Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Optimal Liouville-type theorems for a parabolic system

Pages: 399 - 409, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.399

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Quoc Hung Phan - Institute of Research and Development, Duy Tan University, Da Nang, Vietnam (email)

Abstract: We prove Liouville-type theorems for a parabolic system in dimension $N=1$ and for radial solutions in all dimensions under an optimal Sobolev growth restriction on the nonlinearities. This seems to be the first example of a Liouville-type theorem in the whole Sobolev subcritical range for a parabolic system (even for radial solutions). Moreover, this also seems to be the first application of the Gidas-Spruck technique to a parabolic system.

Keywords:  Liouville-type theorem, non-cooperative parabolic system, decay estimate, singularity estimate.
Mathematics Subject Classification:  Primary: 35B53, 35B33; Secondary: 35K55.

Received: September 2013;      Revised: June 2014;      Available Online: August 2014.