We shall study the asymptotic behavior at infinite-time of the Sobolev flow. The Sobolev flow describes the gradient flow associated with the Sobolev inequality and is given as a doubly nonlinear parabolic equation. We present the global existence for Cauchy-Dirichlet problem for the Sobolev flow, a boundedness, a positivity and a regularity of the solution. The behavior at infinity-time of the Sobolev flow is studied using the local boundedness of the solution. The local boundedness is the new ingredient obtained for the doubly nonlinear parabolic equation. The concentration phenomenon of volume and energy at infinite-time is shown by the local boundedness.
This is based on a collaborative work with Tuomo Kuusi in University of Helsinki, Finland and Kenta Nakamura in Kumamoto University.
References
[1] T. Kuusi, M. Misawa, K. Nakamura: J. Geom. Anal. 30 (2020) 1918-1964;
[2] T. Kuusi, M. Misawa, K. Nakamura: J. Differ. Equ. 279 (2021) 245-281;
[3] M. Misawa, K. Nakamura: Adv. Calc. Var. (2021);
[4] J. Geom. Anal. 33: 33 (2023);
[5] M. Misawa, K. Nakamura, Md Abu Hanif Sarkar: Nonlinear Differ. Eqn. Appl. 30 ; 43 (2023);
[6] M. Misawa: Calc. Var. (to appear)
