Subcritical Case Research Articles

On the Kolmogorov constant explicit form in the theory of discrete-time stochastic branching systems

AbstractWe consider a discrete-time population growth system called the Bienaymé–Galton–Watson stochastic branching system. We deal with a noncritical case, in which the per capita offspring mean $m\neq1$ . The famous Kolmogorov theorem asserts that the expectation of the population size in the subcritical case $m<1$ on positive trajectories of the system asymptotically stabilizes and approaches ${1}/\mathcal{K}$ , where $\mathcal{K}$ is called the Kolmogorov constant. The paper is devoted to the search for an explicit expression of this constant depending on the structural parameters of the system. Our argumentation is essentially based on the basic lemma describing the asymptotic expansion of the probability-generating function of the number of individuals. We state this lemma for the noncritical case. Subsequently, we find an extended analogue of the Kolmogorov constant in the noncritical case. An important role in our discussion is also played by the asymptotic properties of transition probabilities of the Q-process and their convergence to invariant measures. Obtaining the explicit form of the extended Kolmogorov constant, we refine several limit theorems of the theory of noncritical branching systems, showing explicit leading terms in the asymptotic expansions.

Gravity–capillary wave-making resistance on deep water

For supercritical cases (forcing speed > the minimum phase speed, 0.23 m/s), the problem of two-dimensional linear, inviscid gravity–capillary waves generated by a moving delta-function type pressure source is well known. Using harmonic functions or Fourier transform, Lamb [Hydrodynamics, 6th ed. (Cambridge University Press, 1993)] and Rayleigh [Proc. London Math. Soc. s1-15(1), 69–78 (1883)] detailed the steady-state full-space wave-profile solution using an artificial viscosity. Whitham [Linear and Nonlinear Waves (Wiley-Interscience, 1974)] presented the same solution for the region that is far-from-the-forcing using a slowly varying exponential function. For the same problem, but, considering not only supercritical but also subcritical cases, and, without using the artificial viscosity, the present work provides a detailed solution procedure to find full-space wave-profile solutions based on Fourier transform where complex integration is needed; different analytical expressions of the same wave profile will be provided depending on different paths. The associated wave-making resistance is shown to be equal to the integral of the product of a moving pressure source, and the resultant wave slope and is calculated in two ways. One is a direct calculation in the physical domain, which requires the wave-profile solution, and the other is an indirect calculation in the wavenumber domain, which does not require the wave-profile solution. For supercritical cases, short and long sinusoidal waves are calculated ahead of and behind the pressure source. The associated wave-making resistance decreases toward a certain constant as the forcing speed increases and the associated required power features a minimum at the forcing speed of 0.3027 m/s. For subcritical cases, a simple symmetric dimple is calculated and the wave-making resistance becomes zero due to its symmetry.

Pitchfork bifurcation and vibrational resonance in a neutral delayed system

Abstract The pitchfork bifurcation and vibrational resonance phenomenon are analytically and numerically investigated in the neutral delayed system. The results show that the neutral delayed time induces the pitchfork bifurcation transiting between supercritical and subcritical cases while the neutral delayed strength may induce supercritical pitchfork bifurcation. The neutral delayed time can also induce the periodic or quasi-periodic vibrational resonance. This period simultaneously contains the period of the low-frequency harmonic excitation and period of the high-frequency harmonic excitation. The results extend the investigations of the pitchfork bifurcation and vibrational resonance.

2D Surface Quasi Geostrophic Equations and its Regularity, A Numerical Study

In this article, we present some results on the numerical solutions of the 2-D Surface Quasi Geostrophic Equation (SQG) using the pseudospectral method along with an exponential filter. The global regularity of the solution of the inviscid SQG equation for general data remains an outstanding open problem. Our computations mainly focus on the inviscid and supercritical cases. We monitored the regions where the level curves come significantly close to one another, the L2 norm, and the growth of |∇⊥θ| throughout our computations. Our numerical findings show that there is no significant difference among the solutions of the supercritical, critical, and subcritical cases as we vary the values of the parameter α in the interval (0, 1).

