Conditions For Uniqueness Of Solutions Research Articles

Supercritical Poincaré–Andronov–Hopf Bifurcation in a Mean-Field Quantum Laser Equation

We deal with the dynamical system properties of a Gorini–Kossakowski–Sudarshan–Lindblad equation with mean-field Hamiltonian that models a simple laser by applying a mean-field approximation to a quantum system describing a single-mode optical cavity and a set of two-level atoms, each coupled to a reservoir. We prove that the mean-field quantum master equation has a unique regular stationary solution. In case a relevant parameter $$C_\mathfrak {b} $$ , i.e., the cavity cooperative parameter, is less than 1, we prove that any regular solution converges exponentially fast to the equilibrium, and so the regular stationary state is a globally asymptotically stable equilibrium solution. We obtain that a locally exponential stable limit cycle is born at the regular stationary state as $$C_\mathfrak {b} $$ passes through the critical value 1. Then, the mean-field laser equation has a Poincare–Andronov–Hopf bifurcation at $$C_\mathfrak {b} =1 $$ of supercritical-like type. Namely, we derive rigorously, at the level of density matrices—for the first time—the transition from a global attractor quantum state, where the light is not emitted, to a locally stable set of coherent quantum states producing coherent light. Moreover, we establish the local exponential stability of the limit cycle in case a relevant parameter is between the first and second laser thresholds appearing in the semiclassical laser theory. Thus, we get that the coherent laser light persists over time under this condition. In order to prove the exponential convergence of the quantum state, as the time goes to $$+ \infty $$ , we develop a new technique for proving the exponential convergence in open quantum systems that is based on a new variation of constant formula, which is obtained by combining probabilistic techniques with classical arguments from the semigroup theory. Furthermore, applying our main results we find the long-time behavior of the von Neumann entropy, the photon number statistics, and the quantum variance of the quadratures.

On stationary solutions and inviscid limits for generalized Constantin–Lax–Majda equation with O(1) forcing

The generalized Constantin–Lax–Majda (gCLM) equation was introduced to model the competing effects of advection and vortex stretching in hydrodynamics. Recent investigations revealed possible connections with the two-dimensional turbulence. With this connection in mind, we consider the steady problem for the viscous gCLM equations on : where is the parameter measuring the relative strength between advection and stretching, ν > 0 is the viscosity constant, and f is a given O(1)-forcing independent of ν. For some range of parameters, we establish existence and uniqueness of stationary solutions. We then numerically investigate the behaviour of solutions in the vanishing viscosity limit, where bifurcations appear, and new solutions emerge. When the parameter a is away from [−1/2, 1], we verify that there is convergence towards smooth stationary solutions for the corresponding inviscid equation. Moreover, we analyse the inviscid limit in the fractionally dissipative case, as well as the behaviour of singular limiting solutions.

Existence and uniqueness of solutions for a coupled system of sequential fractional differential equations with initial conditions

In this paper, we consider a coupled system of sequential fractional differential equations associated with initial conditions. The main theorems provide new existence and uniqueness conditions for solutions of the proposed coupled system. We conclude an immediate consequence that establishes weaker conditions to ensure the existence and uniqueness of solutions for the corresponding sequential fractional differential equation. Meanwhile, an iterative sequence is constructed in terms of solution operator that converges to the unique fixed point which corresponds to the unique solution. The consistency of the main results is verified by presenting two numerical examples. For the sake of completeness, we end the paper with a concluding remark.

