2021/08/26 by Mohandas Pillai, Pillai, Mohandas
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2108.12024
openalex publication_date 2021/08/26 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We consider the quintic, focusing semilinear wave equation on\n\ℝ1+3, in the radially symmetric setting, and construct infinite\ntime blow-up, relaxation, and intermediate types of solutions. More precisely,\nwe first define an admissible class of time-dependent length scales, which\nincludes a symbol class of functions. Then, we construct solutions which can be\ndecomposed, for all sufficiently large time, into an Aubin-Talentini (soliton)\nsolution, re-scaled by an admissible length scale, plus radiation (which solves\nthe free 3 dimensional wave equation), plus corrections which decay as time\napproaches infinity. The solutions include infinite time blow-up and relaxation\nwith rates including, but not limited to, positive and negative powers of time,\nwith exponents sufficiently small in absolute value. We also obtain solutions\nwhose soliton component has oscillatory length scales, including ones which\nconverge to zero along one sequence of times approaching infinity, but which\ndiverge to infinity along another such sequence of times. The method of proof\nis similar to a recent wave maps work of the author, which is itself inspired\nby matched asymptotic expansions.\n