Parabolic Equations with Quadratic Growth in \(\mathbb {R}^{n}\)

  • Alain BensoussanEmail author
  • Jens Frehse
  • Shige Peng
  • Sheung Chi Phillip Yam
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 47)


We study here quasi-linear parabolic equations with quadratic growth in \(\mathbb {R}^{n}\). These parabolic equations are at the core of the theory of PDE; see Friedman (Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, 1964) [6], Ladyzhenskaya et al. (Translations of Mathematical Monographs. AMS, 1968) [4] for details. However, for the applications to physics and mechanics, one deals mostly with boundary value problems. The boundary is often taken to be bounded and the solution is bounded. This brings an important simplification. On the other hand, stochastic control theory leads mostly to problems in \(\mathbb {R}^{n}\). Moreover, the functions are unbounded and the Hamiltonian may have quadratic growth. There may be conflicts which prevent solutions to exist. In stochastic control theory, a very important development deals with BSDE (Backward Stochastic Differential Equations). There is a huge interaction with parabolic PDE in \(\mathbb {R}^{n}\). This is why, although we do not deal with BSDE in this paper, we use many ideas from Briand and Hu (Probab Theory Relat Fields 141(3–4):543–567, 2008) [1], Da Lio and Ley (SIAM J Control Optim 45(1):74–106, 2006) [2], Karoui et al. (Backward stochastic differential equations and applications, Princeton BSDE Lecture Notes, 2009) [3], Kobylanski (Ann Probab 28(2):558–602, 2000) [5]. Our presentation provided here is slightly innovative.



Alain Bensoussan’s research is supported by the National Science Foundation under grants DMS-1303775, DMS-1612880 and the Research Grants Council of the Hong Kong Special Administrative Region (CityU 500113, HKGRF CityU 11303316: Mean Field Control with Partial Information).

Phillip Yam acknowledges the financial supports from HKRGC GRF14301015 with the project title: Advance in Mean Field Theory, and Direct Grant for Research 2015/16 with project code: 4053141 offered by CUHK. He also acknowledges the financial support from Department of Statistics of Columbia University in the City of New York during the period he was a visiting faculty member.


  1. 1.
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    Da Lio F, Ley O (2006) Uniqueness results for second-order Bellman-Isaacs equations under quadratic growth assumptions and applications. SIAM J Control Optim 45(1):74–106MathSciNetCrossRefGoogle Scholar
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    Ladyzhenskaya OA, Solonnikov VA, Ural’ceva NN (1968) Linear and quasi-linear equations of parabolic type. Transl Math Monogr. AMSGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Alain Bensoussan
    • 1
    • 2
    Email author
  • Jens Frehse
    • 3
  • Shige Peng
    • 4
  • Sheung Chi Phillip Yam
    • 5
  1. 1.International Center for Decision and Risk Analysis, Jindal School of ManagementUniversity of Texas at DallasRichardsonUSA
  2. 2.College of Science and Engineering, Systems Engineering and Engineering ManagementCity University Hong KongKowloon TongHong Kong
  3. 3.Insitute for Applied MathematicsUniversity of BonnBonnGermany
  4. 4.Shandong UniversityJinanPeople’s Republic of China
  5. 5.Department of StatisticsThe Chinese University of Hong KongSha TinHong Kong

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