We consider the Cauchy problem for a semilinear parabolic equation with a nonlinearity which is critical in the Joseph-Lundgren sense. We find the grow-up rate of solutions that approach a singular steady state from below as $t\to\infty$. The grow-up rate in the critical case contains a logarithmic term which does not appear in the Joseph-Lundgren supercritical case, making the calculations more delicate.

We consider, in a bounded and smooth domain, Maxwell's equations with a delay term in the boundary or in the internal feedbacks. Under suitable assumptions we obtain exponential stability results. Some instability examples are also given.

We study in this work the existence of minimizing solutions to the critical-power type equation $\triangle _{\textbf{g}}u+h.u=f .u^{\frac{n+2}{n-2}} $ on a compact Riemannian manifold in the limit case normally not solved by variational methods. For this purpose, we use a concept of "critical function" that was originally introduced by E. Hebey and M. Vaugon for the study of the second best constant in the Sobolev embeddings. Along the way, we prove an important estimate concerning concentration phenomenas when $f$ is a non-constant function.