Parabolic equations with critical nonlinearities

Jan W. Cholewa, Tomasz Dłotko

DOI: http://dx.doi.org/10.12775/TMNA.2003.019

Abstract


As well known the problem of global continuation of solutions
to semilinear parabolic equations is completely solved when the nonlinear
term is subordinated to an $\alpha$-power of the main linear
operator with $\alpha\in[0,1)$. In this paper we study three
examples of {\it critical problems} in which the mentioned subordination takes place
with $\alpha=1$, i.e. the nonlinearity has the same {\it order of magnitude}
as the linear main part. We use specific techniques of proving global
solvability that fit well the considered examples for which
general abstract methods fail.

Keywords


Global solutions; critical nonlinearity; Navier-Stokes system

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