Global Solution Research Articles

Global existence and blowup of smooth solutions to the semilinear wave equations in FLRW spacetime

We are interested in the semilinear wave equations evolving in the expanding spacetimes with Friedmann–Lemaître–Robertson–Walker (FLRW) metric. By the weighted energy estimate, we show that when the nonlinearity depends on the time derivative of the unknown, the equation admits a global smooth solution if the spacetime is undergoing accelerated expansion. While the solution will blowup in the sense of some averaged quantity if the expanding rate is not fast enough. When the nonlinearity depends on the space derivatives of the unknown or the unknown itself, we can show that the solution will blowup in finite time even though the expanding rate is fast enough (accelerated expansion). Our results show that the semilinear wave equations in FLRW spacetimes have different properties from the famous Glassey and Strauss conjectures in flat or asymptotically flat spacetimes.

An initialization approach for Metaheuristic Algorithms by using Gibbs Sampling

Recently, several new initialization techniques have been proposed. Despite their good results in very low dimensions, their performance deteriorated significantly with an increase in the number of dimensions. This paper introduces a new method for initializing metaheuristic algorithms, based on the Gibbs sampling approach. The proposed method samples according to the Gibbs method, a multidimensional Gaussian that completely covers the search space defined by the objective function. In this process, each decision variable is sequentially sampled to ensure that the value of each is dependent only on the value of the variable sampled before it. This process leads to the generation of initial positions with significantly low mutual correlation, which is an advantageous feature that prevents the aggregation of initial solutions in specific areas of the search space. This issue is particularly prevalent in optimization problems with a greater number of dimensions. To test the effectiveness of this method, it was applied in conjunction with a Differential Evolution algorithm. The complete approach has been evaluated using a selection of pertinent and challenging functions in the field. The outcome of these experiments showed that the algorithm can establish a superior set of initial solutions, which allows consistent determination of the global solution, even as the complexity of the problem increases with more dimensions.

Bilinear optimization of protein structure prediction: An exact approach via AB off-lattice model

Protein structure prediction (PSP) remains a central challenge in computational biology due to its inherent complexity and high dimensionality. While numerous heuristic approaches have appeared in the literature, their success varies. The AB off-lattice model, which characterizes proteins as sequences of A (hydrophobic) and B (hydrophilic) beads, presents a simplified perspective on PSP. This work presents a mathematical optimization-based methodology capitalizing on the off-lattice AB model. Dissecting the inherent non-linearities of the energy landscape of protein folding allowed for formulating the PSP as a bilinear optimization problem. This formulation was achieved by introducing auxiliary variables and constraints that encapsulate the nuanced relationship between the protein’s conformational space and its energy landscape. The proposed bilinear model exhibited notable accuracy in pinpointing the global minimum energy conformations on a benchmark dataset presented by the Protein Data Bank (PDB). Compared to traditional heuristic-based methods, this bilinear approach yielded exact solutions, reducing the likelihood of local minima entrapment. This research highlights the potential of reframing the traditionally non-linear protein structure prediction problem into a bilinear optimization problem through the off-lattice AB model. Such a transformation offers a route toward methodologies that can determine the global solution, challenging current PSP paradigms. Exploration into hybrid models, merging bilinear optimization and heuristic components, might present an avenue for balancing accuracy with computational efficiency.

Weak Solutions to an Initial-Boundary Value Problem of a Consistent Three-phase-field Model

Global warming has caused the melting of glaciers, especially the Greenland sea ice which is a great threat to human beings. The phase-field method is used to describe the evolution of multi-phase microstructure of Greenland sea ice. This paper shall investigate the global existence, uniqueness and regularity of weak solutions to an initial-boundary value problem for a three-phase-field model based on criteria that lead to both physical and mathematical consistency. For global solutions, it is calculated by applying the method of continuation of local solutions. This model allows for the formulation of spurious phases, which is completely consistent with physical reality.

