Viscosity solutions of discontinuous Hamilton-Jacobi equations (Q998753)

The initial value problem for the Hamilton-Jacobi equation \[ \varphi_t(x,t)=\nu(x,t)H(\nabla\varphi),x\in\mathbb{R}^N,\;t>0,\quad \varphi(x,0)= \varphi_0(x),\;x\in\mathbb{R}^N,\;t=0,\tag{1} \] where \(\nu\) is positive, bounded and measurable, and \(H(q)\) is non-negative and Lipschitz continuous is considered. The various properties of the viscosity solutions are proved and a few examples which illustrate the difficulties in studying the existence, uniqueness and continuous dependence of solutions of (1) are given. Under certain assumptions, the existence and uniqueness of Lipschitz continuous viscosity solutions are established. Finally, the blow-up property is discussed and a few more examples are provided.