The evolution problem associated with eigenvalues of the Hessian

In this paper we study the evolution problem \[ \left\lbrace\begin{array}{ll} u_t (x,t)- \lambda_j(D^2 u(x,t)) = 0, & \text{in } \Omega\times (0,+\infty), \\ u(x,t) = g(x,t), & \text{on } \partial \Omega \times (0,+\infty), \\ u(x,0) = u_0(x), & \text{in } \Omega, \end{array}\right. \] where $\Omega$ is a bounded domain in $\mathbb{R}^N$ (that verifies a suitable geometric condition on its boundary) and $\lambda_j(D^2 u)$ stands for the $j-$st eigenvalue of the Hessian matrix $D^2u$. We assume that $u_0 $ and $g$ are continuous functions with the compatibility condition $u_0(x) = g(x,0)$, $x\in \partial \Omega$. We show that the (unique) solution to this problem exists in the viscosity sense and can be approximated by the value function of a two-player zero-sum game as the parameter measuring the size of the step that we move in each round of the game goes to zero. In addition, when the boundary datum is independent of time, $g(x,t) =g(x)$, we show that viscosity solutions to this evolution problem stabilize and converge exponentially fast to the unique stationary solution as $t\to \infty$. For $j=1$ the limit profile is just the convex envelope inside $\Omega$ of the boundary datum $g$, while for $j=N$ it is the concave envelope. We obtain this result with two different techniques: with PDE tools and and with game theoretical arguments. Moreover, in some special cases (for affine boundary data) we can show that solutions coincide with the stationary solution in finite time (that depends only on $\Omega$ and not on the initial condition $u_0$).

The structure of stationary solutions to a micro-electro mechanical system with fringing field

We study the structure of stationary solutions of a micro-electro mechanical system with fringing field. It is known that there is a positive critical value such that no stationary solutions exist for the applied voltage beyond this critical value, at least two stationary solutions exist for the applied voltage below this critical value and there is a unique stationary solution at this critical value. In this paper, we verify that there are exactly two solutions below this critical value analytically. Moreover, the stability of the smaller stationary solutions is derived.

On the asymptotic behavior of the solutions to parabolic variational inequalities

Abstract We consider various versions of the obstacle and thin-obstacle problems, we interpret them as variational inequalities, with non-smooth constraint, and prove that they satisfy a new constrained Łojasiewicz inequality. The difficulty lies in the fact that, since the constraint is non-analytic, the pioneering method of L. Simon ([22]) does not apply and we have to exploit a better understanding on the constraint itself. We then apply this inequality to two associated problems. First we combine it with an abstract result on parabolic variational inequalities, to prove the convergence at infinity of the strong global solutions to the parabolic obstacle and thin-obstacle problems to a unique stationary solution with a rate. Secondly, we give an abstract proof, based on a parabolic approach, of the epiperimetric inequality, which we then apply to the singular points of the obstacle and thin-obstacle problems.

Bifurcation for a free boundary problem modeling the growth of multilayer tumors with ECM and MDE interactions

We study a free boundary problem modeling the growth of multilayer tumors. This model describes the invasion of tumors: the tumor cells produce matrix degrading enzymes (MDEs) to degrade the extracellular matrix (ECM) which provides structural support of the surrounding tissue. As in Pan and Xing, the influence of ECM and MDE interactions is considered in this paper. The model equations include two diffusion equations for the nutrient concentration and MDE concentration and an ordinary differential equation for ECM concentration. The tumor cell proliferation is at the rate , which characterizes the aggressiveness of the tumor. In contrast to Pan and Xing, we consider “flat stationary solution” in this paper. We first show that there exists a unique flat stationary solution for any . Then, we prove that there are infinite branches of bifurcation solutions from the flat stationary solutions at the bifurcation points ( ).

Restoration of fast moving objects.

If an object is photographed at motion in front of a static background, the object will be blurred while the background sharp and partially occluded by the object. The goal is to recover the object appearance from such blurred image. We adopt the image formation model for fast moving objects and consider objects undergoing 2D translation and rotation. For this scenario we formulate the estimation of the object shape, appearance, and motion from a single image and known background as a constrained optimization problem with appropriate regularization terms. Both similarities and differences with blind deconvolution are discussed with the latter caused mainly by the coupling of the object appearance and shape in the acquisition model. Necessary conditions for solution uniqueness are derived and a numerical solution based on the alternating direction method of multipliers is presented. The proposed method is evaluated on a new dataset.

