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September  2007, 19(3): 515-529. doi: 10.3934/dcds.2007.19.515

On unique continuation for the modified Euler-Poisson equations

A. Alexandrou Himonas 1, , Gerard Misiołek 2, and  Feride Tiǧlay 3,

1. 

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States
2. 

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-5683, United States
3. 

Department of Mathematics, University of New Orleans, New Orleans, LA 70148, United States

Received  May 2007 Revised  June 2007 Published  July 2007

It is shown that if a classical solution $(u, n)$ of the modified Euler-Poisson equation (mEP) in one space dimension is such that $u$, $u_x$ and $n$ are initially decaying exponentially and for some later time the first component $u$ is also decaying exponentially, then $n$ must be identically equal to zero and $u$ must be a solution to the Burgers equation. In particular, if $n$ and $u$ are initially compactly supported then $n$ can not be compactly supported at any later time, unless $n$ is identically equal to zero and $u$ is a solution to the Burgers equation. It is also shown that the mEP equations are locally well-posed in $H^s \times H^{s-1}$ for $s>5/2$.
Keywords: local well-posedness, exponential decay, unique continuation, Burgers equation, Camassa-Holm equation, Modified Euler-Poisson equations, Sobolev spaces..
Mathematics Subject Classification: Primary: 35Q5.
Citation: A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete & Continuous Dynamical Systems, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515
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