Abstract
We consider the energy critical semilinear heat equation
and give a complete classification of the flow near the ground state solitary wave
in dimension \({d \ge 7}\), in the energy critical topology and without radial symmetry assumption. Given an initial data \({Q + \varepsilon_0}\) with \({\|\nabla \varepsilon_0\|_{L^2} \ll 1}\), the solution either blows up in the ODE type I regime, or dissipates, and these two open sets are separated by a codimension one set of solutions asymptotically attracted by the solitary wave. In particular, non self similar type II blow up is ruled out in dimension \({d \ge 7}\) near the solitary wave even though it is known to occur in smaller dimensions (Schweyer, J Funct Anal 263(12):3922–3983, 2012). Our proof is based on sole energy estimates deeply connected to Martel et al. (Acta Math 212(1):59–140, 2014) and draws a route map for the classification of the flow near the solitary wave in the energy critical setting. A by-product of our method is the classification of minimal elements around Q belonging to the unstable manifold.
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Collot, C., Merle, F. & Raphaël, P. Dynamics Near the Ground State for the Energy Critical Nonlinear Heat Equation in Large Dimensions. Commun. Math. Phys. 352, 215–285 (2017). https://doi.org/10.1007/s00220-016-2795-4
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DOI: https://doi.org/10.1007/s00220-016-2795-4