Finite-time singularity versus global regularity for hyper-viscous Hamilton-Jacobi-like equations

Abstract

The global regularity for the two- and three-dimensional Kuramoto-Sivashinsky equations is one of the major open questions in nonlinear analysis. Inspired by this question, we introduce in this paper and family of hyper-viscous Hamilton-Jacobi-like equations parametrized by the exponent in the nonlinear term, p, where in the case of the usual Hamilton-Jacobi nonlinearity, p = 2. Under certain conditions on the exponent p we prove the short-time existence of weak and strong solutions to this family of equations. We also show the uniqueness of strong solutions. Moreover, we prove the blow-up in finite time of certain solutions to this family of equations when the exponent p > 2. Furthermore, we discuss the difference in the formation and structure of the singularity between the viscous and hyper-viscous versions of this type of equation.