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Some backward uniqueness results. (English) Zbl 0622.35029

This paper establishes backward uniqueness and unique continuation results for nonlinear parabolic equations and inequalities. There are two principal results. One establishes backward uniqueness for inequalities of the type \[ \| du/dt+\mu A(t)u(t)\|_ H\leq n(t)\| u(t)\|_{D[A^{1/2}(t)]}\quad, \] where \(\{A(t)\}_{t\geq 0}\) is a family of self-adjoint unbounded linear operators on a Hilbert space H and n(t) is a square integrable function. The second result shows that for u not identically zero then there exist \(a,b>0\), such that \(| u(t)| \geq C \exp [-Re \mu (1+b)at]\); this is the unique continuation result.
The results are applied to the Navier-Stokes equations, equations for a heat conducting viscous fluid, MHD equations, and the Kuramoto- Sivashinsky equations on both bounded and unbounded spatial domains; \(L^ 2\) integrability is always assumed.
Reviewer: B.Straughan

MSC:

35K55 Nonlinear parabolic equations
35Q30 Navier-Stokes equations
35B60 Continuation and prolongation of solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
34G10 Linear differential equations in abstract spaces
35B40 Asymptotic behavior of solutions to PDEs
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