Sublinear Growth at Infinity

• Sublinear growth at infinity is characterized by functions or solutions that grow slower than linear benchmarks, defined by precise asymptotic conditions.
• It plays a crucial role in unifying differential, integral, and complex analysis by establishing critical thresholds for qualitative changes in behavior.
• Applications range from rigidity in Liouville-type theorems and delay equations to refined frequency analysis in infinite-dimensional Hamiltonian systems.

Sublinear growth at infinity describes a regime in which a function, sequence, operator, or solution to a differential or integral equation exhibits asymptotic growth that is strictly less than a prescribed benchmark—usually a linear, polynomial, or exponential reference scale. This concept is central across harmonic analysis, complex analysis, nonlinear PDE theory, functional and Volterra equations, infinite-dimensional Hamiltonian systems, and the paper of entire functions, with rigorously specified thresholds according to context. Sublinear growth conditions generally appear as structural assumptions in existence, rigidity, classification, and Liouville-type theorems, or as sharp dividing lines for the emergence or absence of complex dynamical and regularity phenomena.

1. Definitions and Prototype Examples

Sublinear growth at infinity typically refers to the property that, for some class of objects (functions, solutions, frequencies), growth does not outpace a given sublinear reference. Precise definitions depend on the context:

• Real/complex functions: A function $f:\mathbb{R}\to\mathbb{R}$ satisfies sublinear growth if $\lim_{x\to\infty} f(x)/x=0$. In complex analysis, an entire function $f:\mathbb{C}\to\mathbb{C}$ has sublinear growth if its order $\rho$ satisfies $0\le \rho < 1$, i.e., for any $\epsilon>0$ there exists $C = C(\epsilon)$ such that $|f(z)| \le C e^{|z|^{\rho+\epsilon}}$ for large $|z|$ (Lastra, 2021).
• Integro-differential equations: For solutions $u$ of an operator equation $L u = f(u)$ on $\mathbb{R}^n$, sublinear growth can be specified as $|u(x)| \le K_0(1+|x|^k)$ for some $k<k_{*}$, with $k_{*}$ a critical exponent determined by the order of the operator (Farina et al., 2014).
• Nonlinear functionals: For nonlinearities $g$ or $f$ in ODEs/PDEs, the condition $g(u)/u \to 0$ as $u\to\infty$ is sublinear at infinity. For instance, $g(s) = s^\beta$ with $0<\beta<1$ is sublinear (Boscaggin et al., 2015).
• Normal frequencies in infinite-dimensional systems: A frequency mapping $\Omega_n = |n|^\alpha + 1 + \widetilde{\Omega}_n$, $0<\alpha<1$, is sublinear in $n$ (Xu, 2018).

Prototype examples include $f(x) = x^\beta$ with $0<\beta<1$; $f(x) = x/\log x$; $f(x) = \log(1+x)$; and in the complex case, entire functions of order less than 1 such as the Mittag–Leffler function $E_{1/s}(z)$.

2. Sublinear Growth in Differential and Integral Equations

2.1 Functional and Volterra Differential Equations

For scalar autonomous functional differential equations (FDEs) and Volterra equations, sublinear growth of the nonlinearity $f$ (i.e., $f(x)/x\to0$ as $x\to\infty$) directly governs the qualitative and quantitative growth rate of solutions. The principal result (Appleby et al., 2016, Appleby et al., 2014, Appleby et al., 2016, Appleby et al., 2016) is that, under regularity and monotonicity hypotheses, if $f$ is sublinear and exhibits "regular variation," the solution $x(t)$ of the delay or Volterra equation

$$x'(t) = \int_{-\tau}^0 f(x(t+s))\,\mu(ds), \quad \text{or} \quad x(t) = x(0) + \int_0^t K(t-s) f(x(s))\,ds$$

is asymptotic to the solution of the corresponding surrogate ODE $y'(t) = M f(y(t))$, $M = \int \mu(ds)$. More precisely, defining

$F(x) = \int_1^x \frac{du}{f(u)},$

solutions satisfy $F(x(t)) \sim M t$ as $t\to\infty$, with $x(t) \sim F^{-1}(Mt)$ under mild additional constraints. All such solutions then display subexponential growth: $\log x(t)/t \to 0$ as $t\to\infty$ (Appleby et al., 2016, Appleby et al., 2014).

An exact threshold, $f_c(x)=x/\log x$, emerges: If $f(x)/f_c(x) \to 0$ as $x\to\infty$, delay is asymptotically negligible, and $x(t)\sim y(t)$; if $f(x)\sim \lambda f_c(x)$, the FDE and ODE grow at the same order but with a non-unit constant factor; if $f(x)/f_c(x)\to\infty$, delay suppresses the ODE growth so that $x(t)=o(y(t))$ (Appleby et al., 2016, Appleby et al., 2014).

