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On a class of infinite-dimensional Hamiltonian systems with asymptotically periodic nonlinearities

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Abstract

The authors study the existence of homoclinic type solutions for the following system of diffusion equations on ℝ × ℝN

$$ \left\{ \begin{gathered} \partial _t u - \Delta _x u + b \cdot \Delta _x u + au + V(t,x)v = H_v (t,x,u,v), \hfill \\ - \partial _t v - \Delta _x v - b \cdot \Delta _x v + av + V(t,x)u = H_u (t,x,u,v), \hfill \\ \end{gathered} \right. $$

where z = (u, v): ℝ × ℝN → ℝm × ℝm, a > 0, b = (b 1, …, b N ) is a constant vector and V ε C(ℝ × ℝN, ℝ), H ε C 1 (ℝ × ℝN × ℝ2m, ℝ). Under suitable conditions on V(t,x) and the nonlinearity for H(t, x, z), at least one non-stationary homoclinic solution with least energy is obtained.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, China

    Minbo Yang

  2. Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing, 100190, China

    Zifei Shen & Yanheng Ding

Authors
  1. Minbo Yang
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  2. Zifei Shen
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  3. Yanheng Ding
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Correspondence to Minbo Yang.

Additional information

Project supported by the National Natural Science Foundation of China (No. 10971194), the Zhejiang Provincial Natural Science Foundation of China (Nos. Y7080008, R6090109) and the Zhejiang Innovation Project (No. T200905).

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Yang, M., Shen, Z. & Ding, Y. On a class of infinite-dimensional Hamiltonian systems with asymptotically periodic nonlinearities. Chin. Ann. Math. Ser. B 32, 45–58 (2011). https://doi.org/10.1007/s11401-010-0625-0

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  • DOI: https://doi.org/10.1007/s11401-010-0625-0

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