Existence, continuation, persistence and dynamics of solutions for a generalized 0-Holm-Staley equation

TL;DR

This paper investigates the existence, uniqueness, and dynamics of solutions for a family of non-local equations including the 0-Holm-Staley equation, revealing non-compact support, special solution behaviors, and conditions for global existence.

Contribution

It provides new results on the non-existence of compactly supported solutions, unique continuation, and global existence for the 0-Holm-Staley equation family.

Findings

Non-trivial solutions lack compact support.

Unique continuation results are established.

Global existence of solutions is proved.

Abstract

We consider a family of non-local evolution equations including the $0-$Holm-Staley equation. We show that the family considered does not posses compactly supported solutions as long as the initial data is non-trivial. Also, we prove different unique continuation results for the solutions of the family studied. In addition, some special solutions, such as peakons and kinks, are studied and their dynamics are analyzed. Persistence properties of the solutions are also investigated as well as we describe the scenario for the global existence of solutions of the $0-$Holm-Staley equation. In particular, the prove of global existence of solutions as well as our demonstrations for unique continuation results of solutions partially answer some questions pointed out in [A. A. Himonas and R. C. Thompson, Persistence properties and unique continuation for a generalized Camassa-Holm equation, J. Math. Phys., vol. 55, paper 091503, (2014)].