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We consider the following parabolic system whose nonlinearity has no gradientstructure: $$\left\{\begin{array}{ll} \partial_t u = \Delta u + |v|^{p-1}v,\quad & \partial_t v = \mu \Delta v + |u|^{q - 1}u,\\ u(\cdot, 0) = u_0, \quad& v(\cdot, 0) = v_0, \end{array}\right. $$ in the whole space $\mathbb{R}^N$,where $p, q > 1$ and $\mu > 0$. We show the existence of initial data such thatthe corresponding solution to this system blows up in finite time $T(u_0, v_0)$simultaneously in $u$ and $v$ only at one blowup point $a$, according to thefollowing asymptotic dynamics: $$\left\{\begin{array}{c} u(x,t)\sim\Gamma\left[(T-t) \left(1 + \dfrac{b|x-a|^2}{(T-t)|\log(T-t)|}\right)\right]^{-\frac{(p + 1)}{pq - 1}},\\ v(x,t)\sim \gamma\left[(T-t)\left(1 + \dfrac{b|x-a|^2}{(T-t)|\log (T-t)|}\right)\right]^{-\frac{(q + 1)}{pq- 1}}, \end{array}\right.$$ with $b = b(p,q,\mu) > 0$ and $(\Gamma, \gamma) =(\Gamma(p,q), \gamma(p,q))$. The construction relies on the reduction of theproblem to a finite dimensional one and a topological argument based on theindex theory to conclude. Two major difficulties arise in the proof: thelinearized operator around the profile is not self-adjoint even in the case$\mu = 1$; and the fact that the case $\mu \ne 1$ breaks any symmetry in theproblem. In the last section, through a geometrical interpretation ofquantities of blowup parameters whose dimension is equal to the dimension ofthe finite dimensional problem, we are able to show the stability of theseblowup behaviors with respect to perturbations in initial data.

Construction and stability of blowup solutions for a non-variational semilinear parabolic system