Hopf-type formula defines viscosity solution for Hamilton–Jacobi equations with tt-dependence Hamiltonian

We prove that under some assumptions, the Hopf-type formula u(t,x)=maxq∈Rn{〈x,q〉−σ∗(q)−∫0tH(τ,q)dτ} defines a viscosity solution of the Cauchy problem for Hamilton–Jacobi equation (H,σ)(H,σ) where the initial condition σσ is convex but the Hamiltonian H=H(t,p)H=H(t,p) is not necessarily convex in pp.