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Periodic and almost periodic evolution flows and their stability on non-compact Einstein manifolds and applications

Tom 129 / 2022

Thieu Huy Nguyen, Thi Van Nguyen, Truong Xuan Pham, Thi Ngoc Ha Vu Annales Polonici Mathematici 129 (2022), 147-174 MSC: Primary 35Q30; Secondary 35B10, 58J35. DOI: 10.4064/ap210811-25-6 Opublikowany online: 28 September 2022

Streszczenie

Considering the evolution equations of parabolic type on a non-compact Einstein manifold $(\mathcal M,g)$ with negative Ricci curvature tensor, we establish results on the existence, uniqueness of time-periodic (on the time half-axis) and almost periodic (on the whole time axis) mild solutions to such equations. We develop a general framework for evolution equations on $(\mathcal M,g)$. Using certain dispersive and smoothing properties of semigroups, we construct a bounded (in time) mild solution and prove a Massera-type theorem for the linearized equations on the half-line using a mean-ergodic method to obtain the existence and uniqueness of periodic {and almost periodic solutions} to evolution equations. Next, using fixed point arguments, we can pass from linear equations to semilinear equations to prove the existence, uniqueness and stability of periodic solutions. Lastly, we apply our abstract results to establish the existence and stability of periodic and almost periodic solutions to Stokes and Navier–Stokes equations as well as to semilinear vectorial heat equations with rough coefficients on $(\mathcal M,g)$.

Autorzy

  • Thieu Huy NguyenSchool of Applied Mathematics and Informatics
    Hanoi University of Science and Technology
    Vien Toan ung dung va Tin hoc
    Dai hoc Bach khoa Hanoi
    1 Dai Co Viet
    Hanoi, Vietnam
    e-mail
  • Thi Van NguyenFaculty of Information Technology
    Department of Mathematics
    Thuyloi University
    Khoa Cong nghe Thong tin
    Bo mon Toan, Dai hoc Thuy loi
    175 Tay Son, Dong Da
    Hanoi, Vietnam
    e-mail
  • Truong Xuan PhamFaculty of Pedagogy
    VNU University of Education
    Vietnam National University
    144 Xuan Thuy, Cau Giay
    Hanoi, Vietnam
    e-mail
  • Thi Ngoc Ha VuSchool of Applied Mathematics and Informatics
    Hanoi University of Science and Technology
    Vien Toan ung dung va Tin hoc, Dai hoc Bach khoa Hanoi
    1 Dai Co Viet
    Hanoi, Vietnam
    e-mail

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