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

Abstract

We consider in this note the semilinear heat system $${\partial }_{t}u=\Delta u+f(v),{\partial }_{t}v=\mu \Delta v+g(u),\mu >0,$$ where the nonlinearity has no gradient structure taking of the particular form $$f(v)=v|v{|}^{p-1}\text{and}g(u)=u|u{|}^{q-1}\text{with}p,q>1,$$ or $$f(v)=e^{pv}\text{and}g(u)=e^{qu}\text{with}p,q>0.$$ We exhibit type I blowup solutions for this system and give a precise description of its blowup profiles. The method relies on two-step procedure: the reduction of the problem to a finite dimensional one via a spectral analysis, 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 [Ann. IHP 2018] and [JDEs 2018].