Normalized solutions for the Schrödinger-Poisson system with doubly critical growth

In this paper we are concerned with normalized solutions to the Schrödinger-Poisson system with doubly critical growth \[ \begin{cases} -\Delta u-\phi |u|^3u=\lambda u+\mu|u|^{q-2}u+|u|^4u, &x \in \R^{3},\\ -\Delta \phi=|u|^5, &x \in \R^{3}, \end{cases} \] and prescribed mass \[ \int_{\R^3}|u|^2dx=a^2,\] where $a> 0$ is a constant, $\mu> 0$ is a parameter and $2< q< 6$. In the $L^2$-subcritical case, we study the multiplicity of normalized solutions by applying the truncation technique, and the genus theory; and in the $L^2$-supercritical case, we obtain a couple of normalized solutions by developing a fiber map. Under both cases, to recover the loss of compactness of the energy functional caused by the critical growth, we need to adopt the concentration-compactness principle. Our results complement and improve some related studies for the Schrödinger-Poisson system with nonlocal critical term in the literature.

On the Cauchy problem for semilinear σ‐evolution equations with time‐dependent damping

In this paper, we would like to consider the Cauchy problem for semilinear ‐evolution equations with time‐dependent damping for any . Motivated strongly by the classification of damping terms in some previous papers, the first main goal of the present work is to make some generalizations from to and simultaneously to investigate decay estimates for solutions to the corresponding linear equations in the so‐called effective damping cases. For the next main goals, we are going not only to prove the global well‐posedness property of small data solutions but also to indicate blow‐up results for solutions to the semilinear problem. In this concern, the novelty which should be recognized is that the application of a modified test function combined with a judicious choice of test functions gives blow‐up phenomena and upper bound estimates for lifespan in both the subcritical case and the critical case, where is assumed to be any fractional number. Finally, lower bound estimates for lifespan in some spatial dimensions are also established to find out their sharp results.

Semilinear elliptic Schrödinger equations with singular potentials and absorption terms

AbstractLet () be a bounded domain and be a compact, submanifold without boundary, of dimension with . Put in , where and is a parameter. We investigate the boundary value problem (P) in with condition on , where is a nondecreasing, continuous function, and and are positive measures. The complex interplay between the competing effects of the inverse‐square potential , the absorption term and the measure data discloses different scenarios in which problem (P) is solvable. We provide sharp conditions on the growth of for the existence of solutions. When is a power function, namely with , we show that problem (P) admits several critical exponents in the sense that singular solutions exist in the subcritical cases (i.e. is smaller than a critical exponent) and singularities are removable in the supercritical cases (i.e. is greater than a critical exponent). Finally, we establish various necessary and sufficient conditions expressed in terms of appropriate capacities for the solvability of (P).

Speed of extinction for continuous-state branching processes in a weakly subcritical Lévy environment

AbstractWe continue with the systematic study of the speed of extinction of continuous-state branching processes in Lévy environments under more general branching mechanisms. Here, we deal with the weakly subcritical regime under the assumption that the branching mechanism is regularly varying. We extend recent results of Li and Xu (2018) and Palau et al. (2016), where it is assumed that the branching mechanism is stable, and complement the recent articles of Bansaye et al. (2021) and Cardona-Tobón and Pardo (2021), where the critical and the strongly and intermediate subcritical cases were treated, respectively. Our methodology combines a path analysis of the branching process together with its Lévy environment, fluctuation theory for Lévy processes, and the asymptotic behaviour of exponential functionals of Lévy processes. Our approach is inspired by the last two previously cited papers, and by Afanasyev et al. (2012), where the analogue was obtained.

Solutions for the Navier-Stokes equations with critical and subcritical fractional dissipation in Lei-Lin and Lei-Lin-Gevrey spaces

In this article, we prove the existence of a unique global solution for the critical case of the generalized Navier-Stokes equations in Lei-Lin and Lei-Lin-Gevrey spaces, by assuming that the initial data is small enough. Moreover, we obtain a unique local solution for the subcritical case of this system, for any initial data, in these same spaces. It is important to point out that our main result is obtained by discussing some properties of the solutions for the heat equation with fractional dissipation.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/78/abstr.html

Concentrating ground state for linearly coupled Schrödinger systems involving critical exponent cases

We study the following linearly coupled Schrödinger system:{−Δu+(μV1(x)+a)u=f(x)|u|p−2u+β(x)v,x∈RN,−Δv+(μV2(x)+b)v=g(x)|v|2⁎−2v+β(x)u,x∈RN, where N≥3,2<p≤2⁎, V1,V2∈C(RN,R+) are potential wells with bottoms Ωi=intVi−1(0), the parameters a>−λ1(Ω1),b>−λ1(Ω2) and λ1(Ωi) is the first eigenvalue of −Δ in H01(Ωi). Under some suitable assumptions on β(x) which relate to the potentials V1,V2 and the parameters a,b, the existence of positive ground states is obtained by variational method. Some interesting phenomena are that we relax the upper control condition of the coupling function β(x) and we do not need the weight function f(x) to be integrable or bounded in the subcritical case 2<p<2⁎. Moreover, from the concentration phenomenon of solutions, we obtain some results of the existence of positive ground states to a linearly coupled Schrödinger system in a bounded domain, which extend the recent results of Peng, et al. (2017), [25].

