On inf-type extremal solution on sphere \(S^N\) (Q2782068)

The paper deals with Hamiltonian-Jacobi equation \(\frac{\partial u}{\partial t} + \frac 12 \langle\nabla u,\nabla u\rangle = 0\) with initial data \(u|_{t=0^+} = v: \mathbb{R}^{N+1}\to \mathbb{R}\), where \(\langle\cdot,\rangle\) is ordinary scalar product in \(\mathbb{R}^{N+1}\). It is proved that the solution this equation is bounded on sphere \(S^N\) for Cauchy data \( v\in BSC(S^N)\). The authors also present an extremal solution of Lax inf-type.