Acta mathematica scientia,Series B ›› 2019, Vol. 39 ›› Issue (3): 645-668.doi: 10.1007/s10473-019-0303-6

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Le CHEN1, Kunwoo KIM2   

  1. 1. Department of Mathematical Sciences, University of Nevada, Las Vegas, 4505 S. Maryland Pkwy, Las Vegas, Nevada, 89154-4020, USA;
    2. Department of Mathematics, Pohang University of Science and Technology, 77 Cheongam-Ro, Nam-Gu, Pohang, Gyeongbuk, 37673, Korea
  • Received:2018-02-08 Online:2019-06-25 Published:2019-06-27
  • Supported by:
    The second author is supported by the National Research Foundation of Korea (NRF-2017R1C1B1005436) and the TJ Park Science Fellowship of POSCO TJ Park Foundation.

Abstract: In this article, we study the nonlinear stochastic heat equation in the spatial domain Rd subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and nonnegative-definite function that satisfies Dalang's condition. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. With these moment bounds, we first establish some necessary and sufficient conditions for the phase transition of the moment Lyapunov exponents, which extends the classical results from the stochastic heat equation on Zd to that on Rd. Then, we prove a localization result for the intermittency fronts, which extends results by Conus and Khoshnevisan[9] from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al[17] using different techniques.

Key words: Stochastic heat equation, moment estimates, phase transition, intermittency, intermittency front, measure-valued initial data, moment Lyapunov exponents

CLC Number: 

  • 60H15