Layered viscosity solutions of nonautonomous Hamilton–Jacobi equations: Semiconvexity and relations to characteristics

We construct an explicit representation of viscosity solutions of the Cauchy problem for the Hamilton–Jacobi equation (H,σ)(H,σ) on a given domain Ω=(0,T)×RnΩ=(0,T)×Rn. It is known that, if the Hamiltonian H=H(t,p)H=H(t,p) is not a convex (or concave) function in p  , or H(⋅,p)H(⋅,p) may change its sign on (0,T)(0,T), then the Hopf-type formula does not define a viscosity solution on Ω  . Under some assumptions for H(t,p)H(t,p) on the subdomains (ti,ti+1)×Rn⊂Ω(ti,ti+1)×Rn⊂Ω, we are able to arrange “partial solutions” given by the Hopf-type formula to get a viscosity solution on Ω. Then we study the semiconvexity of the solution as well as its relations to characteristics.