We construct an explicit representation of viscosity solutions of the Cauchy problem for the Hamilton-Jacobi equation $(H,\sigma)$ on a given domain $\Omega= (0,T)\times \R^n.$ It is known that, if the Hamiltonian $H = H(t,p)$ is not a convex (or concave) function in $p$, or $H(\cdot, p)$ may change its sign on $(0,T)$, then the Hopf-type formula does not define a viscosity solution on $\Omega.$ Under some assumptions for $H(t,p)$ on the subdomains $(t_i, t_{ i+1})\times \R^n\subset \Omega$, we are able to arrange "partial solutions" given by the Hopf-type formula to get a viscosity solution on $\Omega.$ Then we study the semiconvexity of the solution as well as its relations to characteristics.
Categories: math.AP