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

Abstract

We consider in this note the semilinear heat system $$\partial_t u = \Delta u + f(v), \quad \partial_t v = \mu\Delta v + g(u), \quad \mu > 0,$$ where the nonlinearity has no gradient structure taking of the particular form $$f(v) = v|v|^{p-1} \quad \text{and}\quad g(u) = u|u|^{q-1} \quad \text{with} \quad p, q > 1, $$ or $$f(v) = e^{pv}\quad \text{and} \quad g(u) = e^{qu} \quad \text{with} \quad p,q > 0.$$ 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].