Critical Mass Phenomenon Research Articles

Critical mass phenomenon for a parabolic–elliptic multispecies chemotaxis system in a two‐dimensional disk

A parabolic–elliptic Keller–Segel model for multipopulations in a two‐dimensional unit ball will be considered in this paper. For the initial boundary value problem with mutually attractive populations, the global existence of bounded solutions has been proved through the logarithmic Hardy–Littlewood–Sobolev inequality for system when the initial masses satisfy the subcritical condition. Moreover, based on the moments technique, there exists a blow‐up solution when the initial masses satisfy the supercritical case. In a word, a critical mass phenomenon has been established in this multispecies chemotaxis system.

Global existence of solutions to the 4D attraction–repulsion chemotaxis system and applications of Brezis–Merle inequality

We consider the Cauchy problem for an attraction–repulsion chemotaxis system in the four space dimension. One of main topics in the study of such a system is the presence of L 1 threshold. In fact, the critical mass phenomenon called 8π-problem is well-known in the two-dimensional setting. In this paper, we show the global existence of solutions to the Cauchy problem for the four-dimensional subcritical case, that is, the total mass of the initial data is less than the four-dimensional L 1 threshold value . The key ingredients are the four-dimensional Brezis–Merle type inequality of the 4th-order elliptic equation and some inequalities derived from rearrangement arguments.

Critical mass for Keller–Segel systems with supercritical nonlinear sensitivity

This paper is concerned with the following radially symmetric Keller–Segel systems with nonlinear sensitivity [Formula: see text] and [Formula: see text], posed on [Formula: see text] [Formula: see text] and subjected to homogeneous Neumann boundary conditions. It is well-known that [Formula: see text] is the critical exponent of the systems in the sense that all solutions exist globally if [Formula: see text] and there exist finite-time blowup solutions if [Formula: see text]. Here we consider the supercritical case [Formula: see text] and show a critical mass phenomenon. Precisely, we prove that there exists a critical mass [Formula: see text] such that (1) for arbitrary nonincreasing nonnegative initial data [Formula: see text] with [Formula: see text] and [Formula: see text], the corresponding solution blows up in finite time if [Formula: see text], and if [Formula: see text] we can only prove that the solution blows up in finite time or infinite time; (2) for some nonincreasing nonnegative initial data with [Formula: see text], the corresponding solutions are globally bounded. Our results extend that of Winkler’s paper [M. Winkler, How unstable is spatial homogeneity in Keller–Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic–elliptic cases, Math. Ann. 373 (2019) 1237–1282], where he proved similar results for the system with [Formula: see text].

Critical mass phenomenon in a chemotaxis fluid system

In this paper, we study the following parabolic–parabolic–elliptic attraction–repulsion chemotaxis fluid system driven by a friction force n∇(χ1c−χ2v): (0.1)nt+u⋅∇n=Δn−∇⋅(χ1n∇c)+∇⋅(χ2n∇v),x∈Ω,t>0,ct=Δc−λ1c+n,x∈Ω,t>0,0=Δv−λ2v+n,x∈Ω,t>0,ut+(u⋅∇)u+∇P=Δu+n∇(χ1c−χ2v),∇⋅u=0,x∈Ω,t>0,∂n∂ν=∂c∂ν=∂v∂ν=0,u=0,x∈∂Ω,t>0,n(x,0)=n0(x),c(x,0)=c0(x),u(x,0)=u0(x),x∈Ω,in a bounded domain Ω⊂R2 with smooth boundary. With the help of a free energy functional, it is proved that (0.1) admits a global classical solution with arbitrary large initial data in the repulsive dominant or balance case χ1≤χ2. Moreover, there exists a critical mass phenomenon in the attractive dominant case χ1>χ2. Specifically, when the initial mass (χ1−χ2)∫Ωn0(x)dx≤4π, the solution exists globally; while if the initial mass (χ1−χ2)∫Ωn0(x)dx>4π and (χ1−χ2)∫Ωn0(x)dx∉{4πm:m∈N+}, the solution may blow up. This work extends the conclusions of the critical mass phenomenon in the Patlak–Keller–Segel-Navier–Stokes system (Gong and He, 2021) to the parabolic–parabolic–elliptic attraction–repulsion chemotaxis fluid system.

