Blowup solutions for a reaction–diffusion system with exponential nonlinearities.

We consider the following parabolic system whose nonlinearity has no gradient structure: { ∂ t u = Δ u + e p v , ∂ t v = μ Δ v + e q u , u ( ⋅ , 0 ) = u 0 , v ( ⋅ , 0 ) = v 0 , p , q , μ > 0 , in the whole space R N . We show the existence of a stable blowup solution and obtain a complete description of its singularity formation. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. In particular, our analysis uses neither the maximum principle nor the classical methods based on energy-type estimates which are not supported in this system. The stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem. [ABSTRACT FROM AUTHOR]