Upper-semicontinuity of uniform attractors for the non-autonomous viscoelastic Kirchhoff plate equation with memory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911331869786112 |
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| author | Qin, Yuming Wang, Hongli |
| author_facet | Qin, Yuming Wang, Hongli |
| contents | This paper delves into the long-time dynamics of a non-autonomous viscoelastic Kirchhoff plate equation with memory effects, described by
$$
u_{t t}-Δu_{t t}+a_ε(t) u_t+αΔ^2 u-\int_0^{\infty} μ(s) Δ^2 u(t-s) \mathrm{d} s-Δu_t+f(u)=g(x,t),
$$
in bounded domain $Ω\subset \mathbb{R}^N$ with smooth boundary and nonlinear terms. Initially, the global existence of a weak solution that induces a continuous process is established. Subsequently, the existence of a uniform attractor is demonstrated in both subcritical and critical growth scenarios, utilizing operator techniques and an innovative analytical approach. Finally, the upper semicontinuity of the family of uniform attractors as the pert parameterurbation $ε\to 0^+$ is proven through delicate energy estimates and a contradiction argument. Our results not only extend classical attractor theory to more general non-autonomous viscoelastic systems but also resolve open questions regarding the limiting behavior of attractors in the presence of both memory and critical nonlinearity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19079 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Upper-semicontinuity of uniform attractors for the non-autonomous viscoelastic Kirchhoff plate equation with memory Qin, Yuming Wang, Hongli Analysis of PDEs This paper delves into the long-time dynamics of a non-autonomous viscoelastic Kirchhoff plate equation with memory effects, described by $$ u_{t t}-Δu_{t t}+a_ε(t) u_t+αΔ^2 u-\int_0^{\infty} μ(s) Δ^2 u(t-s) \mathrm{d} s-Δu_t+f(u)=g(x,t), $$ in bounded domain $Ω\subset \mathbb{R}^N$ with smooth boundary and nonlinear terms. Initially, the global existence of a weak solution that induces a continuous process is established. Subsequently, the existence of a uniform attractor is demonstrated in both subcritical and critical growth scenarios, utilizing operator techniques and an innovative analytical approach. Finally, the upper semicontinuity of the family of uniform attractors as the pert parameterurbation $ε\to 0^+$ is proven through delicate energy estimates and a contradiction argument. Our results not only extend classical attractor theory to more general non-autonomous viscoelastic systems but also resolve open questions regarding the limiting behavior of attractors in the presence of both memory and critical nonlinearity. |
| title | Upper-semicontinuity of uniform attractors for the non-autonomous viscoelastic Kirchhoff plate equation with memory |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.19079 |