Table: Delay Equation Growth Regimes

Growth regime for $f$	Asymptotic solution behavior
$f(x)/f_c(x) \to 0$	$x(t) \sim y(t)$ (delay negligible)
$f(x)\sim \lambda f_c(x)$	$x(t)/y(t) \to e^{-\lambda C} \in (0,1)$
$f(x)/f_c(x)\to\infty$	$x(t) = o(y(t))$ (delay dominant)

2.2 Nonlocal and PDE Rigidity (Liouville-Type Theorems)

Rigidity results for solutions with sublinear (or "subcritical") growth at infinity are established for entire solutions to nonlocal semilinear or fractional PDEs (Farina et al., 2014). For an integro-differential operator $L$ of order $2s$ (e.g., fractional Laplacian), if $u$ grows like $|x|^k$ with $k<2s$, any solution to $L u = f(u)$ (with $f$ nondecreasing) must be affine or, for nontrivial $f$, must be constant. The critical exponent $2s$ is sharp: growth exactly like $|x|^{2s}$ may admit nontrivial solutions.

2.3 Gradient Systems and Sublinear Perturbations

In nonlinear gradient systems, sublinear growth of a perturbing Nemytskii operator $N$ (i.e., $\|N(u)\| \le a + b \|u\|^\alpha$ with $0<\alpha<1$ for large $\|u\|$) is essential in guaranteeing maximal $L^2$-regularity, existence, and a priori bounds of solutions to perturbed evolution equations. The perturbation remains "tame" at infinity, ensuring the applicability of fixed-point arguments (Schaefer’s theorem) and quantitative estimates (Arendt et al., 2019).

3. Sublinear Growth for Entire Functions and Moment-Differential Systems

In the context of entire functions and moment-differential equations, sublinear growth at infinity invokes the theory of function order. An entire function $f$ has order $$\rho(f)=\limsup_{r\to\infty} \frac{\ln\ln M_f(r)}{\ln r}$$ where $M_f(r) = \max_{|z|=r} |f(z)|$. Sublinear growth corresponds to $0\le\rho<1$, i.e., no exponential-type term governs the large-$|z|$ behavior (Lastra, 2021).

For linear moment-differential systems $\partial_m y = Ay$, if the underlying moment growth kernel $E$ has order $\rho(E)<1$, all entire solutions exhibit sublinear growth and, in the language of proximate order, this is both necessary and sufficient (Lastra, 2021). Classical Mittag–Leffler functions $E_{1/s}(z)$ with $s>1$ appear as canonical examples.

4. Sublinear Frequency Growth in Infinite-Dimensional Hamiltonian Systems

Sublinear growth of normal frequencies is central in KAM theory for Hamiltonian PDEs and infinite-dimensional dynamical systems (Xu, 2018). A frequency sequence $\Omega_n = |n|^\alpha + 1 + \widetilde{\Omega}_n$, $\alpha\in(0,1)$, is sublinear. In such systems, KAM schemes require delicate small-divisor analysis since the spectral gaps $\Omega_n - \Omega_{n+1}\to 0$ as $n\to\infty$, which presents substantial obstacles absent in the linear-frequency ($\alpha=1$) case. Toeplitz–Lipschitz and exponential decay properties of perturbations are leveraged to ensure the convergence of an iterative normal-form procedure and reducibility to pure-point, time-independent dynamics.

5. Sublinear Growth in Multiplicity and Bifurcation Theory

The condition of sublinear growth at infinity for nonlinearities $g$ in second-order ODEs (e.g., $g(s)/s\to0$ as $s\to\infty$) underpins high multiplicity results for nonlinear indefinite boundary-value problems (Boscaggin et al., 2015). In combination with superlinear growth at zero, it allows for the application of Leray–Schauder degree arguments to establish the existence of at least $3^m-1$ positive periodic or subharmonic solutions, for $m$ positive humps in the sign-changing weight function. Sublinear growth at infinity is crucial to keep the solution set compact and control the amplitude of solutions, preventing escape to infinity and ensuring the nonlinearity becomes too weak to sustain unbounded solutions.

6. Liouville-Type Rigidity Theorems and Critical Exponents

Sublinear growth at infinity is tightly linked to Liouville-type theorems and critical exponents in elliptic and parabolic PDEs. For nonlocal and fractional equations (order $2s$), the exponent $k=2s$ is critical: solutions growing strictly slower than this are forced to be constant or affine, depending on the form of the nonlinearity, whereas critical growth may still admit nontrivial (e.g., polynomial or explicit) solutions (Farina et al., 2014). In planar stationary MHD equations, sublinear growth of velocity at infinity combined with suitable decay/smallness of the magnetic field leads to rigidity, enforcing trivial (constant) solutions (Wang, 2019).

7. Summary and Significance

Sublinear growth at infinity provides a unifying asymptotic constraint that organizes solution behavior, regularity, multiplicity, rigidity, and existence theory in several domains:

• It dictates the asymptotic matching of complicated (functional, Volterra, PDE) equations to corresponding ODE or algebraic surrogates.
• It establishes sharp thresholds (critical exponents, delay-negligibility) for qualitative changes in global solution structure.
• In infinite dimensions, sublinear frequency growth necessitates refined analytical frameworks in KAM and normal-form theory.
• In Liouville-type results, it produces rigidity phenomena that are fundamental to classification and symmetry analysis.

Across these settings, sublinear growth is not merely a technical assumption but often a sharp and indispensable threshold for the emergence of fundamental mathematical structures and phenomena (Farina et al., 2014, Lastra, 2021, Appleby et al., 2014, Appleby et al., 2016, Arendt et al., 2019, Xu, 2018, Appleby et al., 2016, Boscaggin et al., 2015, Appleby et al., 2016, Wang, 2019).