Effects of multiplicative noise on the fractional Hartree equation

This paper is dedicated to radial solutions to the Cauchy problem for the fractional Hartree equation with multiplicative noise. First, we establish a stochastic Strichartz estimate related to the fractional Schrödinger propagator. Local well-posedness for the Cauchy problem is proved by using stochastic and radial deterministic Strichartz estimates. Then, based on Itô’s formula and stopping time arguments, the existence of a global solution is studied. Finally, we investigate the blow-up phenomenon and give a criterion via localized virial estimates.

Multi-strategy learning-based particle swarm optimization algorithm for COVID-19 threshold segmentation

With advancements in science and technology, the depth of human research on COVID-19 is increasing, making the investigation of medical images a focal point. Image segmentation, a crucial step preceding image processing, holds significance in the realm of medical image analysis. Traditional threshold image segmentation proves to be less efficient, posing challenges in selecting an appropriate threshold value. In response to these issues, this paper introduces Inner-based multi-strategy particle swarm optimization (IPSOsono) for conducting numerical experiments and enhancing threshold image segmentation in COVID-19 medical images. A novel dynamic oscillatory weight, derived from the PSO variant for single-objective numerical optimization (PSOsono) is incorporated. Simultaneously, the historical optimal positions of individuals in the particle swarm undergo random updates, diminishing the likelihood of algorithm stagnation and local optima. Moreover, an inner selection learning mechanism is proposed in the update of optimal positions, dynamically refining the global optimal solution. In the CEC 2013 benchmark test, PSOsono demonstrates a certain advantage in optimization capability compared to algorithms proposed in recent years, proving the effectiveness and feasibility of PSOsono. In the Minimum Cross Entropy threshold segmentation experiments for COVID-19, PSOsono exhibits a more prominent segmentation capability compared to other algorithms, showing good generalization across 6 CT images and further validating the practicality of the algorithm.

Asymptotic Behavior of Global Solution of an Anomalous Gray Scott Model

The object of this paper is to prove that existence global and asymptotic behavior of solutions for anomalous coupled reaction diffusion system (Gray Scott model) with homogeneous Neumann boundary conditions. The existence and uniqueness of the local solution are given by the Banach fixed point theorem. However, the asymptotic behavior is investiged by technique semi group estimates and the Sobolev embedding theorem

Mathematical Models for the Single-Channel and Multi-Channel PMU Allocation Problem and Their Solution Algorithms

Phasor measurement units (PMUs) are deployed at power grid nodes around the transmission grid, determining precise power system monitoring conditions. In real life, it is not realistic to place a PMU at every power grid node; thus, the lowest PMU number is optimally selected for the full observation of the entire network. In this study, the PMU placement model is reconsidered, taking into account single- and multi-capacity placement models rather than the well-studied PMU placement model with an unrestricted number of channels. A restricted number of channels per monitoring device is used, instead of supposing that a PMU is able to observe all incident buses through the transmission connectivity lines. The optimization models are declared closely to the power dominating set and minimum edge cover problem in graph theory. These discrete optimization problems are directly related with the minimum set covering problem. Initially, the allocation model is declared as a constrained mixed-integer linear program implemented by mathematical and stochastic algorithms. Then, the 0/1 integer linear problem is reformulated into a non-convex constraint program to find optimality. The mathematical models are solved either in binary form or in the continuous domain using specialized optimization libraries, and are all implemented in YALMIP software in conjunction with MATLAB. Mixed-integer linear solvers, nonlinear programming solvers, and heuristic algorithms are utilized in the aforementioned software packages to locate the global solution for each instance solved in this application, which considers the transformation of the existing power grids to smart grids.