О единственности решения обратной задачи атмосферного электричества

We investigate the inverse problem of determination of two unknown numerical parameters occuring linearly and nonlinearly in the higher coefficient of a linear second order elliptic equation of the diffusion–reaction type in a domain Ω diffeomorphic to a ball layer under special boundary conditions by observation in neighborhoods of the correspondent amount of points. For an analogous inverse problem under Dirichlet boundary conditions, sufficient conditions of solution uniqueness was obtained by the author formerly, but they had an abstract character and so were inconvenient for practical usage. In the paper, these conditions are extended to the case of different boundary conditions and rendered concrete for the case of the exponential type higher coefficient. The inverse problem investigated in the paper refers to research of electric processes in the Earth atmosphere in the frame of global electric circuit in the stationary approximation and arises from needs of recovering the unknown higher coefficient of the equation on the base of observation data obtained from two local transmitters.

Convergence of transport noise to Ornstein–Uhlenbeck for 2D Euler equations under the enstrophy measure

We consider the vorticity form of the 2D Euler equations which is perturbed by a suitable transport type noise and has white noise initial condition. It is shown that stationary solutions of this equation converge to the unique stationary solution of the 2D Navier–Stokes equation driven by the space-time white noise.

Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity

The paper is devoted to investigating a semilinear parabolic equation with a nonlinear gradient source term: \begin{document}$ u_t = u_{xx}+x^m|u_x|^p, t>0, 0 where \begin{document}$ p>m+2 $\end{document} , \begin{document}$ m\geq0 $\end{document} . Zhang and Hu [Discrete Contin. Dyn. Syst. 26 (2010) 767-779] showed that finite time gradient blowup occurs at the boundary and the accurate blowup rate is also obtained for super-critical boundary value. Throughout this paper, we present a complete large time behavior of a classical solution \begin{document}$ u $\end{document} : \begin{document}$ u $\end{document} is global and converges to the unique stationary solution in \begin{document}$ C^1 $\end{document} norm for subcritical boundary value, and \begin{document}$ u_x $\end{document} blows up in infinite time for critical boundary value. Gradient growup rate is also established by the method of matched asymptotic expansions. In addition, gradient estimate of solutions is obtained by the Bernstein-type arguments.

Exponential-Type GARCH Models With Linear-in-Variance Risk Premium

One of the implications of the intertemporal capital asset pricing model is that the risk premium of the market portfolio is a linear function of its variance. Yet, estimation theory of classical GARCH-in-mean models with linear-in-variance risk premium requires strong assumptions and is incomplete. We show that exponential-type GARCH models such as EGARCH or Log-GARCH are more natural in dealing with linear-in-variance risk premia. For the popular and more difficult case of EGARCH-in-mean, we derive conditions for the existence of a unique stationary and ergodic solution and invertibility following a stochastic recurrence equation approach. We then show consistency and asymptotic normality of the quasi-maximum likelihood estimator under weak moment assumptions. An empirical application estimates the dynamic risk premia of a variety of stock indices using both EGARCH-M and Log-GARCH-M models. Supplementary materials for this article are available online.

Distributed Power Control for Large Energy Harvesting Networks: A Multi-Agent Deep Reinforcement Learning Approach

In this paper, we develop a multi-agent reinforcement learning (MARL) framework to obtain online power control policies for a large energy harvesting (EH) multiple access channel, when only causal information about the EH process and wireless channel is available. In the proposed framework, we model the online power control problem as a discrete-time mean-field game (MFG), and analytically show that the MFG has a unique stationary solution. Next, we leverage the fictitious play property of the mean-field games, and the deep reinforcement learning technique to learn the stationary solution of the game, in a completely distributed fashion. We analytically show that the proposed procedure converges to the unique stationary solution of the MFG. This, in turn, ensures that the optimal policies can be learned in a completely distributed fashion. In order to benchmark the performance of the distributed policies, we also develop a deep neural network (DNN) based centralized as well as distributed online power control schemes. Our simulation results show the efficacy of the proposed power control policies. In particular, the DNN based centralized power control policies provide a very good performance for large EH networks for which the design of optimal policies is intractable using the conventional methods such as Markov decision processes. Further, performance of both the distributed policies is close to the throughput achieved by the centralized policies.

