Technical Summary: Nonautonomous Systems of Evolution Inclusions

1. Problem Statement

The paper investigates the existence of global solutions for coupled systems of partially nonautonomous evolution inclusions. The system is formulated in a separable real Hilbert space $H$ (or a product of such spaces) and takes the form:

$$\begin{cases} \frac{d}{dt}u + Eu \in F(u, v) & \text{on } (0, T), \\ \frac{d}{dt}v + A(\cdot)v \in G(u, v) & \text{on } (0, T), \\ (u(0), v(0)) = (u_0, v_0) & \text{in } H \times H, \end{cases}$$

where:

• $-E$ is the generator of a $C_0$-semigroup with compact resolvent on $H$.
• $A(t)$ is a time-dependent, univalued (single-valued) maximal monotone operator of subdifferential type, i.e., $A(t) = \partial \phi_t$, where $\phi_t$ is a real potential satisfying specific regularity conditions (based on Yotsutani's results).
• $F$ and $G$ are multivalued (set-valued) maps representing the coupling terms. These maps are assumed to be Hausdorff-continuous, half weakly continuous (with respect to the first variable), and to take nonempty, bounded, closed, and convex (bcc) values. They may also satisfy a linear growth bound.

The system is motivated by applications in electrorheological fluids, image processing, magnetostatics, and capillarity phenomena, specifically extending previous work on autonomous Schrödinger–Debye systems to the nonautonomous case with variable exponent Sobolev spaces.

2. Methodology

The authors employ a combination of semigroup theory, monotone operator theory, and set-valued analysis. The methodology addresses the lack of maximal regularity in the semigroup part (common in Schrödinger-type equations) by imposing stronger continuity conditions on the set-valued terms.

Key methodological steps include:

1. Decoupling and Solvability of Single-Valued Problems:

   • The authors first establish solvability for the individual components: the abstract Cauchy problem driven by the semigroup generator $-E$ and the monotone evolution problem driven by $A(t)$.
   • For the semigroup part, they utilize extrapolation spaces ($H^{-1}(E)$) and Yosida approximations to handle the lack of maximal regularity, proving that solutions converge uniformly under compact resolvent assumptions.
   • For the monotone part, they rely on existing results (Yotsutani) for strong solutions of evolution equations with time-dependent subdifferentials.

2. Selection Theory and Approximation:

   • A central technical challenge is handling the multivalued right-hand sides $F$ and $G$. The authors prove a new existence result for measurable selections (Theorem 3.3.1).
   • This theorem establishes that if a sequence of functions $u_n$ converges uniformly to $u$, and $F$ is Hausdorff-continuous and bcc, then for any selection $f \in \text{Sel}_F(u)$, there exist selections $f_n \in \text{Sel}_F(u_n)$ that converge to $f$ in the $L^2$-norm. This result is crucial for constructing approximating solutions in the fixed-point argument.

3. Fixed Point Argument:

   • The existence of solutions is proven using a multivalued fixed point theorem (Theorem 5.1.1) by Vrabie et al.
   • The authors define a solution map $\Phi$ on a bounded set of $L^2$ functions. They verify that $\Phi$ is well-defined, has bcc values, and possesses a weakly $\times$ weakly sequentially closed graph.
   • The proof of the closed graph property relies heavily on the uniform convergence of solutions to the decoupled problems (established in Section 4) and the approximation of selections (Theorem 3.3.1).

4. Global Existence:

   • Local existence is established first. Global existence is then derived by assuming a linear growth bound on $F$ and $G$ and applying Grönwall's inequality to prevent blow-up in finite time.

3. Key Contributions

• Extension to Nonautonomous Systems: The paper generalizes previous results on autonomous systems (specifically those in [30]) to the nonautonomous case, where the operator $A(t)$ depends on time and can be defined on Sobolev spaces with variable exponents.
• Handling Lack of Maximal Regularity: The authors address the specific difficulty that Schrödinger-type operators do not possess maximal regularity. Instead of relying on maximal regularity (which fails for these operators), they strengthen the continuity requirements on the set-valued terms (requiring Hausdorff-continuity rather than just upper semicontinuity) and utilize Yosida approximations to recover necessary convergence properties.
• New Selection Approximation Theorem: Theorem 3.3.1 provides a novel technical tool for approximating measurable selections in infinite-dimensional Hilbert spaces under Hausdorff-continuity, filling a gap in the literature regarding the stability of selections under uniform convergence of arguments.
• Correction of Previous Statements: The authors note that they correct certain statements in [30] regarding the maximal regularity of Schrödinger equation solutions, clarifying that such regularity does not generally hold and adjusting the proof strategy accordingly.

4. Main Results

• Local Existence (Theorem 5.2.2): Under the assumptions of Hausdorff-continuity, half weak continuity, and bcc values for $F$ and $G$, and assuming $-E$ has a compact resolvent, the system admits at least one strong local solution on a time interval $[0, T_0]$.
• Global Existence (Theorem 5.3.2): If the multivalued maps $F$ and $G$ additionally satisfy a linear growth bound, the local solution can be extended to a global solution defined on $[0, T_m]$ (where $T_m$ can be $\infty$).
• Uniform Convergence (Theorem 4.1.4 & Proposition 4.2.1): The paper establishes that sequences of solutions to the decoupled problems converge uniformly (in the appropriate topology) when the right-hand sides converge weakly, a prerequisite for the fixed point argument.

5. Significance and Scope

The paper claims significance primarily in its methodological contribution to the theory of coupled evolution inclusions. By extending the approach of Vrabie et al. to the nonautonomous setting and developing new tools for selection approximation, the authors provide a framework for analyzing systems where:

• One component is governed by a non-parabolic operator (like the Schrödinger operator) lacking maximal regularity.
• The other component is a time-dependent monotone operator (potentially with variable exponents).
• The coupling is multivalued, modeling uncertainties or discontinuities.

The authors explicitly state that their results cover nonautonomous generalized Schrödinger–Debye systems with variable exponents, as well as hyperbolic-parabolic systems such as Maxwell–parabolic systems. The paper does not propose new experimental applications but rather provides the rigorous analytical foundation for existing models in electrorheological fluids, image processing, and electromagnetism. The results are presented as a theoretical extension and correction of prior work, ensuring well-posedness under conditions that accommodate the specific regularity limitations of the operators involved.