Blowup solutions for a reaction-diffusion system with exponential nonlinearities

We consider the following parabolic system whose nonlinearity has no gradient structure: {partial derivative(t)u = Lambda u + e(pv), partial derivative(t)v = mu Lambda v + e(qu), u(., 0) = u(0), v(., 0) = v(0), p, q, mu > 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. (C) 2018 Elsevier Inc. All rights reserved.