Liouville theorems and blow up behaviour in semilinear reaction diffusion systems
Daniele Andreucci, Miguel A. Herrero, Juan J. L. Velázquez · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 1997
This paper is concerned with positive solutions of the semilinear system: \tag{S} \left\{\begin{matrix} u_{t} = \Delta u + \upsilon ^{p}, & p \geq 1, \\ \upsilon _{t} = \Delta \upsilon + u^{q}, & q \geq 1, \\ \end{matrix}\right. which blow up at x = 0 and t = T 0 . We then use (1) to derive a complete classification of blow up patterns. This last result is achieved by means of a parabolic Liouville theorem which we retain to be of some independent interest. Finally, we prove the existence of solutions of (S) exhibiting a type of asymptotics near blow up which is qualitatively different from those that hold for the scalar case.