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 ) on a given domain Ω = ( 0 , T ) × R n . It is known that, if the Hamiltonian H = H ( t , p ) is not a convex (or concave) function in p, or H ( ⋅ , p ) may change its sign on ( 0 , T ) , then the Hopf-type formula does not define a viscosity solution on Ω. Under some assumptions for H ( t , p ) on the subdomains ( t i , t i + 1 ) × R n ⊂ 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.