Critical mass phenomena in higher dimensional quasilinear Keller–Segel systems with indirect signal production

In this paper, we deal with quasilinear Keller–Segel systems with indirect signal production, complemented with homogeneous Neumann boundary conditions and suitable initial conditions, where is a bounded smooth domain, and We show that in the case , there exists such that if either or , then the solution exists globally and remains bounded, and that in the case , if either or , then there exist radially symmetric initial data such that and the solution blows up in finite or infinite time, where the blow‐up time is infinite if . In particular, if , there is a critical mass phenomenon in the sense that is a finite positive number.

A family of mass‐critical Keller–Segel systems

The no-flux initial-boundary value problem for the quasilinear Keller–Segel system u t = ∇ · ( D ( u ) ∇ u ) − ∇ · ( S ( u ) ∇ v ) , v t = Δ v − v + u , \begin{equation} \hspace*{6pc}{\left\lbrace \def\eqcellsep{&}\begin{array}{l}u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \hspace*{-6pc}\\[3pt] v_t=\Delta v-v+u, \end{array} \right.} \end{equation} (*)is considered in smoothly bounded domains Ω ⊂ R n $\Omega \subset \mathbb {R}^n$ , n ⩾ 3 $n\geqslant 3$ , where D ∈ C 2 ( [ 0 , ∞ ) ) $D\in C^2([0,\infty ))$ and S ∈ C 2 ( [ 0 , ∞ ) ) $S\in C^2([0,\infty ))$ are such that D > 0 $D>0$ on [ 0 , ∞ ) $[0,\infty )$ and that S ( 0 ) = 0 < S ( s ) $S(0)=0<S(s)$ for all s > 0 $s>0$ . A particular focus is on cases in which there exist κ > 0 , C S D > 0 $\kappa >0, C_{SD}>0$ and f ∈ L 1 ( ( 1 , ∞ ) ) $f\in L^1((1,\infty ))$ such that − f ( s ) ⩽ D ( s ) S ( s ) − κ s 2 / n ⩽ C S D s for all s ⩾ 1 . \begin{equation} \hspace*{6pc}- f(s) \leqslant \frac{D(s)}{S(s)} - \frac{\kappa }{s^{2/n}} \leqslant \frac{C_{SD}}{s} \quad \mbox{for all } s\geqslant 1.\hspace*{-6pc} \end{equation} (**)It is first shown that then there exists m 0 > 0 $m_0>0$ such that whenever u 0 $u_0$ and v 0 $v_0$ are reasonably regular and nonnegative with ∫ Ω u 0 < m 0 $\int _\Omega u_0 < m_0$ , within a suitably generalized concept of solvability one can always find a global solution for which u $u$ remains bounded with respect to the norm in L 2 ( n − 1 ) / n ( Ω ) $L^{2(n-1)/{n}}(\Omega )$ . Second, in radially symmetric settings this is complemented by a converse result on nonexistence of such solutions for some appropriately large initial data, hence leading to the conclusion that any pair ( D , S ) $(D,S)$ from the family of nonlinearities fulfilling (**) gives rise to a critical mass phenomenon in (*). Remarkably, this does not only include cases of arbitrarily strong diffusion degeneracies due to fast decay of D ( s ) $D(s)$ as s → ∞ $s\rightarrow \infty$ , but according to the considerably wide funnel described by (*) this moreover indicates that mass criticality in Keller–Segel systems is far more than a nongeneric phenomenon limited to precise functional forms of model ingredients hardly to be found in nature. Applications to concrete scenarios include the detection of mass-critical probability distribution functions of the form 0 ⩽ s ↦ exp ( − s ( n − 2 ) / n ) $0\leqslant s \mapsto \exp (-s^{(n-2)/{n}})$ in the version of (**) accounting for so-called volume-filling effects.

