带时间依赖记忆核和非局部扩散的热传导方程解的长时间行为
Long-Term Behavior of Heat Conduction Equation Solutions with Time-Dependent Memory and Non-Local Diffusion
摘要: 本文分析了带时间依赖记忆核和非局部扩散的热传导方程解的长时间行为。 当非线性项满足次临界增长条件时，在时间依赖空间中，首先利用 Galerkin 逼近法得到了解的适定性和正则性，进而利用分解技巧和积分估计法证明了时间依赖全局吸引子的存在性。
Abstract: In this paper,we analysize the long-time behavior of solutions for the non-local diffu- sion equation with time-dependent memory kernel. When nonlinear term adheres to subcritical growth condition and in the time-dependent space by use of the Galerkin approximation method, the well-posedness and the regularity of the solution arise achieved. And then the existence of the time-dependent global attractors is proved by applying the delicate integral estimation method and decom- position technique.
文章引用：黄晶朋, 汪璇. 带时间依赖记忆核和非局部扩散的热传导方程解的长时间行为[J]. 应用数学进展, 2023, 12(12): 5267-5291. https://doi.org/10.12677/AAM.2023.1212517

参考文献

[1] Caraballo, T., Garrido-Atienza, M.J., Schmalfuß, B. and Valero, J. (2010) Global Attrac- tor for a Non-Autonomous Integrodifferential Equation in Materials with Memory. Nonlinear Analysis, 73, 183-201. [Google Scholar] [CrossRef] 
[2] Caraballo, T. and Real, J. (2004) Attractors for 2D-Navier-Stokes Models with Delays. Journal of Differential Equations, 205, 271-297. [Google Scholar] [CrossRef] 
[3] Giorgi, C., Pata, V. and Marzochi, A. (1998) Asymptotic Behavior of a Semilinear Problem in Heat Conduction with Memory. Nonlinear Differential Equations and Applications, 5, 333-354. [Google Scholar] [CrossRef] 
[4] Chipot, M. and Rodrigues, J.F. (1992) On a Class of Nonlocal Nonlinear Elliptic Problems. ESAIM, 26, 447-467. [Google Scholar] [CrossRef] 
[5] Chipot, M., Valente, V. and Caffarelli, G.V. (2003) Remarks on a Nonlocal Problem Involving the Dirichlet Energy. Rendiconti del Seminario Matematico della Universita` di Padova, 110, 199-220.
[6] Xu, J.H., Zhang, Z.C. and Caraballo, T. (2021) Non-Autonomous Nonlocal Partial Differential Equations with Delay and Memory. Journal of Differential Equations, 270, 505-546. [Google Scholar] [CrossRef] 
[7] Dafermos, C.M. (1970) Asymptotic Stability in Viscoelasticity. Archive for Rational Mechanics and Analysis, 37, 297-308. [Google Scholar] [CrossRef] 
[8] Xu, J.H., Caraballo, T. and Valero, J. (2022) Asymptotic Behavior of a Semilinear Problem in Heat Conduction with Long Time Memory and Non-Local Diffusion. Journal of Differential Equations, 327, 418-447. [Google Scholar] [CrossRef] 
[9] Conti, M., Danese, V., Giorgi, C. and Pata, V. (2018) A Model of Viscoelasticity with Time- Dependent Memory Kernels. American Journal of Mathematics, 140, 349-389. [Google Scholar] [CrossRef] 
[10] Sun, Y. and Yang, Z.J. (2022) Longtime Dynamics for a Nonlinear Viscoelastic Equation with Time-Dependent Memory Kernel. Nonlinear Analysis: Real World Applications, 64, Article 103432. [Google Scholar] [CrossRef] 
[11] Li, Y.N. and Yang, Z.J. (2023) Exponential Attractor for the Viscoelastic Wave Model with Time-Dependent Memory Kernels. Journal of Dynamics and Differential Equations, 35, 679- 707. [Google Scholar] [CrossRef] 
[12] Conti, M., Pata, V. and Temam, R. (2013) Attractors for Processes on Time-Dependent Spaces. Applications to Wave Equations. Journal of Differential Equations, 255, 1254-1277. [Google Scholar] [CrossRef] 
[13] Borini, S. and Pata, V. (1999) Uniform Attractors for a Strongly Damped Wave Equation with Linear Memory. Asymptotic Analysis, 20, 263-277.
[14] [ Dafermos, C.M. (1970) Asymptotic Stability in Viscoelasticity. Archive for Rational Mechanics and Analysis, 37, 297-308. [Google Scholar] [CrossRef] 
[15] Pata, V. and Squassina, M. (2005) On the Strongly Damped Wave Equation. Communications in Mathematical Physics, 253, 511-533. [Google Scholar] [CrossRef] 
[16] Conti, M. and Pata, V. (2014) Asymptotic Structure of the Attractor for Processes on Time- Dependent Spaces. Nonlinear Analysis: Real World Applications, 19, 1-10. [Google Scholar] [CrossRef] 
[17] Gatti, S., Miranville, A., Pata, V. and Zelik, S. (2008) Attractors for Semi-Linear Equations of Viscoelasticity with Very Low Dissipation. Rocky Mountain Journal of Mathematics, 38, 1117-1138. [Google Scholar] [CrossRef] 
[18] Pata, V. and Zucchi, A. (2001) Attractors for a Damped Hyperbolic Equation with Linear Memory. Advances in Mathematical Sciences and Applications, 11, 505-529.
[19] Chepyzhov, V.V. and Vishik, M.I. (2002) Attractors for Equations of Mathematical Physics. In: American Mathematical Society Colloquium Publications, Vol. 49, American Mathematical Society, Providence, RI. [Google Scholar] [CrossRef] 
[20] Simon, J. (1987) Compact Sets in the Space Lp(0, T ; B). Annali di Matematica Pura ed Ap- plicata, 146, 65-96. [Google Scholar] [CrossRef] 
[21] Meng, F.J., Wu, J. and Zhao, C.X. (2019) Time-Dependent Global Attractor for Extensible Berger Equation. Journal of Mathematical Analysis and Applications, 469, 1045-1069. [Google Scholar] [CrossRef] 
[22] Liu, Y.F. (2015) Time-Dependent Global Attractor for the Nonclassical Diffusion Equations. Applicable Analysis, 94, 1439-1449. [Google Scholar] [CrossRef] 

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