带有Carathéodory强迫的非线性增生演化的存在性与定性理论
Existence and qualitative theory for nonlinear accretive evolutions with Carathéodory forcing
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中文总结 AI 辅助
本文研究实Banach空间中带Carathéodory强迫的非线性增生演化的约束问题，通过两层逼近等方法建立其温和解的存在性与适定性，并将理论应用于反应-扩散系统等问题。
中文摘要 AI 辅助
设A为实Banach空间X中的m-增生算子。给定闭集K(t)⊂X，我们针对约束问题的温和解建立存在性理论：u'(t)+Au(t)∋f(t,u(t))，其中u(t)∈K_A(t):=K(t)∩$\bar{D(A)}$，该问题的强迫项在有效管K_A上是Carathéodory型的。我们还假设f在移动图上具有联合可测性，或可延拓为固定柱上的Carathéodory型函数。在线性增长假设以及相应的正则时间和异常时间次切条件下，我们将问题简化为具有公共可积界的有界有效管。首个核心结果构造了一个闭的可分预解不变子空间，其保持到移动集的距离以及次切条件，从而允许将Scorza–Dragoni性质应用于简化后的强迫项。第二个核心要素是两层逼近：由w_ε驱动的温和解u_ε伴随有辅助路径v_ε和当前时间选择子x_ε，满足x_ε(t)∈K_A(t)，且对于几乎全测度的闭正则集上的几乎每个t，有w_ε(t)=f(t,x_ε(t))；对每个t，还存在σ_ε(t)∈[(t−ε)^+,t]，使得v_ε(σ_ε(t))∈K_A(σ_ε(t))。当前时间选择子确定极限强迫项，而滞后节点保证生存性。当这类强迫项f关于状态变量局部Lipschitz时，这在随时间变化的约束下得到了适定性；在各种紧性条件下还可得到进一步的生存性结果。将该抽象生存性理论应用可得到比较原理、非自治Lyapunov对、周期解以及抽象反应-扩散系统的时变界。
英文摘要
Let $A$ be $m$-accretive in a real Banach space $X$. Given closed sets $K(t)\subset X$, we develop an existence theory for mild solutions of the constrained problem \[ u'(t)+Au(t)\ni f(t,u(t)), \qquad u(t)\in K_A(t):=K(t)\cap\overline{D(A)}, \] where the forcing is Carathéodory on the effective tube $K_A$. We also assume joint measurability of $f$ on the moving graph or a Carathéodory extension to a fixed cylinder. Under a linear-growth hypothesis and the corresponding regular- and exceptional-time subtangential conditions, we reduce the problem to a bounded effective tube with a common integrable bound. % A first central result constructs a closed separable resolvent-invariant subspace preserving distances to the moving sets and the subtangential conditions, permitting use of the Scorza--Dragoni property for the reduced forcing. A second central ingredient is a two-level approximation. Mild solutions $u_\varepsilon$ driven by $w_\varepsilon$ are accompanied by auxiliary paths $v_\varepsilon$ and current-time selectors $x_\varepsilon$ satisfying $x_\varepsilon(t)\in K_A(t)$ and $w_\varepsilon(t)=f(t,x_\varepsilon(t))$ for almost every $t$ in a closed regularity set of almost full measure. For every $t$ there is also $σ_\varepsilon(t)\in[(t-\varepsilon)^+,t]$ with $v_\varepsilon(σ_\varepsilon(t))\in K_A(σ_\varepsilon(t))$. The current-time selectors identify the limiting forcing, while the lagged nodes guarantee viability. % This yields well-posedness under time-dependent constraints for such forcings $f$ when they are locally Lipschitz in the state variable. Further viability results follow under various compactness conditions. Application of the abstract viability theory yields comparison principles, nonautonomous Lyapunov pairs, periodic solutions, and time-dependent bounds for abstract reaction--diffusion systems.