Integrated optimization of traffic signal timings and vehicle trajectories considering mandatory lane-changing at isolated intersections

Intersections are the predominant bottlenecks in urban areas due to the intricate conflicts among vehicles with different movements. With the advent of connected and automated vehicles (CAVs), it is more effective to reorganize urban traffic in a coordinated manner, and integrate the planning of both the traffic signal timings and vehicle trajectories. Most existing studies in this field assume that all vehicles are in their intended lanes initially, and only optimize the trajectories in the longitudinal dimension for simplicity, which is ideal and unrealistic. Therefore, this paper considers mandatory lane-changing for the integrated optimization of traffic signal timings and vehicle trajectories at isolated intersections. The entire planning horizon is segmented into a series of time-slots, based on which we select phases and plan the trajectories. First, a single-layer mixed integer programming (MIP) model is formulated to obtain global optimal solutions. Nevertheless, for large-scale cases, the MIP model is not applicable due to its computational complexities. Thus, we regulate vehicles into multi-lane platoons that correspond to green phases to reduce the solution space, and propose a beam-search-based platoon formation planning (BPFP) framework for efficient optimization. A beam search procedure is proposed to select phases for each time-slot, which incorporates a vehicle assignment algorithm to design the platoon formation as a subroutine. The two-dimensional vehicle trajectories are subsequently planned according to the platoon formation profiles. Numerical experiments illustrate that BPFP is capable of acquiring near-optimal solutions and considerably outperforms the baselines, while the computation time is within seconds even in over-saturated scenarios.

Improved stochastic fractal search algorithm involving design operators for solving parameter extraction problems in real-world engineering optimization problems

The characteristic parameters of renewable energy sources (RESs) are changes over time under changing operating conditions. Therefore, the accuracy of the model is important in order to establish appropriate control and operating plans for the stable operation of such systems. Determining the parameters of RESs and creating an accurate model of them is an inevitable reality and plays an important role in the success of system modeling. Many methods have been used to determine these parameters in the literature. Furthermore, optimization algorithms are now considered to be among the most significant approaches for resolving problems pertaining to system parameter extraction in many scientific areas. In the study, this parameter extraction is done by a new optimization method called stochastic fractal search algorithm (FDB-NSM-SFS-OBLs), which includes natural survivor updating (NSM) and fitness distance balance (FDB) guiding mechanisms, and opposite based learning (OBL) methods. The performance of the proposed FDB-NSM-SFS-OBL algorithm has been tested in two different experimental studies. In the first experimental study, the proposed approach is tested on CEC2020 benchmark test functions and the algorithm structure containing the best OBL approach is determined using nonparametric Wilcoxon and Friedman statistical analysis methods. The second experimental study is carried out with the proposed optimization algorithm to estimate the parameters of the photovoltaic cell used in the modeling, which is the building block of photovoltaic panels used in renewable energy systems, proton-exchange membrane fuel cell (PEMFC) that is another renewable energy source, and Li-Ion battery used as a backup power unit. The FDB-NSM-SFS-OBL algorithm, which is used both in the CEC2020 benchmark test functions and in determining the parameters of photovoltaic (PV) cells, PEMFC and Li-Ion batteries, performed better in searching and finding the global solution point during the optimization process.

Existence and regularity results for a class of singular parabolic problems with L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} data

In this paper we prove existence and regularity results for a class of parabolic problems with irregular initial data and lower order terms singular with respect to the solution. We prove that, even if the initial datum is not bounded but only in L1(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1(\\Omega )$$\\end{document}, there exists a solution that “instantly” becomes bounded. Moreover we study the behavior in time of these solutions showing that this class of problems admits global solutions which all have the same behavior in time independently of the size of the initial data.

Global well-posedness for the inhomogeneous incompressible MHD equations with variable viscosity coefficient and electrical conductivity

In this paper, we study the Cauchy problem for the inhomogeneous incompressible MHD system with variable viscosity coefficient and electrical conductivity. It is shown that the 3-D inhomogeneous incompressible MHD system has unique global solutions with small initial data conditions. Furthermore, we establish the global well-posedness of the two-dimensional MHD equations in critical spaces, given certain conditions on the density.