Comparison methods for a Keller–Segel-type model of pattern formations with density-suppressed motilities

This paper is concerned with global existence of classical solutions as well as occurrence of infinite-time blowups to the following fully parabolic system 1 $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\varDelta (\gamma (v)u)\\ v_t-\varDelta v+v=u \end{array}\right. } \end{aligned}$$ in a smooth bounded domain $$\varOmega \subset {\mathbb {R}}^n$$ , $$n\ge 1$$ with no-flux boundary conditions. This model was recently proposed in Fu et al. (Phys Rev Lett 108:198102, 2012) and Liu et al. (Science 334:238, 2011) to describe the process of stripe pattern formations via the so-called self-trapping mechanism. The system features a signal-dependent motility function $$\gamma (\cdot )$$ , which is decreasing in v and will vanish as v tends to infinity. An essential difficulty in analysis comes from the possible degeneracy as $$v\nearrow \infty .$$ In this work we develop a novel comparison method to tackle the degeneracy issue, which greatly differs from the conventional energy method in literature. An explicit point-wise upper-bound estimate for v is obtained for the first time, which shows that v(x, t) grows point-wisely at most exponentially in time. An intrinsic mechanism is then unveiled that the finite-time degeneracy is prohibited in any spatial dimension with a generic decreasing $$\gamma $$ . With such new findings, we further study global existence of classical solutions when $$n\le 3$$ and discuss uniform-in-time boundedness when $$\gamma (\cdot )$$ decreases algebraically at large signal concentrations. Besides, a new critical-mass phenomenon in dimension two is observed if $$\gamma (v)=e^{-v}$$ . Indeed, we prove that the classical solution always exists globally and remains uniformly-in-time bounded in the sub-critical case, while in the super-critical case a blowup may take place in infinite time rather than finite time.

Critical mass on the Keller-Segel system with signal-dependent motility

This paper is concerned with the global boundedness and blow-up of solutions to the Keller-Segel system with density-dependent motility in a two-dimensional bounded smooth domain with Neumman boundary conditions. We show that if the motility function decays exponentially, then a critical mass phenomenon similar to the minimal Keller-Segel model will arise. That is, there is a number m ∗ &gt; 0 m_*&gt;0 , such that the solution will globally exist with uniform-in-time bound if the initial cell mass (i.e., L 1 L^1 -norm of the initial value of cell density) is less than m ∗ m_* , while the solution may blow up if the initial cell mass is greater than m ∗ m_* .

Boundedness of solutions to the critical fully parabolic quasilinear one‐dimensional Keller–Segel system

AbstractIn this paper we consider a one‐dimensional fully parabolic quasilinear Keller–Segel system with critical nonlinear diffusion. We show uniform‐in‐time boundedness of solutions, which means, that unlike in higher dimensions, there is no critical mass phenomenon in the case of critical diffusion. To this end we utilize estimates from a well‐known Lyapunov functional and a recently introduced new Lyapunov‐like functional in .

