On the uniqueness for the heat equation on complete Riemannian manifolds
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On the uniqueness for the heat equation on complete Riemannian manifolds Fei He1 · Man‑Chun Lee2 Received: 16 April 2020 / Accepted: 8 September 2020 © Springer Nature B.V. 2020
Abstract We prove some uniqueness result for solutions to the heat equation on Riemannian manifolds. In particular, we prove the uniqueness of Lp solutions with 0 < p < 1 and improves the L1 uniqueness result of Li (J Differ Geom 20:447–457, 1984) by weakening the curvature assumption. Keywords Uniqueness problem · Heat equation on manifolds · Complete noncompact manifolds
1 Introduction In this article, we consider the uniqueness problem for solutions to the heat equation on complete Riemannian manifolds (M, g):
(𝜕t − Δ)f = 0, where Δ is the Laplace–Beltrami operator with respect to the metric g. It is well-known that uniqueness may fail in general unless we restrict the solutions on some suitable class of functions. A example is the set of functions bounded from below. In [7], the uniqueness of nonnegative solutions to the heat equation has been established under the quadratic Ricci lower bound assumption
Ric(x) ≥ −C(r(x) + 1)2 ,
(1.1)
where r(x) is the geodesic distance from some fixed point and C is a nonnegative constant. Another typical of class where uniqueness holds is the set of functions with appropriate growth rate in the spirit of [10]. For solutions with L2 integrals on geodesic balls or * Man‑Chun Lee [email protected] Fei He [email protected] 1
School of Mathematical Science, Xiamen University, 422 S. Siming Rd., Xiamen 361000, Fujian, People’s Republic of China
2
Mathematics Department, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA
13
Vol.:(0123456789)
Annals of Global Analysis and Geometry
parabolic cylinders growing under certain rate, the uniqueness was proved in [1, 3]. The same result holds if L2 is replaced by Lp with 1 < p ≤ 2 , and for a special class of manifolds when p = 1 [8]. These results imply uniqueness for solutions with suitable pointwise growth rate, provided that the manifold has some volume growth constraint. A case of particular interest is for bounded solutions, see [2] for a survey. Our first theorem is an improvement in the results in [1, 3]. Namely, we allow the integral to be weighted by a positive power of the time variable. We will also demonstrate an example in Sect. 3.
Theorem 1.1 Let M be a complete Riemannian manifold, and let f(x, t) be a nonnegative subsolution to the heat equation on M × (0, 1] with initial data f (x, 0) = 0 in the sense of 2 Lloc (M) . Suppose for some point q ∈ M , and constant a > 0, �0
1
ta
�Bq (r)
f 2 ≤ eL(r) ,
∀r > 0,
where L(r) is a positive nondecreasing function satisfying
Then f ≡ 0 on M × (0, 1].
∫1
∞
r dr = ∞. L(r)
(1.2)
In [5], Li considered the uniqueness for Lp solutions to the heat equation. When p > 1 , the uniqueness holds without further assumption. However when p = 1 the uniqueness may fail on sufficiently negatively curved manifolds, it was proved in [5] that the uniqueness for L1 soluti
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