Document Type



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.

Publication Date



This is an Accepted Manuscript of an article published in Music Reference Services Quarterly in 2004, Copyright 2004 by The Haworth Press, Inc., Binghamton, NY.

Original Citation

Bellout, H, S. Benachour and E.S. Titi, "Finite-time singularity versus global regularity for hyper-viscous Hamilton-Jacobi-like equations" Nonlinearity 16 (November 2003) 1967-1989.

Legacy Department

Department of Mathematical Sciences






Institute of Physics



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