On Global Solutions and Blow-Up for a Short-Ranged Chemical Signaling Loop

This paper aims at providing a substantial step toward the global boundedness and blow-up of solutions to a two-species and two-stimuli chemotaxis model, in which the process of the species results in a short-ranged chemical signaling loop. More precisely, we consider the following Neumann initial-boundary value problem $$\begin{aligned} \left\{ \begin{array}{llll} u_t=\Delta u-\chi _1\nabla \cdot (u\nabla v), &{} \quad x\in \varOmega , &{}t>0,\\ 0=\Delta v-\mu _2+w, &{} \quad x\in \varOmega ,&{} t>0,\\ w_t=\Delta w-\chi _2\nabla \cdot ( w\nabla z)-\chi _3\nabla \cdot ( w\nabla v), &{} \quad x\in \varOmega , &{}t>0,\\ 0=\Delta z-\mu _1+u, &{} \quad x\in \varOmega , &{}t>0\\ \end{array} \right. \end{aligned}$$ in the unit disk $$\varOmega :=B_1(0)\subset \mathbb {R}^2$$ with chemotactic sensitivities $$\chi _1>0,\chi _2>0$$ and $$\chi _3\ge 0$$ and radially symmetric nonnegative initial data $$u_0$$ and $$w_0$$ , where $$\mu _1 and \mu _2$$ are given, respectively, by $$\mu _1:=\frac{m_1}{|\varOmega |}$$ and $$\mu _2:=\frac{m_2}{|\varOmega |}$$ with $$m_1=\int _{\varOmega }u_0$$ , $$m_2=\int _{\varOmega }w_0$$ . Explicit conditions on $$\chi _i, \mu _i$$ , $$u_0$$ and $$w_0$$ are given for the simultaneous global boundedness and simultaneous finite-time blow-up of classical solutions. Specifically, when the effect of $$\chi _3>0$$ is strong enough in the sense that the dynamical properties of the above system behave like one single-species Keller–Segel chemotaxis system, it is shown that if only the total mass $$m_2<\frac{4\pi }{\chi _1}$$ and $$m_2<\frac{4\pi }{\chi _3}$$ , the solutions are globally bounded, while blow-up may occur provided that $$m_2>\frac{4\pi }{\chi _1}$$ and $$m_2>\frac{8\pi }{\chi _3}$$ . Moreover, in view of the chemotactic signaling loop, one can find a critical mass phenomenon: with some finite time $$T>0$$ . The simultaneous blow-up phenomenon for two species is also given in this paper. This, in particular, shows, when $$\chi _3=0$$ , that smallness of mass of each species implies global solvability, whereas largeness of masses induces blow-up to occur.

How unstable is spatial homogeneity in Keller-Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic cases

The parabolic-elliptic Keller-Segel system $$\begin{aligned} \left\{ \begin{array}{ll} u_t = \Delta u - \nabla \cdot (u\nabla v),&{} \\ 0 = \Delta v - \mu + u, &{}\quad \mu :=\frac{1}{|\Omega |} \int _\Omega u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ is considered under homogeneous Neumann boundary conditions in the ball $$\Omega =B_R(0)\subset {\mathbb {R}}^n$$ . The main objective is to reveal that in the context of radially symmetric solutions, this problem exhibits an apparently novel type of critical mass phenomenon: It is shown, namely, that for any choice of $$n\ge 2$$ and $$R>0$$ there exists a positive number $$m_c=m_c(n,R)$$ with the following properties: In consequence, precisely at mass levels above $$m_c$$ the constant steady states of ( $$\star $$ ) possess the extreme instability property of repelling arbitrary concentration-increasing perturbations in such a drastic sense that corresponding trajectories collapse in finite time.

Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production

We study the Neumann initial-boundary problem for the chemotaxis system \left\{ \begin{array}{ll} u_t= \Delta u - \nabla \cdot (u\nabla v), &amp; x\in \Omega, \, t&gt;0, \\ 0=\Delta v - \mu(t)+w, &amp; x\in \Omega, \, t&gt;0, \\ \tau w_t + \delta w = u, &amp; x\in \Omega, \, t&gt;0, \end{array} \right. \qquad \qquad (\star) in the unit disk \Omega:=B_1(0)\subset R^2 , where \delta\ge 0 and \tau&gt;0 are given parameters and \mu(t):=\frac{1}{|\Omega|} \int_\Omega w(x,t)dx , t&gt;0 . It is shown that this problem exhibits a novel type of critical mass phenomenon with regard to the formation of singularities, which drastically differs from the well-known threshold property of the classical Keller-Segel system, as obtained upon formally taking \tau\to 0 , in that it refers to blow-up in infinite time rather than in finite time. Specifically, it is first proved that for any sufficiently regular nonnegative initial data u_0 and w_0 , ( \star ) possesses a unique global classical solution. In particular, this shows that in sharp contrast to classical Keller-Segel-type systems reflecting immediate signal secretion by the cells themselves, the indirect mechanism of signal production in ( \star ) entirely rules out any occurrence of blow-up in finite time. However, within the framework of radially symmetric solutions it is next proved that \|u(\cdot,t)\|_{L^\infty(\Omega)} \to \infty \qquad \text{as } t\to\infty.

