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

Abstract

AbstractIn this note, we consider the semilinear heat system\partial_{t}u=\Delta u+f(v),\quad\partial_{t}v=\mu\Delta v+g(u),\quad\mu>0,where the nonlinearity has no gradient structure taking of the particular formf(v)=v\lvert v\rvert^{p-1}\quad\text{and}\quad g(u)=u\lvert u\rvert^{q-1}\quad% \text{with }p,q>1,orf(v)=e^{pv}\quad\text{and}\quad g(u)=e^{qu}\quad\text{with }p,q>0.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, 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, 3, 381–450].