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Backward uniqueness for general parabolic operators in the whole space

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Abstract

We prove the backward uniqueness for general parabolic operators of second order in the whole space under assumptions that the leading coefficients of the operator are Lipschitz and their gradients satisfy certain decay conditions. The point is that the decay rate is related to the exponential growth rate of the solution, which is quite different from the case of the half-space (Wu and Zhang in Commun Contemp Math 18(1):1550011, 2016). This result extends in some ways a classical result of Lions and Malgrange (Math Scand 8:277–286, 1960 ) and a recent result of Wu and Zhang (2016).

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Acknowledgements

We wish to thank the referee for valuable comments, which are very helpful to this paper. J. Wu is supported by NSFC under grant 11601373. L. Zhang is partially supported by NSFC under grant 11471320 and 11631008.

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Authors and Affiliations

  1. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

    Jie Wu

  2. The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

    Jie Wu

  3. Hua Loo-Keng Key Laboratory of Mathematics, Institute of Mathematics, AMSS, and School of Mathematical Sciences, UCAS, Beijing, 100190, China

    Liqun Zhang

Authors
  1. Jie Wu
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Correspondence to Jie Wu.

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Communicated by L. Caffarelli.

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Wu, J., Zhang, L. Backward uniqueness for general parabolic operators in the whole space. Calc. Var. 58, 155 (2019). https://doi.org/10.1007/s00526-019-1593-1

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  • DOI: https://doi.org/10.1007/s00526-019-1593-1

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