Value functions for certain class of Hamilton Jacobi equations

We consider a class of Hamilton Jacobi equations (in short, HJE) of type

$ u_t + \frac{1}{2}\big(|u_{x_1}|^2+ \cdots +|u_{x_{n-1}}|^2\big) + \frac{\mathrm{e}^u}{m}|u_{x_n}|^m=0, $

in ? ⁿ ×?_?+? and m?>?1, with bounded, Lipschitz continuous initial data. We give a Hopf-Lax type representation for the value function and also characterize the set of minimizing paths. It is shown that the minimizing paths in the representation of value function need not be straight lines. Then we consider HJE with Hamiltonian decreasing in u of type

$ u_t + H_1\big(u_{x_1},\ldots,u_{x_i}\big) + \mathrm{e}^{-u}H_2\big(u_{x_{i+1}},\ldots, u_{x_n}\big)=0 $

where H ₁,H ₂ are convex, homogeneous of degree n,m?>?1 respectively and the initial data is bounded, Lipschitz continuous. We prove that there exists a unique viscosity solution for this HJE in Lipschitz continuous class. We also give a representation formula for the value function.