex Hamiltonians $H$ which grow faster than the identity, the unique continuous viscosity solution can be obtained via the Lax-Hopf formula $$ u(t,x) = \inf_{y\in \Bbb R} \left\lbrace t L \big(\tfrac{x-y}{t}\big) - \cos y\right\rbrace $$ where the Lagrangian $L$ is the Legendre-Fenchel transform of $H$: $$ L(q) = \sup_{p\in\Bbb R} \lbrace qp - H(p)\rbrace = \frac{4}{27}|q|^3 . $$ However, the Hamiltonian is nonconvex here, which invalidates the previous approach. One may have a look at this post for complements.