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Unique continuation for parabolic equations. (English) Zbl 0852.35055

From introduction: “Let \(u\) be a solution of the heat equation \[ {\partial u\over \partial t}= \Delta u+ \sum^n_{i= 1} b_i (x, t)\cdot {\partial u\over \partial x_i}+ c(x, t) u \] for \(x\in \mathbb{R}^n\) and \(t> 0\). Let \(c\) and \(b= (b_1, b_2,\dots, b_n)\) be uniformly bounded functions, i.e., \[ \sup_{x\in \mathbb{R}^n, t> 0} |c(x, t)|+ |b(x, t)|\leq M \] for some constant \(M> 0\). We show
Theorem. Suppose that both \(u\) and \(\nabla u\) are uniformly bounded. If \(u\) vanishes of infinite order in space-time at any point in \(\mathbb{R}^n\times (0, \infty)\), then \(u\) is identically zero.
We say that a function \(u\) vanishes of infinite order in space-time at \((x_0, t_0)\) if \(u(x, t)= O(|x- x_0|^2+ |t- t_0|)^K\) for any integer \(K> 0\), for \((x, t)\) near \((x_0, t_0)\), \(t< t_0\).”
A similar result holds if \(u\) is defined in a bounded convex domain in \(\mathbb{R}^n\) and satisfies zero boundary conditions.
Reviewer: V.Mustonen (Oulu)

MSC:

35K10 Second-order parabolic equations
35B60 Continuation and prolongation of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
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References:

[1] Almgren F.J., in ”Minimal Submanifolds and Geodesics” (1979)
[2] Garofalo N., Indiana Univ. Math.J. 35 pp 254– (1986)
[3] Garogalo N., A gemetric–variational approach, Comm. Pure Appl. Math. 40 pp 347– (1987) · Zbl 0674.35007
[4] Hamilton, R.S., In Analysis and Geomatery 1 pp 88– (1993)
[5] Hamilton, R.S., In Analysis and Geometry, 1 pp 100– (1993)
[6] Lin, F.H., Pure Appl. Math. 43 pp 125– (1990)
[7] Russell, D.L., In Appl. Math. 52 (1973) pp 198–
[8] Struwe, M., Diff. Geom. 28 pp 485– (1988)
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