### Parabolic equations with critical nonlinearities

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.

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

#### Full Text:

FULL TEXT### Refbacks

- There are currently no refbacks.