Response to “Comment on ‘Lock-in in forced vibration of a circular cylinder’” [Phys. Fluids 29, 109101 (2017)

The map of the energy transfer between the fluid and structure, on the A*− f* plane, for forced vibrations at Re = 100 was presented in our earlier work. The free vibration response of the cylinder for m* = 10 and zero damping, at the same Re, was overlaid on the energy map. In this work we add free vibration data to the plot for more values of m*. This data has been recently presented by Navrose and Mittal [“The critical mass phenomenon in vortex-induced vibration at low Re,” J. Fluid Mech. 820, 159–186 (2017)] for zero structural damping and a wide range of mass ratio up to very low values of m*.

The critical mass phenomenon in vortex-induced vibration at low

Vortex-induced vibration of a circular cylinder with low mass ratio ($0.05\leqslant m^{\ast }\leqslant 10.0$) is investigated, via a stabilized space–time finite element formulation, in the laminar flow regime where$m^{\ast }$is defined as the ratio of the mass of the oscillating structure to the mass of the fluid displaced by it. Computations are carried out over a wide range of reduced speed,$U^{\ast }$, which is defined as$U/f_{N}D$, where$U$is the free-stream speed,$f_{N}$the natural frequency of the spring mass system in vacuum and$D$the diameter of the cylinder. In particular, the situation where the lock-in regime extends up to infinite reduced speed is explored. Studies at large$Re$, in the past, have shown that the normalized amplitude of cylinder oscillation at infinite reduced speed,$A_{\infty }^{\ast }$, exhibits a sharp increase when$m^{\ast }$is reduced below the critical mass ratio ($m_{crit}^{\ast }$). This jump signifies a shift from desynchronized response to lock-in state. In this work it is shown that in the laminar regime, a jump in$A_{\infty }^{\ast }$occurs only beyond a certain$Re$($=Re_{j}\sim 108$). For$Re&lt;Re_{j}$, the response increases smoothly with decrease in$m^{\ast }$with no discernible jump. In this situation, therefore, the identification of$m_{crit}^{\ast }$based on jump in response at$U^{\ast }=\infty$is not possible. The difference in the$A^{\ast }-m^{\ast }$variation on the two sides of$Re=Re_{j}$, is attributed to the difference in the transition between the lower branch of cylinder response and desynchronization regime. This transition is brought out more clearly by plotting$A^{\ast }$with$f_{v_{o}}/f$, where$f_{v_{o}}$is the vortex shedding frequency for the flow past a stationary cylinder and$f$is the cylinder vibration frequency. In the$A^{\ast }-f_{v_{o}}/f$plane, the response data as well as other quantities related to free vibrations, for different$m^{\ast }$, collapse on a curve. Unlike at high$Re$, the collapsed curves show a dependence on$Re$in the laminar regime. The transition between the lock-in and desynchronized state, as seen from the collapsed curves, is qualitatively different for$Re$on either side of$Re_{j}$. The collapsed curves, at a certain$Re$, are utilized to estimate$A^{\ast }$for the limiting case of$(U^{\ast },m^{\ast })=(\infty ,0)$. Interestingly, unlike at large$Re$, this limit value is found to be lower than the peak amplitude of cylinder vibration at a given$Re$. Hysteresis in the cylinder response, near the higher-$U^{\ast }$end of the lock-in regime, is explored. It is observed that the range of$U^{\ast }$with hysteretic response increases with decrease in$m^{\ast }$. Interestingly, for a certain range of$m^{\ast }$, the response is hysteretic from a finite$U^{\ast }$up to$U^{\ast }=\infty$. We refer to this phenomenon as hysteresis forever. It occurs because of the existence of multiple response states of the system at$U^{\ast }=\infty$, for a certain range of$m^{\ast }$. The study brings out the significant differences in the response of the fluid–structure system associated with the critical mass phenomenon between the low- and high-$Re$regime.

