Relaxation, Perturbation and Entropy Methods

Abstract

Solutions of nonlinear parabolic equations and systems, presented in Chap. 1, can exhibit various and sometimes complex behavior, a phenomena usually called pattern formation. In which circumstances can such complex behavior happen? A first answer is given in this chapter by indicating some conditions for relaxation to trivial steady states; then nothing interesting can happen! We present relaxation results by perturbation methods (small nonlinearity) and entropy methods. Indeed, before we can understand how patterns occur in parabolic systems, a necessary step is to understand why patterns should not appear in principle! Solutions of parabolic equations undergo regularization effects that lead them to constant or simple states. Several asymptotic results are stated in this direction, with Dircihlet or Neumann boundary conditions. Important tools are Poincare inequalities and spectral gap.