Strong unique continuation for variable coefficient parabolic operators with Hardy type potential

In this paper, we prove the strong unique continuation property at the origin for solutions of the following scaling critical parabolic differential inequality|div(A(x,t)∇u)−ut|≤M|x|2|u|, where the coefficient matrix A is Lipschitz continuous in x and t. Our main result sharpens a previous one of Vessella concerned with the subcritical case as well as extends an earlier result of one of us with Garofalo and Manna for the heat operator.

Small perturbations of critical nonlocal equations with variable exponents

Abstract In this article, we are concerned with the following critical nonlocal equation with variable exponents: ( − Δ ) p ( x , y ) s u = λ f ( x , u ) + ∣ u ∣ q ( x ) − 2 u in Ω , u = 0 in R N \ Ω , \left\{\begin{array}{ll}{\left(-\Delta )}_{p\left(x,y)}^{s}u=\lambda f\left(x,u)+{| u| }^{q\left(x)-2}u&amp; \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0&amp; \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right,\end{array}\right. where Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with Lipschitz boundary, N ≥ 2 N\ge 2 , p ∈ C ( Ω × Ω ) p\in C(\Omega \times \Omega ) is symmetric, f : C ( Ω × R ) → R f:C\left(\Omega \times {\mathbb{R}})\to {\mathbb{R}} is a continuous function, and λ \lambda is a real positive parameter. We also assume that { x ∈ R N : q ( x ) = p s ∗ ( x ) } ≠ ∅ \left\{x\in {{\mathbb{R}}}^{N}:q\left(x)={p}_{s}^{\ast }\left(x)\right\}\ne \varnothing , and p s ∗ ( x ) = N p ˜ ( x ) ⁄ ( N − s p ˜ ( x ) ) {p}_{s}^{\ast }\left(x)=N\tilde{p}\left(x)/\left(N-s\tilde{p}\left(x)) is the critical Sobolev exponent for variable exponents. We prove the existence of non-trivial solutions in the case of low perturbations ( λ \lambda small enough) by using the mountain pass theorem, the concentration-compactness principles for fractional Sobolev spaces with variable exponents, and the Moser iteration method. The features of this article are the following: (1) the function f f does not satisfy the usual Ambrosetti-Rabinowitz condition and (2) this article contains the presence of critical terms, which can be viewed as a partial extension of the previous results concerning the the existence of solutions to this problem in the case of s = 1 s=1 and subcritical case.

On Liouville theorems of a Hartree–Poisson system

AbstractIn this paper, we are concerned with the non-existence of positive solutions of a Hartree–Poisson system: \begin{equation*} \left\{ \begin{aligned} &amp;-\Delta u=\left(\frac{1}{|x|^{n-2}}\ast v^p\right)v^{p-1},\quad u \gt 0\ \text{in} \ \mathbb{R}^{n},\\ &amp;-\Delta v=\left(\frac{1}{|x|^{n-2}}\ast u^q\right)u^{q-1},\quad v \gt 0\ \text{in} \ \mathbb{R}^{n}, \end{aligned} \right. \end{equation*} where $n \geq3$ and $\min\{p,q\} \gt 1$. We prove that the system has no positive solution under a Serrin-type condition. In addition, the system has no positive radial classical solution in a Sobolev-type subcritical case. In addition, the system has no positive solution with some integrability in this Sobolev-type subcritical case. Finally, the relation between a Liouville theorem and the estimate of boundary blowing-up rates is given.

The Cauchy–Dirichlet problem for the fast diffusion equation on bounded domains

The Fast Diffusion Equation (FDE) ut=Δum, with m∈(0,1), is an important model for singular nonlinear (density dependent) diffusive phenomena. Here, we focus on the Cauchy–Dirichlet problem posed on smooth bounded Euclidean domains. In addition to its physical relevance, there are many aspects that make this equation particularly interesting from the pure mathematical perspective. For instance: mass is lost and solutions may extinguish in finite time, merely integrable data can produce unbounded solutions, classical forms of Harnack inequalities (and other regularity estimates) fail to be true, etc.In this paper, we first provide a survey (enriched with an extensive bibliography) focussing on the more recent results about existence, uniqueness, boundedness and positivity (i.e., Harnack inequalities, both local and global), and higher regularity estimates (also up to the boundary and possibly up to the extinction time). We then prove new global (in space and time) Harnack estimates in the subcritical regime. In the last section, we devote a special attention to the asymptotic behaviour, from the first pioneering results to the latest sharp results, and we present some new asymptotic results in the subcritical case.

