Well-posedness and blow-up for a two-component Degasperis-Procesi equation with infinitely fast propagating solutions

TL;DR

This paper studies a two-component Degasperis-Procesi equation, proving local well-posedness, analyzing blow-up scenarios, and showing infinite speed of propagation with exponential decay of solutions.

Contribution

It establishes the local well-posedness in Sobolev spaces and characterizes the blow-up behavior and infinite propagation speed for the two-component Degasperis-Procesi equation.

Findings

Solutions do not remain compactly supported for positive times.

Solutions propagate with infinite speed from compactly supported initial data.

Solutions decay exponentially fast during their existence.

Abstract

In this paper, a two-component variant of the Degasperis-Procesi equation on the real line is discussed. Applying Kato's theory, we first prove the local well-posedness for the equation under consideration in $H^s\times H^{s-1}$, for $s\geq 2$. Second we establish the precise blow-up scenario. For compactly supported initial data, we show that the associated solution does not have compact support for any positive time; the localized initial disturbance propagates with an infinite speed. Although the solution is no longer compactly supported we prove that it decays at an exponentially fast rate for the duration of its existence.