Boundedness, blowup and critical mass phenomenon in competing chemotaxis

We consider the following attraction–repulsion Keller–Segel system:{ut=Δu−∇⋅(χu∇v)+∇⋅(ξu∇w),x∈Ω,t>0,vt=Δv+αu−βv,x∈Ω,t>0,0=Δw+γu−δw,x∈Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω, with homogeneous Neumann boundary conditions in a bounded domain Ω⊂R2 with smooth boundary. The system models the chemotactic interactions between one species (denoted by u) and two competing chemicals (denoted by v and w), which has important applications in Alzheimer's disease. Here all parameters χ, ξ, α, β, γ and δ are positive. By constructing a Lyapunov functional, we establish the global existence of uniformly-in-time bounded classical solutions with large initial data if the repulsion dominates or cancels attraction (i.e., ξγ≥αχ). If the attraction dominates (i.e., ξγ<αχ), a critical mass phenomenon is found. Specifically speaking, we find a critical mass m⁎=4παχ−ξγ such that the solution exists globally with uniform-in-time bound if M<m⁎ and blows up if M>m⁎ and M∉{4πmθ:m∈N+} where N+ denotes the set of positive integers and M=∫Ωu0dx the initial cell mass.

New critical exponents in a fully parabolic quasilinear Keller–Segel system and applications to volume filling models

We carry on our studies related to the fully parabolic quasilinear Keller–Segel system started in [6] and continued in [7]. In the above mentioned papers we proved finite-time blowup of some radially symmetric solutions to the quasilinear Keller–Segel system if the nonlinear chemosensitivity is strong enough and an adequate relation between nonlinear diffusion and chemosensitivity holds. On the other hand we proved that once chemosensitivity is weak enough solutions exist globally in time. The present paper is devoted to looking for critical exponents distinguishing between those two behaviors. Moreover, we apply our results to the so-called volume filling models with a power-type jump probability function. The most important consequence of our investigations of the latter is a critical mass phenomenon found in dimension 2. Namely we find a value m⁎ such that when the solution to the two-dimensional volume filling Keller–Segel system starts with mass smaller than m⁎, then it is bounded, while for some initial data with mass exceeding m⁎ solutions are unbounded, though being defined for any time t>0.

Exponential speed of uniform convergence of the cell density toward equilibrium for subcritical mass in a Patlak–Keller–Segel model

This paper is concerned with a chemotaxis aggregation model for cells, more precisely with a parabolic–elliptic semilinear Patlak–Keller–Segel system in a ball of RN, exhibiting a critical mass phenomenon for N≥2. The main result of this paper is the exponential speed of uniform convergence of radial solutions toward the unique steady state in the subcritical case. We stress that this covers in particular the classical Keller–Segel system with N=2, well known for its critical mass 8π, and that the result improves on the known results even for this most studied problem. A key tool is an associated one-dimensional degenerate parabolic problem. The proof exploits its formal gradient flow structure ut=−∇F[u(t)] on an “infinite dimensional Riemannian manifold”. In particular, we show a new Hardy type inequality, equivalent to the strict convexity of F at any steady state of subcritical mass, which heuristically explains the exponential speed of convergence.

