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.
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机译：我们考虑以下非线性无梯度结构的抛物线系统：$$ \ left \ {\ begin {array} {ll} \ partial_t u = \ Delta u + e ^ {pv}，\ quad＆\ partial_t v = \ mu \ Delta v + e ^ {qu}，u（\ cdot，0）= u_0，\ quad＆v（\ cdot，0）= v_0，\ end {array} \ right。 \ quad p，q，\ mu> 0，$$在整个空间$ \ mathbb {R} ^ N $中。我们显示了稳定的爆破解决方案的存在，并获得了其奇异性形成的完整描述。构造依赖于将问题简化为有限维数和基于索引理论得出的拓扑参数来得出结论。特别是，我们的分析既不使用最大原理，也不使用基于能量类型估计的经典方法，而该方法在该系统中不受支持。稳定性是通过对尺寸等于有限维问题的维的爆破参数的数量进行几何解释来证明存在的结果。
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