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Well-posedness and large time behavior for Cahn–Hilliard–Oono equation

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Abstract

In this paper, we are concerned with the well-posedness and large time behavior of Cauchy problem for Cahn–Hilliard equation in \(\mathbb {R}^n\) (\(n\in \mathbb {Z}^+,~n\ge 3\)). First, based on the higher-order norm estimates of solutions and the mollifier technique, we obtain the local well-posedness of strong solutions. Then, by using pure energy method, standard continuity argument together with negative Sobolev norm estimates, one proves the global well-posedness and time decay estimates provided that the \(H^{\left[ \frac{n}{2}\right] +1}\)-norm of initial data is sufficiently small.

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The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

The authors would like to thank the anonymous referee for helpful comments.

Funding

This work is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. N2205009).

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Authors and Affiliations

  1. College of Sciences, Northeastern University, Shenyang, 110004, People’s Republic of China

    Ning Duan & Jing Wang

  2. College of Sciences, Northeastern University, Shenyang, 110004, People’s Republic of China

    Xiaopeng Zhao

  3. College of Mathematics and Physics, WenZhou University, Wenzhou, 325035, People’s Republic of China

    Xiaopeng Zhao

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  1. Ning Duan
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  2. Jing Wang
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ND, JW and XZ wrote the main manuscript text. All authors reviewed the manuscript.

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Correspondence to Xiaopeng Zhao.

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Duan, N., Wang, J. & Zhao, X. Well-posedness and large time behavior for Cahn–Hilliard–Oono equation. Z. Angew. Math. Phys. 74, 226 (2023). https://doi.org/10.1007/s00033-023-02119-1

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  • DOI: https://doi.org/10.1007/s00033-023-02119-1

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