Asymptotic behavior for a time-dependent semilinear reaction–diffusion equation with a mixed local–nonlocal operator

Guoqiang Wang; Yanbin Tang; Guoqiang Wang; Yanbin Tang

doi:10.3934/nhm.2026072

Networks and Heterogeneous Media

2026, Volume 21, Issue 5: 1924-1949. doi: 10.3934/nhm.2026072

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Research article

Asymptotic behavior for a time-dependent semilinear reaction–diffusion equation with a mixed local–nonlocal operator

Guoqiang Wang ,
Yanbin Tang ^,

School of Mathematics and Statistics, Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

Received: 19 June 2026 Revised: 27 August 2026 Accepted: 11 September 2026 Published: 20 September 2026

In this paper, we consider a semilinear reaction–diffusion equation with a mixed local–nonlocal spatial operator and a time integral operator. The equation involves the mixed local–nonlocal Laplacian operator $\mathcal{L} = -\Delta+(-\Delta)^{\alpha/2}$ and the Riemann–Liouville fractional integral operator $I^{1-\gamma}_{0|t}$. Our main goal is to characterize the large–time asymptotic behavior of global mild solutions.
- global mild solution,
- anomalous diffusion,
- reaction–diffusion,
- mixed local–nonlocal Laplacian operator,
- nonlinear memory
Citation: Guoqiang Wang, Yanbin Tang. Asymptotic behavior for a time-dependent semilinear reaction–diffusion equation with a mixed local–nonlocal operator[J]. Networks and Heterogeneous Media, 2026, 21(5): 1924-1949. doi: 10.3934/nhm.2026072

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Abstract

In this paper, we consider a semilinear reaction–diffusion equation with a mixed local–nonlocal spatial operator and a time integral operator. The equation involves the mixed local–nonlocal Laplacian operator $\mathcal{L} = -\Delta+(-\Delta)^{\alpha/2}$ and the Riemann–Liouville fractional integral operator $I^{1-\gamma}_{0|t}$. Our main goal is to characterize the large–time asymptotic behavior of global mild solutions.

References

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