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We consider the following parabolic system whose nonlinearity has no gradientstructure: $$\left\{\begin{array}{ll} \partial_t u = \Delta u + e^{pv}, \quad &\partial_t v = \mu \Delta v + e^{qu}, u(\cdot, 0) = u_0, \quad & v(\cdot, 0) =v_0, \end{array}\right. \quad p, q, \mu > 0, $$ in the whole space$\mathbb{R}^N$. We show the existence of a stable blowup solution and obtain acomplete description of its singularity formation. The construction relies onthe reduction of the problem to a finite dimensional one and a topologicalargument based on the index theory to conclude. In particular, our analysisuses neither the maximum principle nor the classical methods based onenergy-type estimates which are not supported in this system. The stability isa consequence of the existence proof through a geometrical interpretation ofthe quantities of blowup parameters whose dimension is equal to the dimensionof the finite dimensional problem.

Blowup solutions for a reaction-diffusion system with exponential nonlinearities