On the well-posedness of linear evolution equations under unbounded nonautonomous perturbations

Abstract: We study conditions for the well-posedness of nonautonomous perturbation of evolution equations of the form [ u'(t)=(A+B(t))u(t), \quad t \in [a,b], ] where $A$ generates a $\mathrm{C}<em>0$-semigroup $\left (T(t)\right )</em>{t\ge 0}$ with $| T(t)| \le Me^{ω_0 t}$, $t\ge 0$, in a Banach space $\mathbb{X}$ and $B(t)$ are $t$-dependent (unbounded) linear operators in $\mathbb{X}$. The unbounded perturbation operators $B(t)$ are assumed to belong to a normed space (denoted by $\mathcal{GL}<em>A (\mathbb{X})$) of unbounded linear operators $C$ in $\mathbb{X}$ such that $D(A) \subset D(C)$ with norm [ | C|_A:= (1/M) \sup{μ>ω0 } | (μ-ω_0) CR(μ,A)| <\infty. ] We prove that the above-mentioned evolution equation admits an evolution family if $| B(\cdot)|_A$ is continuous in $[a,b]$. The evolution family is unique if $B(\cdot)R(μ, A)$ as a function $[a,b]\to \mathcal{L}(\mathbb{X})$ is continuously differentiable, and [ \limsup{μ\to\infty} \sup_{t\in [a,b]} \left | \frac{d}{dt}[B(t)R(μ,A)]\right | <\infty. ] Examples are given to illustrate the obtained results.

• The paper demonstrates that linear evolution equations in Banach spaces remain well-posed under unbounded, time-varying perturbations if continuity and differentiability conditions in the A-norm are satisfied.
• It introduces operator metric tools like the Yosida distance and piecewise constant approximations to effectively manage unbounded nonautonomous perturbations.
• The analysis extends classical semigroup methods, ensuring stability and exponential dichotomy for evolution families with applications to parabolic PDEs and control systems.

Well-Posedness of Linear Evolution Equations with Unbounded Nonautonomous Perturbations

Overview and Motivation

This paper addresses the well-posedness of nonautonomous linear evolution equations in Banach spaces under unbounded, time-dependent perturbations. Specifically, the authors consider equations of the form

$u'(t) = [A + B(t)] u(t), \quad t \in [a, b],$

where $A$ is the generator of a (possibly non-exponentially bounded) $C_0$-semigroup on a Banach space $\mathbb{X}$ and $B(t)$ is a family of (potentially unbounded) linear operators depending on time. The principal challenge is extending the classical semigroup and perturbation methods—which are well-understood for bounded and certain relatively bounded (autonomous) perturbations—to the setting where $B(t)$ may be unbounded and nonautonomous. The analysis departs from existing Miyadera-type or variation-of-constants frameworks by adapting the Yosida distance and normed spaces naturally tied to the generator, thus broadening the class of admissible perturbations.

Technical Framework

The authors introduce and work in the space $\mathcal{GL}_A(\mathbb{X})$ of unbounded linear operators $C$ with domain $D(C)\supset D(A)$, possessing a norm

$$\|C\|_A = (1/M) \sup_{\mu > \omega_0} \|(\mu-\omega_0)\, C R(\mu, A)\| < \infty,$$

where $A$0 is the resolvent of $A$1, $A$2 is the semigroup bound, and $A$3 the associated exponential type. This class encompasses both bounded and certain unbounded perturbations. For time-dependent $A$4, the norm continuity and additional differentiability with respect to $A$5 in this topology form the core technical assumptions.

The methodology pivots on several metric and operator-theoretic tools:

• Yosida Distance: Used to metrize unbounded generators, providing fine control over convergence and approximation of families of operators.
• Piecewise Constant Approximations and Euler Polygon Method: The nonautonomous evolution equation is solved by breaking $A$6 into small intervals and replacing $A$7 with a time-frozen $A$8. The solution family is then defined as the strong limit of operator products as the mesh refines.
• Perturbed Semigroup Generation: Under suitable boundedness in the $A$9-norm, unbounded perturbations yield a well-posed generator, and quantitative operator-norm estimates are provided.

