Dynamics near the ground state for the Sobolev critical Fujita type heat equation in 6D

2024/12/05 by Junichi Harada, Harada, Junichi
Mathematics · #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.2412.04049

Abstract

This paper investigates the asymptotic behavior of solutions to u_t=Δu+|u|^p-1u in the Sobolev critical case. Our main result is a classification of the dynamics near the ground states in the six dimensional case. It is shown that if the initial data u₀∈ H¹(ℝ⁶) satisfies ‖u₀-\sf Q‖ _H¹(ℝ⁶)≪1, then the solution falls into one of the following three scenarios: 1) It is globally defined and converge to one of the ground states as t→∞. 2) It is globally defined and converge to 0 in H¹(ℝ⁶) as t→∞. 3) It exhibits finite time blowup with a type I rate. This paper extends the classification result in the case n≥7, previously obtained by Collot-Merle-Raphaël, to the borderline case n=6.

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