David Sedlak. Water for all: global solutions for a changing climate

David Sedlak. Water for all: global solutions for a changing climate

Contribution of PRIDE VLBI products to the joint JUICE-Europa Clipper moons’ ephemerides solution

In the coming decade, JUICE and Europa Clipper radio-science will yield the most accurate estimation to date of the Galilean moons’ physical parameters and ephemerides. JUICE’s PRIDE (Planetary Radio Interferometry and Doppler Experiment) will help achieve such a solution by providing VLBI (Very Long Baseline Interferometry) observations of the spacecraft’s lateral position, complementing nominal radio-science measurements. In this paper, we quantify how PRIDE VLBI can contribute to the moons’ ephemerides determination, in terms of attainable solution improvement and validation opportunities. To this end, we simulated VLBI data for JUICE, but also investigated the possibility to perform simultaneous tracking of JUICE and Europa Clipper, thus ultimately generating both single- and dual-spacecraft VLBI. We considered various tracking and data quality scenarios for both VLBI types, and compared the formal uncertainties provided by covariance analyses with and without VLBI. These analyses were performed for both global and local (i.e. per-flyby) estimations of the moons’ states, as eventually achieving a global solution first requires proceeding arc-per-arc. We showed that both single- and multi-spacecraft VLBI measurements only bring limited improvement to the global state estimation, but significantly contribute to the moons’ normal points (i.e. local states at flyby times), most notably in the out-of-plane direction. Additionally, we designed a validation plan exploiting PRIDE VLBI to progressively validate the classical radio-science solution, whose robustness and statistical realism is sensitive to modelling inconsistencies. By improving the local state estimations and offering various validation opportunities, PRIDE will be invaluable in overcoming possible dynamical challenges. It can therefore play a key role in reconstructing a global solution for the Galilean moons’ dynamics with the uncertainty levels promised by JUICE-Europa Clipper analyses. This, in turn, is critical to the accurate characterisation of tidal dissipation in the Jovian system, holding the key to the long-term evolution of the Galilean moons.

Existence of a global smooth solution for a degenerate problem of the hyperbolic conservation law systems

In this paper, we consider the existence of a global smooth solution for a degenerate problem of the hyperbolic conservation law systems. We use the characteristic decomposition to establish the global existence of smooth solutions to the degenerate Goursat problem up to a boundary and prove that the vacuum bubble is a point. This problem comes from the expansion of a wedge of gas with constant velocity into vacuum in gas dynamics.

Flat flow solution to the mean curvature flow with volume constraint

Abstract In this paper I will revisit the construction of a global weak solution to the volume preserving mean curvature flow via discrete minimizing movement scheme by Mugnai, Seis and Spadaro [L. Mugnai, C. Seis and E. Spadaro, Global solutions to the volume-preserving mean-curvature flow, Calc. Var. Partial Differential Equations 55 2016, 1, Article ID 18]. This method is based on the gradient flow approach due to Almgren, Taylor and Wang [F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim. 31 1993, 2, 387–438] and Luckhaus and Sturzenhecker [S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations 3 1995, 2, 253–271] and my aim is to replace the volume penalization with the volume constraint directly in the discrete scheme, which from practical point of view is perhaps more natural. A technical novelty is the proof of the density estimate which is based on second variation argument.