Orbital Stability of Ground States for a Mass Subcritical Fractional Schrodinger-Poisson Equation

In this paper, we study the existence of ground state standing waves with prescribed mass constraints for the fractional Schrodinger-Poisson equation \(\mathit{i}\partial_t\psi\) - (- \(\Delta\))\(\\^S\)\(\psi\) - (|\(\mathit{x}\)|\(\\^2\\^t\\^-\\^3\) * |\(\psi\)|\(\\^2\))\(\psi\) + |\(\psi\)|\(\\^p\\^-\\^2\)\(\psi\) = 0, where \(\psi\) : \(\mathbb{R}\\^3\) x \(\mathbb{R}\) \(\to\) \(\mathbb{C}\), s, t \(\in\) (0,1), 2s + 2t &gt; 3 and \(\mathit{p}\) \(\in\) \((2 , {4s \pm 2t \over s+t})\). In particular, in the mass subcritical case but \(p \neq \frac{4 s+2 t}{s+t}\), that is, \(p \in\left(2,2+\frac{4 s}{3}\right) \backslash\left\{\frac{4 s+2 t}{s+t}\right\}\),we prove that the solution with initial datum 0 exists globally and the set of ground states is orbitally stable.

Global existence and blow-up in higher-dimensional Patlak-Keller-Segel system for multi populations

This work is concerned with a degenerate parabolic-elliptic Patlak-Keller-Segel system for multi populations in space dimensions d≥3. We first provide a sufficient condition for the global existence of weak solution to the Cauchy problem. Then the global results are obtained both for any large initial data in the sub-critical case and for small initial data in the super-critical case. Finally, the finite-time blow-up solutions are constructed for large initial data in the super-critical case.

On Attractors of Ginzburg–Landau Equations in Domain with Locally Periodic Microstructure: Subcritical, Critical, and Supercritical Cases

On Attractors of Ginzburg–Landau Equations in Domain with Locally Periodic Microstructure: Subcritical, Critical, and Supercritical Cases

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.

Normalized solutions for Schrödinger-Poisson equation with prescribed mass: The Sobolev subcritical case and the Sobolev critical case with mixed dispersion

In this paper, we study the following Schrödinger Poisson equation{−Δu−λu−γ(|x|−1⁎|u|2)u=f(u),x∈R3;∫R3u2dx=c, where c>0, λ∈R and f∈C(R,R). When γ<0 and f satisfies some weaker L2-supercritical conditions in Sobolev subcritical case, we show the existence of normalized solutions. When f(u)=μ|u|q−2u+|u|4u with μ>0, γ>0 and 2<q<103, a more complicated situation, we obtain the existence of multiple solutions. Our main tools are some technically simpler than the Ghoussoub minimax principle [18], which not only allow us to construct a bounded (PS) sequence under some weaker conditions on f than before, but help to yield richer results. In particular, our results generalize and improve some ones in [21] and some other related literatures.

Comparative molecular dynamics study of liquid hydrogen annular jets in subcritical and supercritical environments

In this paper, the injection characteristics of a liquid hydrogen propellant jet under subcritical and supercritical operating conditions are comparatively studied at the molecular scale using molecular dynamics methods in the context of spacecraft propulsion systems. To control the temperature and pressure within the environment and nozzle respectively, two sub-computational domains are set up for separate modeling. The sub-computational domain merging process eliminates the broken molecular bonds by using the computational domain reduction in the x-direction. The jet velocity, density and other physical property distributions at different temperatures and pressures were comparatively studied. It is found that the rate of change of the radial density of the jet under supercritical conditions is small. Based on this, the surface area and volume of the liquid phase region of the jet were extracted and calculated. Compared to result of subcritical case, the volume of the liquid hydrogen jet in the supercritical environment is reduced by 1.9%, while the surface area is increase by 6.7%. To explain the variation in the surface area due to molecular diffusion, the molecular stress results revealed that the stress in the hydrogen jet in supercritical environment is only 88.9% of that in subcritical environment.