an:04025971
Zbl 0631.35044
Saut, Jean-Claude; Scheurer, Bruno
Unique continuation for some evolution equations
EN
J. Differ. Equations 66, 118-139 (1987).
00152622
1987
j
35G10 35K25 35B60
evolution operator; unique continuation property; Carleman estimates; dispersive-dissipative equations
Let L be an evolution operator acting on functions defined on some connected open set \({\mathcal O}\) of \({\mathbb{R}}^{n+1}={\mathbb{R}}^ n_ x\times {\mathbb{R}}_ t\). L is said to have the unique continuation property if every solution u of \(Lu=0\) which vanishes one some nonempty open set \(\omega\) of \({\mathcal O}\) vanishes in the horizontal component of \(\omega\) in \({\mathcal O}\) which is the union of all open segments \(t=cons\tan t\) in \({\mathcal O}\) which contains a point of \(\omega\). In the first section of this paper the authors prove a unique continuation theorem when L is a second order parabolic equation with coefficients not always smooth. The proof is the application of extended Carleman estimates for second order elliptic operators. Next sections, they discuss the parabolic equations with \(\Delta^ m\) as principal part and a more general class of dispersive-dissipative equations.
Y.Ebihara