Mathematics – Analysis of PDEs
Scientific paper
2009-02-02
Mathematics
Analysis of PDEs
34 pages
Scientific paper
The initial boundary-value problem (IBVP) and the Cauchy problem for the Kuramoto--Sivashinsky equation and other related $2m$th-order semilinear parabolic partial differential equations in one and N dimensions are considered. Global existence and blow-up as well as uniform bounds are reviewed by using: (i) classic tools of interpolation theory and Galerkin methods, (ii) eigenfunction and nonlinear capacity methods, (iii) Henry's version of weighted Gronwall's inequalities, and (vi) two types of scaling (blow-up) arguments. For the IBVPs, existence of global solutions is proved for both Dirichlet and "Navier" boundary conditions. For some related higher-order PDEs in N dimensions uniform boundedness of global solutions of the Cauchy problem are established. As another related application, the well-posed Burnett-type equations, which are a higher-order extension of the classic Navier-Stokes equations, are studied. As a simple illustration, a generalization of the famous Leray-Prodi-Serrin-Ladyzhenskaya regularity results is obtained.
Galaktionov Artem V.
Mitidieri Enzo
Pohozaev S. I.
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