Unique continuation for some evolution equations. (English) Zbl 0631.35044

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.
Reviewer: Y.Ebihara


35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35B60 Continuation and prolongation of solutions to PDEs
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