A semilinear parabolic–elliptic chemotaxis system with critical mass in any space dimension

We study radial solutions in a ball of of a semilinear, parabolic–elliptic Patlak–Keller–Segel system with a nonlinear sensitivity involving a critical power. For N = 2, the latter reduces to the classical ‘linear’ model, well known for its critical mass 8π. We show that a critical mass phenomenon also occurs for N ⩾ 3, but with a strongly different qualitative behaviour. More precisely, if the total mass of cells is smaller or equal to the critical mass , then the cell density converges to a regular steady state that is supported strictly inside the ball as time goes to infinity. In the case of the critical mass, this result is nontrivial since there exists a continuum of stationary solutions and is moreover in sharp contrast with the case N = 2 where infinite-time blow-up occurs. If the total mass of cells is larger than , then all radial solutions blow up in finite time. This actually follows from the existence (unlike for N = 2) of a family of self-similar, blowing-up solutions that are supported strictly inside the ball.

Global Existence and Finite Time Blow-Up for Critical Patlak--Keller--Segel Models with Inhomogeneous Diffusion

The $L^1$-critical parabolic-elliptic Patlak--Keller--Segel system is a classical model of chemotactic aggregation in microorganisms well known to have critical mass phenomena [Blanchet, Dolbeault, and Perthame, Electron. J. Differential Equations, 2006 (2006), pp. 1--23; Blanchet, Carrillo, and Laurençot, Calc. Var., 35 (2009), pp. 133--168]. In this paper we study this critical mass phenomenon in the context of Patlak--Keller--Segel models with spatially varying diffusivity of the chemoattractant. The primary issue is how, if possible, one localizes the presence of the inhomogeneity in the nonlocal term. Our methods also provide new blow-up results for the homogeneous problem with nonlinear diffusion, showing that there exist blow-up solutions with arbitrarily large (positive) initial free energy.

The effect of Reynolds number on the critical mass phenomenon in vortex-induced vibration

In this study, we investigate the critical mass phenomenon in vortex-induced vibration over a wide range of Reynolds numbers (Re=4000–30 000). We consider an elastically mounted cylinder that is able to vibrate transverse to a fluid flow. If we remove the restoring spring (k=0), then above a certain critical mass ratio (m∗ is oscillating body mass/displaced fluid mass), the cylinder will experience almost no motion, despite its unrestrained freedom to move transverse to the flow. However, when the mass ratio is decreased below a special critical value, without altering anything else in the system, we see a catastrophic increase in amplitude, and the body settles into a large amplitude periodic vibration. This corresponds to a change from a desynchronized wake to a 2P mode of vortex formation, where two pairs of vortices are formed per cycle of motion. Since a system with no restoring force represents a case of infinite normalized velocity (U∗→∞), the observation of high amplitude motion indicates that the regime of U∗ giving resonant vibration extends to infinity, for sufficiently small mass ratio. In this work, we measure the critical mass directly from experiments with no spring stiffness, and we show also how the critical mass may be accurately predicted from force measurements from controlled vibration experiments, or from free vibration measurements of elastically mounted cylinders. Critical mass gradually increases with Reynolds number from a value of 0.36 to 0.54, over the regime Re=4000–30 000. The fact that the critical mass is a function of Reynolds numbers should be expected because it depends on the vortex-induced forces, which are influenced by gradual changes in vortex formation as Re increases. The evaluation of critical mass in this configuration, and indeed in other diverse vortex-induced vibration (VIV) systems, is important because it can predict the regime of normalized velocities that will yield large amplitude vibration, and which one may wish to avoid in practice. The fact that critical mass, at moderate Reynolds numbers, in several diverse VIV systems, including cylinders in one or two degrees of freedom, pivoted bodies, cantilevers, and tethered spheres, are all within a small range 0.36–0.6, remains an interesting question.