Parabolic equations with critical nonlinearities

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 critical problems in which the mentioned subordination takes place with $\alpha=1$, i.e. the nonlinearity has the same 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.