Complete infinite-time mass aggregation in a quasilinear Keller–Segel system

AbstractRadially symmetric global unbounded solutions of the chemotaxis system $$\left\{ {\matrix{{{u_t} = \nabla \cdot (D(u)\nabla u) - \nabla \cdot (uS(u)\nabla v),} \hfill & {} \hfill \cr {0 = \Delta v - \mu + u,} \hfill & {\mu = {1 \over {|\Omega |}}\int_\Omega {u,} } \hfill \cr } } \right.$$ { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( u S ( u ) ∇ v ) , 0 = Δ v − μ + u , μ = 1 | Ω | ∫ Ω u , are considered in a ball Ω = BR(0) ⊂ ℝn, where n ≥ 3 and R > 0.Under the assumption that D and S suitably generalize the prototypes given by D(ξ) = (ξ + ι)m−1 and S(ξ) = (ξ + 1)−λ−1 for all ξ > 0 and some m ∈ ℝ, λ >0 and ι ≥ 0 fulfilling $$m + \lambda < 1 - {2 \over n}$$ m + λ < 1 − 2 n , a considerably large set of initial data u0 is found to enforce a complete mass aggregation in infinite time in the sense that for any such u0, an associated Neumann type initial-boundary value problem admits a global classical solution (u, v) satisfying $${1 \over C} \cdot {(t + 1)^{{1 \over \lambda }}} \le ||u( \cdot ,t)|{|_{{L^\infty }(\Omega )}} \le C \cdot {(t + 1)^{{1 \over \lambda }}}\,\,\,{\rm{for}}\,\,{\rm{all}}\,\,t > 0$$ 1 C ⋅ ( t + 1 ) 1 λ ≤ | | u ( ⋅ , t ) | | L ∞ ( Ω ) ≤ C ⋅ ( t + 1 ) 1 λ f o r a l l t > 0 as well as $$||u( \cdot \,,t)|{|_{{L^1}(\Omega \backslash {B_{{r_0}}}(0))}} \to 0\,\,\,{\rm{as}}\,\,t \to \infty \,\,\,{\rm{for}}\,\,{\rm{all}}\,\,{r_0} \in (0,R)$$ | | u ( ⋅ , t ) | | L 1 ( Ω \ B r 0 ( 0 ) ) → 0 as t → ∞ for all r 0 ∈ ( 0 , R ) with some C > 0.

A Kinematic Synthesis Methodology for P-Drivable Spatial Single Degree-of-Freedom Mechanisms to Avoid Singularities

Abstract For a single degree-of-freedom spatial mechanism, a reference frame attached to any of its links produces a continuous motion of this frame. Given the progression of this frame from the start through the end of the mechanism’s motion, this paper seeks to identify specific points relative to this moving reference frame. The points of interest are those that can be coupled with a second point determined in the fixed frame to act as the end joint locations for a spherical–prismatic–spherical (SPS) driving chain. If the selection of the point pair is made such that the change in distance between them as the mechanism moves is strictly monotonic, then the SPS chain they define is potentially capable of driving the mechanism over the desired range of motion. This motion is referred to as locally P-drivable because a global solution is not ensured by the process proposed herein. This synthesis process can avoid singularities encountered by actuating the mechanism at one of its original joints. The proposed approach enables the dimensional synthesis of a single degree-of-freedom mechanism to focus on creating circuit-defect-free solutions without concern for potential singular positions. The actuating chain can then be determined as a separate step in the synthesis process. This paper also considers motions that are not P-drivable and the specialization to planar systems with the synthesis of a P-drivable revolute–prismatic–revolute (RPR) chain.

Solutions to a coupled Cahn–Hilliard-phase-field-crystal model for grain boundary motion

In this paper, we consider an initial–boundary value problem (IBVP) of a coupled Cahn–Hilliard-phase-field-crystal (CH-PFC) system subject to homogeneous Neumann boundary conditions in one spatial dimension. This CH-PFC model couples the composition field of a diffusing species with the crystallographic and can be used to model the diffusion-induced grain boundary migration in crystalline materials. Under suitable assumptions on the coefficients and initial data, we prove that the IBVP possesses a global weak solution. Our existence proof, which contributes to the verification of the model, is only valid in one space dimension.

Longitudinal vibrations of a bar in memoriam to Professor Luiz Adauto Medeiros

This paper is concerned with the deduction of a mathematical model for the small longitudinal vibrations of a bar where one end is clamped and in the other end is glued a concentrated mass. The existence of global solutions of this model perturbed with a general nonlinear internal dissipation and with a general nonlinear boundary dissipation is derived. For a particular case, the exponential decay rate of the energy is obtained.