Exponential convergence of solutions for random Hamilton–Jacobi equations

We show that for a family of randomly kicked Hamilton–Jacobi equations on the torus, almost surely, the solution of an initial value problem converges exponentially fast to the unique stationary solution. Combined with the earlier results of the authors, this completes the program in the multi-dimensional setting started by E, Khanin, Mazel and Sinai in the one-dimensional case.

Analysis of necrotic core formation in angiogenic tumor growth

In this paper we study a free boundary problem for tumor growth with angiogenesis and a necrotic core. The problem has two free boundaries: the outer boundary R(t) and the necrotic core boundary ρ(t). In order to receive nutrients σ, the tumor attracts blood vessels at a rate proportional to α, so that the Robin condition ∂σ∂r+α(σ−σ̄)=0 holds on the outer boundary, where σ̄ is the nutrients concentration outside the tumor. The sufficient conditions for the existence, uniqueness and stability of the stationary solution to the model are given. The results show that there exists a positive constant σ0 such that if the external nutrient supply σ̄ is greater than or equal to σ0, there exists a unique stationary solution with a necrotic core and this stationary solution is stable. Meanwhile, we also discuss the asymptotic behavior of the transient solutions when the external nutrient supply is below the threshold value σ0. The effects of the external nutrients supply and the connection between the non-necrotic stage and the necrotic stage are also discussed.

New Existence Results for Fractional Langevin Equation

Solvability of new form of nonlinear Langevin equation involving three fractional orders is discussed in this article. First, based on the Leray–Schauder nonlinear alternative an existence result for the solution is presented and then using the Banach contraction principle sufficient conditions for unique solution are established. The fractional derivatives are described in Caputo sense.

Analysis of a free boundary problem for tumor growth with angiogenesis and time delays in proliferation

In this paper we consider a free boundary problem for tumor growth with angiogenesis and time delays in the process of proliferation. The model is established by using reaction–diffusion dynamics and taking a time delay into account. In order to get more nutrients the tumor will attract blood vessels. Assume α(t) is the rate at which the tumor attracts blood vessels, so that ∂σ∂r+α(t)(σ−σ̄)=0 holds on the boundary, where σ̄ is the concentration of nutrients externally supplied to the tumor. When α is a constant, the stability of the unique stationary solution is proved. When α depends on time, we show that (i) R(t) will remain bounded if α(t) is bounded; (ii) limt→∞R(t)=0 if limt→∞α(t)=0; (iii) if α(t) is almost periodic and the nutrients supply outside the tumor is sufficient, there exists an almost periodic R(t).

Stability properties and dynamics of solutions to viscous conservation laws with mean curvature operator

In this paper we study the long time dynamics of the solutions to an initial-boundary value problem for a scalar conservation law with a saturating nonlinear diffusion. After discussing the existence of a unique stationary solution and its asymptotic stability, we focus our attention on the phenomenon of metastability, whereby the time-dependent solution develops into a layered function in a relatively short time and subsequently approaches a steady state in a very long time interval. Numerical simulations illustrate the results.

Asymptotic stability of a free boundary problem for the growth of multi-layer tumours in the necrotic phase

In this paper we study a free boundary problem for the growth of multi-layer tumours in the necrotic phase. The tumour region is strip-like and divided into a necrotic region and a proliferating region with two free boundaries. The upper free boundary is the tumour surface and is governed by a Stefan condition. The lower free boundary is the interface separating the necrotic region from the proliferating region, and its evolution is implicit and intrinsically governed by an obstacle problem. By the Nash–Moser implicit function theorem we show that the lower free boundary is smoothly dependent on the upper free boundary, and by using analytic semigroups theory we obtain asymptotic stability of the unique flat stationary solution.

Integral conditions for uniqueness of solutions to degenerate parabolic equations

We investigate uniqueness of solutions to the initial value problem for degenerate parabolic equations, posed in bounded domains, where no boundary conditions are prescribed. In order to obtain uniqueness, we need that the solutions satisfy certain integral growth conditions, which are crucially related to the degeneracy of the operator near the boundary. In particular, such solutions can be unbounded near the boundary. Our hypothesis on the behavior of the operator at the boundary is optimal; in fact, we show that if it fails, then nonuniqueness of solutions prevails.