在均匀扰动的非光滑区域中具有诺伊曼边界条件的反应扩散方程全局吸引子的一致\(L^{\infty}\)有界性
Uniform $L^{\infty}$-Boundedness of Global Attractors for Reaction-Diffusion Equations with Neumann boundary condition in Uniformly Perturbed Non-Smooth Domains
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中文总结 AI 辅助
研究一族在非光滑区域上具诺伊曼边界条件的半线性抛物方程，通过特定条件建立适定性与吸引子存在性，利用多种方法证明吸引子族\(L^\infty\)一致有界，且在区域体积收敛时关于强\(H^1\)拓扑在\(\mu = 0\)处上半连续。
中文摘要 AI 辅助
我们考虑一族在一族变化的非光滑区域\(\{\Omega_\mu\}_{\mu \in \Lambda} \subset \mathbb{R}^n\)上具有齐次诺伊曼边界条件的半线性抛物方程。仅假设区域具有一致有界体积、满足一致琼斯条件且具有一致椭圆性界，我们在适当的分数巴拿赫空间尺度下建立了问题的适定性并证明了全局吸引子的存在。通过结合莫泽 - 阿利卡科斯引导迭代、一致格朗沃尔引理和琼斯延拓算子的一致性质，我们表明吸引子族在\(L^\infty(\Omega_\mu)\)中一致有界。最后，假设区域的体积收敛，\(|\Omega_\mu \triangle \Omega_0| \to 0\)，我们构建一个连接映射框架来证明吸引子族在\(\mu = 0\)处关于强\(H^1\)拓扑是上半连续的。
英文摘要
We consider a family of semilinear parabolic equations with homogeneous Neumann boundary conditions on a family of varying non-smooth domains $\{Ω_μ\}_{μ\in Λ} \subset \mathbb{R}^n $. Assuming only that the domains have uniformly bounded volumes, bounded away from zero, and satisfy a uniform Jones condition, together with the usual structural requirements on the linear and nonlinear terms, we establish the well-posedness of the problem in an appropriate scale of fractional Banach spaces and prove the existence of global attractors. Using a Moser-Alikakos bootstrap iteration in conjunction with the uniform Gronwall lemma and the uniform properties of the Jones extension operator, we show that bounded sets of $L^{q}(Ω_μ)$ are absorbed, in finite time and uniformly in $μ$, into a fixed ball of $L^\infty(Ω_μ)$; the iteration uses only first-order Sobolev embeddings, which is what makes it available on domains where the domain of the elliptic operator is not contained in $H^{2}$. Finally, assuming the volume convergence of the domains, $|Ω_μ\triangle Ω_0| \to 0$, we construct a framework of connecting maps and prove that the family of attractors is upper semicontinuous at $μ= 0$ in $L^2(Ω_μ)$ and hence, by interpolation, in $H^s(Ω_μ)$ for every $s<1$.
发表机构
- Instituto de Matemática e Estatística, Universidade de São Paulo（圣保罗大学数学与统计学院）
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