S. Vernier Piro

Decay bounds for solutions of second order parabolic problems and their derivatives IV

This article continues the derivation of explicit solution and gradient bounds for classes of second order nonlinear parabolic initial-boundary value problems. Earlier results were restricted to domains with boundaries of nonnegative average curvature. The purpose of the present article is to remove the curvature restriction and to extend the results to more general parabolic problems.

On Extinction Phenomena for Parabolic Systems

We investigate the extinction phenomena for some linear combinations of components of the vector-valued solutions to classes of semilinear parabolic systems. The crucial assumption on simultaneous splitting of the matrix-valued elliptic operators and the nonlinear source term allow us to uncouple the systems into a linear part and a scalar nonlinear equation depending on the solutions of the linear part. We propose necessary conditions and sufficient conditions on the existence of the extinction time for the solutions. We recapture as particular case previous results and apply our abstract theorem to a class of 3 × 3 systems appearing as models in chemical engineering.

Continuous dependence results for parabolic problems under Robin boundary conditions

We investigate continuous dependence on initial data for solutions of a nonlinear parabolic problem, when Robin conditions are prescribed on the boundary ∂ Ω × (t > 0), Ω a bounded R 2 domain.