Blowup solutions for a reaction–diffusion system with exponential nonlinearities

We consider the following parabolic system whose nonlinearity has no gradient structure: {∂_tu=Δu+e^pv,∂_tv=μΔv+e^qu,u(⋅,0)=u₀,v(⋅,0)=v₀,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.