Oscillatory behavior of solutions to the critical Fujita equation in 6D

2025/11/22 by Harada, Junichi
Mathematics · #Navier-Stokes equation solutions #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations

paper · doi:10.48550/arxiv.2511.17891

Abstract

Long time dynamics of solutions to the 6D energy critical heat equation u_t=Δu+|u|^p-1u on \R⁶×(0,∞) is investigated. It is shown that there exists a radially symmetric global solution u(x,t)∈ C([0,∞); H¹(\R⁶)) of the form u(x,t) = λ(t)^-(n-2)/(2) \sf Q(\tfracxλ(t)) + error (x,t), where the function \( λ(t) \) satisfies: \beginitemize \item \dislim_t→∞‖error(⋅,t)‖ _Hx¹(\R⁶)=0, \item \dis\liminf_t→∞λ(t)=0, \item \dis\limsup_t→∞λ(t)=∞. \enditemize The solutions constructed here demonstrate that the dynamical behavior in \( H¹(ℝⁿ) \) can differ significantly from the behavior in \( H¹(ℝⁿ) \).

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