Construction and stability of type i blowup solutions for non-variational semilinear parabolic systems

In this note, we consider the semilinear heat system (equation presented) where the nonlinearity has no gradient structure taking of the particular form (equation presented) We exhibit type I blowup solutions for this system and give a precise description of its blowup profiles. The method relies on a two-step procedure: The reduction of the problem to a finite-dimensional one via a spectral analysis, and then solving the finite-dimensional problem by a classical topological argument based on index theory. As a consequence of our technique, the constructed solutions are stable under a small perturbation of initial data. The results and the main arguments presented in this note can be found in our papers [T.-E. Ghoul, V. T. Nguyen and H. Zaag, Construction and stability of blowup solutions for a non-variational semilinear parabolic system, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 6, 1577-1630] and [M. A. Herrero and J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 19 (1992), no. 3, 381-450].