The asymptotic behavior at infinite-time of the Sobolev flow will be studied. The Sobolev flow is the gradient flow associated with the Sobolev inequality and is described 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 local boundedness is the new ingredient obtained for the doubly nonlinear parabolic equation and the key for studying the concentration phenomenon of volume and energy at infinitetime of the Sobolev flow. This is based on a collaborative work with Tuomo Kuusi in University of Helsinki, Finland and Kenta Nakamura in Kumamoto University.

 

References:

T. Kuusi, M. Misawa, K. Nakamura: J. Geom. Anal. 30 (2020) 1918-1964; J. Differ. Equ. 279 (2021) 245-281.

M. Misawa, K. Nakamura: Adv. Calc. Var. (2021); J. Geom. Anal. 33: 33 (2023).

M. Misawa, K. Nakamura, Md Abu Hanif Sarkar: Nonlinear Differ. Eqn.Appl. 30 ; 43 (2023).

M. Misawa: Calc. Var. 62 (2023), no. 9, No. 265.