Main Results

The main theorem establishes both existence and uniqueness of evolution families solving $C_0$0 under the following:

• Existence holds if $C_0$1 is continuous with respect to the $C_0$2 norm.
• Uniqueness is obtained if, in addition, $C_0$3 is continuously differentiable as a map into $C_0$4 uniformly in $C_0$5, with the supremum norm of the derivative tending to zero as $C_0$6.

The authors prove that the limiting evolution family inherits strong continuity and semigroup-like properties and satisfies the necessary differential relations with domain inclusion $C_0$7. The uniqueness analysis relies on a careful analysis of time-differentiated resolvent equations and an application of Laplace transform techniques.

Additionally, the paper demonstrates:

• Roughness of Exponential Dichotomy: Exponential dichotomy and stability persists under sufficiently small, norm-continuous unbounded nonautonomous perturbations, provided the unperturbed semigroup exhibits exponential dichotomy.
• Well-posedness on the Entire Real Line: The framework extends by concatenation and uniformity to give global-in-time evolution families.

Illustrative Examples

The paper presents several nontrivial examples, including:

• Perturbations of the translation group on $C_0$8 by certain unbounded multiplication operators that are closed but not everywhere defined, demonstrating that genuinely unbounded cases are accommodated.
• Parabolic PDEs with unbounded, highly oscillatory time-dependent lower-order (multiplication) perturbations in $C_0$9, verifying both the norm bounds and differentiability requirements.

These examples clarify both the reach and the limitations of the present techniques. Operators outside the relatively bounded class—yet satisfying the $\mathbb{X}$0-conditions—yield well-posed evolution problems not covered by standard frameworks.

Implications and Theoretical Developments

The results extend prior generation and well-posedness theorems for evolution equations to cover a new regime of unbounded, time-dependent perturbations. Crucially, the analysis does not rest on relative boundedness (Kato, Miyadera conditions), but rather on intrinsic metric and norm-topology notions, thus providing new tools for addressing time-inhomogeneous operator families in infinite-dimensional settings.

This has direct implications in abstract parabolic PDEs, control problems (with time-varying unbounded inputs), and the construction of nonautonomous operators arising in applied mathematics contexts. The criterion involving continuity in the $\mathbb{X}$1-norm is, in many cases, verifiable using Laplace transform and convolution estimates, extending robustness results around exponential dichotomy and stability.

On the theoretical front, the work provides a more flexible foundation for nonautonomous evolution, indicating that a wide class of unbounded time-dependent perturbations can be handled by semigroup-type methods, provided appropriate continuity and differentiability are established in the operator-resolvent framework. The extension to the roughness of exponential dichotomy solidifies the practical applicability of these methods in dynamics and spectral theory.

Future Prospects

Potential directions building on this framework include:

• Further relaxation of differentiability assumptions (e.g., weak or measurable continuity in time) to admit rougher perturbations.
• Extension to evolution equations on spaces of distributions or other topological vector spaces.
• Application to non-linear evolution equations via linearization around nonautonomous (possibly unbounded) time-dependent equilibria.
• Quantitative stability bounds for dichotomy/invariant manifold persistence under perturbations measured by the Yosida distance.

Conclusion

This paper rigorously advances the theory of linear evolution equations in Banach spaces by establishing well-posedness and uniqueness results for cases with unbounded, time-varying perturbations, under explicit and checkable continuity and differentiability conditions in metric operator topologies. The techniques developed herein have significant implications for further investigation of dynamical phenomena in infinite dimensions, robustness of spectral properties, and model classes in applied analysis.

Reference: "On the well-posedness of linear evolution equations under unbounded nonautonomous perturbations" (2604.16798)