Hebe A. Biagioni, Todor Gramchev
Multidimensional Kuramoto-Sivashinsky Type Equations: Singular Initial Data and Analytic Regularity
The paper is published: Mat. Contemp. 15 (1998), 21-42

MSC:

35K55 Nonlinear PDE of parabolic type

35R05 PDE with discontinuous coefficients or data

Abstract: We consider the Cauchy problem for multidimensional
Kuramoto-Sivashinsky type equations on $\R^n$ and on the
torus $\T^n$. The initial data could be singular, in
particular, could belong to Sobolev spaces $H_p^r$, with
$r$ negative. We introduce weighted analytic-Gevrey type
spaces, which allow us to obtain new results both on the
critical index of the singularity of the initial data
and on the analytic regularity of the solutions with
respect to $x$ when $t>0$. We also obtain global
well-posedness in $L^2$ in the case when the
nonlinearities are conservative.

Keywords: Kuramoto-Sivashinsky equation, singular initial data, analytic regula rity, global well-posedness