Preamble

This work, following the developments in [11] and [12], investigates the differential equation
$$H = Ax + h(x, t)$$
where $A$ is a linear operator and $h(x, t)$ is a nonlinear term. We consider the case where $A$ is a closed operator with a domain $D(A)$ dense in a Banach space $X$. In Section 1, we establish the fundamental properties of the operator $A$ and the associated semigroup $e^{At}$. Specifically, we analyze the relationship between the spectrum $\sigma(A)$ and the growth exponent of the solution to the linear equation
$$\dot{x} = Ax, \quad x(0) = x_0. \tag{0.1}$$

The characteristic exponent $\chi(x)$ for a solution $x(t)$ is defined as
$$\chi[x] = \limsup_{t \to \infty} \frac{\ln |x(t)|}{t}.$$
We denote the set of all such exponents as $S(A) = {\chi(x) : x \in X, x \neq 0}$. It is well known that for a bounded operator $A$, the supremum of $S(A)$ coincides with the spectral radius, specifically $\sigma(A) = \sup \text{Re } \sigma(A)$. However, for unbounded operators, this relationship is more complex. As shown in [6] and [10], the upper bound of the spectrum $\sigma(A)$ is defined as:
$$\sigma(A) = \limsup_{t \to \infty} \frac{\ln |e^{At}|}{t} \tag{1.1}$$
where $\sigma(A) = \sup S(A) \geq \sup \text{Re } \sigma(A)$.

Section 1. Spectral Properties and Semigroups

We consider the resolvent $R(\lambda; A) = (\lambda I - A)^{-1}$. For an operator $A$ generating a strongly continuous semigroup, the growth of the semigroup is constrained by the resolvent's behavior in the complex plane. If $A$ satisfies the conditions for an analytic semigroup, then for any $\epsilon > 0$, there exists a constant $M_\epsilon$ such that:
$$|e^{At}x| \leq M_\epsilon e^{(\sigma(A) + \epsilon)t} |x| \tag{1.2}$$
for all $t > 0$ and $x \in X$.

In cases where the space $X$ can be decomposed into invariant subspaces $X = X_1 \oplus X_2$ corresponding to different parts of the spectrum $\sigma(A) = \sigma_1 \cup \sigma_2$, the operator $A$ can be represented as a direct sum $A = A_1 \oplus A_2$. If $\text{Re } \sigma_1 < \xi < \text{Re } \sigma_2$, then the solutions $x(t)$ can be partitioned accordingly, allowing for precise asymptotic estimates of the components in $X_1$ and $X_2$.

For $x \in D(A^{m+2})$, the solution to the inhomogeneous problem can be represented using the Dunford integral:
$$e^{At}x = \frac{1}{2\pi i} \int_{\gamma} e^{\lambda t} R(\lambda; A)x \, d\lambda \tag{1.3}$$
where $\gamma$ is a suitable contour in the resolvent set. As demonstrated in [10], if the resolvent satisfies certain growth conditions, specifically $|R(\lambda; A)| \leq K(1 + |\lambda|)^m$, the integral representation converges for $t > 0$. This allows us to define the fractional powers of the operator $A^\alpha$ and establish estimates of the form:
$$|A^\alpha e^{At}x| \leq M t^{-\alpha} e^{\sigma(A)t} |x| \tag{1.5}$$

Section 2. Nonlinear Perturbations and Stability

We now turn to the nonlinear equation:
$$\dot{x} = Ax + h(x, t) \tag{0.2}$$
where $h(x, t)$ satisfies a Lipschitz condition $|h(x_1, t) - h(x_2, t)| \leq L |x_1 - x_2|$. We assume that $h(0, t) = 0$, ensuring that $x=0$ is a trivial solution. Using the variation of constants formula, the solution can be written in the integral form:
$$x(t) = e^{A(t-t_0)}x_0 + \int_{t_0}^t e^{A(t-\tau)} h(x(\tau), \tau) \, d\tau \tag{2.9}$$

Applying the estimates from Section 1, we can prove the existence and uniqueness of solutions in the space $D(A^\alpha)$. Specifically, if $x_0 \in D(A^\alpha)$, then for a sufficiently small time interval $[t_0, t_0 + T]$, there exists a unique solution $x(t)$ such that:
$$|A^\alpha x(t)| \leq K(t-t_0)^{-\alpha} |x_0| + \int_{t_0}^t (t-\tau)^{-\alpha} |h(x(\tau), \tau)| \, d\tau$$
By applying Gronwall's inequality, we establish the stability of the trivial solution under the condition that the linear part $A$ is exponentially stable, i.e., $\sigma(A) < 0$.

Section 3. Asymptotic Behavior

In this section, we analyze the characteristic exponents of the nonlinear system (0.2). Let $\sigma(A)$ be the spectral bound of the linear operator. We show that if the nonlinear term $h(x, t)$ is small in a certain sense, specifically if:
$$|h(x, t)| \leq \gamma(t) |A^\alpha x| \tag{3.1}$$
where $\gamma(t) \to 0$ as $t \to \infty$, then the characteristic exponents of the nonlinear system are governed by the spectrum of $A$.

Theorem 3.1. For any $\epsilon > 0$, there exists a $\delta > 0$ such that if $|x_0| < \delta$, the solution $x(t)$ to (0.2) satisfies:
$$\chi[x] \leq \sigma(A) + \epsilon$$
This result implies that the stability of the nonlinear system is robust to perturbations that satisfy the growth conditions defined in (3.1). The proof utilizes a contraction mapping argument in the space of continuous functions with exponential weight, combined with the fractional power estimates derived in Section 2.

Furthermore, we consider the case where the spectrum $\sigma(A)$ is divided by a gap. If the nonlinear perturbation is sufficiently small, the manifold structure of the linear system persists. Specifically, there exist invariant manifolds in $X$ that are tangent to the spectral subspaces $X_1$ and $X_2$ at the origin. This allows for a local decoupling of the dynamics, facilitating the study of conditional stability and the existence of bounded solutions on the half-line $[t_0, \infty)$.