Article
Mathematics
YanYan Li, Zhuolun Yang
Summary: This paper studies the insulated conductivity problem with inclusions embedded in a bounded domain in R-n. The upper bound for the blow up rate has been improved in dimension n >= 3.
MATHEMATISCHE ANNALEN
(2023)
Article
Computer Science, Artificial Intelligence
Mubashir Qayyum, Aneeza Tahir, Saraswati Acharya
Summary: Fuzzy differential equations have been widely used to model complex systems with uncertainty or imprecise information. This study proposes a dual parametric extension of the He-Laplace algorithm to solve time-fractional fuzzy Fisher models. The obtained solutions are more accurate and validated against existing results. The analysis shows that the uncertain probability density function decreases gradually at the boundaries when the fractional order increases.
INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS
(2023)
Article
Physics, Mathematical
Zhiwen Zhao
Summary: In this paper, an insulated conductivity model with two neighboring m-convex inclusions in R-d is considered, where m >= 2 and d >= 3. We establish the pointwise gradient estimates for the insulated conductivity problem and obtain the gradient blow-up rate of order epsilon(-1/m+beta) as the distance epsilon between these two insulators tends to zero, where beta=[-(d+m-3)+root(d+m-3)(2)+4(d-2)]/(2m) is an element of (0,1/m). The optimality of the blow-up rate is also demonstrated for a class of axisymmetric m-convex inclusions.
JOURNAL OF MATHEMATICAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Youngjin Hwang, Junxiang Yang, Gyeongyu Lee, Seokjun Ham, Seungyoon Kang, Soobin Kwak, Junseok Kim
Summary: In this study, a fast and efficient finite difference method is proposed for solving the AC equation on cubic surfaces. The method unfolds the cubic surface domain in 3D space into 2D space and applies appropriate boundary conditions on the planar sub-domains to calculate numerical solutions. Numerical experiments show the effectiveness of the proposed algorithm.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2024)
Article
Physics, Fluids & Plasmas
Yi Xu, Jan Egedal
Summary: The adiabatic invariance of the magnetic moment during particle motion is fundamental for the dynamics of magnetized plasma. The study investigates the pitch angle scattering rate of fast particles moving through static magnetic perturbations and derives predictions for the scattering rates based on numerical integration of particle orbits in prescribed magnetic fields.
PHYSICS OF PLASMAS
(2022)
Article
Mathematics
Yong-Gwan Ji, Hyeonbae Kang
Summary: In this study, we investigate the field concentration for conductivity equations in the presence of closely located circular inclusions by utilizing the spectral nature of the phenomenon. Our approach not only allows us to recover existing results with new insights but also leads to significant new findings. We obtain optimal estimates for the derivatives of the solution when the conductivities of the inclusions have the same relative signs and when they have different signs. The estimates show that the gradient of the solution is bounded regardless of the distance between the inclusions, but the 2nd and higher derivatives may blow up if one conductivity is 0 and the other is infinity, which we demonstrate through examples.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2022)
Article
Mathematics
Yong-Gwan Ji, Hyeonbae Kang
Summary: We investigate the field concentration for conductivity equations in the presence of closely located circular inclusions by exploiting the spectral nature residing behind the phenomenon of the field concentration. This approach enables us to recover existing results with new insights and produce significant new results. We obtain optimal estimates for the derivatives of the solution when the conductivities of the inclusions have both the same and different relative signs, showing the behavior of the solution near the inclusions as the distance tends to zero.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2023)
Article
Mathematics
Sergey Kashchenko
Summary: This study focuses on the behavior of solutions of the logistic equation with delay and diffusion near equilibrium, considering Andronov-Hopf bifurcation conditions, perturbations of coefficients, and construction of equations on the central manifold. The nonlocal dynamics on the central manifold determine the behavior of solutions in a small neighborhood of the equilibrium state, with the ability to control dynamics using phase change in perturbing force. Numerical and analytical results on parametric perturbations are obtained, along with asymptotic formulas for solutions.
Article
Mathematics, Applied
Liyan Pang, Shi-Liang Wu
Summary: This paper investigates the spreading properties of a reaction-diffusion equation in a cylinder with partially exponentially unbounded initial conditions. It is proven that the level sets of the solutions move infinitely fast as time approaches infinity, and that the locations of the level sets are determined by the maximum initial values in the y direction.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Physics, Multidisciplinary
Esteban Martinez Vargas, Christoph Hirche, Gael Sentis, Michalis Skotiniotis, Marta Carrizo, Ramon Munoz-Tapia, John Calsamiglia
Summary: This paper introduces sequential analysis in quantum information processing, with a focus on the fundamental task of quantum hypothesis testing. The goal is to discriminate between two arbitrary quantum states with a prescribed error threshold, and lower bounds on the average number of copies needed to accomplish this task are obtained. A block-sampling strategy is provided to achieve the lower bound for certain classes of states, and the findings show that a sequential strategy based on fixed local measurements outperforms other procedures for qubit states.
PHYSICAL REVIEW LETTERS
(2021)
Article
Mathematics
Wen Wang, Rulong Xie, Pan Zhang
Summary: The paper presents a new proof and derives new elliptic type (Hamilton type) gradient estimates for fast diffusion equations on a complete noncompact Riemannian manifold with a fixed metric and along the Ricci flow by constructing a new auxiliary function. These results generalize earlier findings in the literature and also obtain some parabolic type Liouville theorems for ancient solutions.
CHINESE ANNALS OF MATHEMATICS SERIES B
(2021)
Article
Mathematics, Applied
Benoit Perthame, Jakub Skrzeczkowski
Summary: In this study, we investigate the fast reaction limit in a reaction-diffusion system with a nonmonotone reaction function and a nondiffusing component. As the reaction speed tends to infinity, the concentration of the nondiffusing component exhibits rapid oscillations. We identify its Young measure precisely, which also leads to the strong convergence of the diffusing component, a result not easily obtained from prior estimates.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Qi Li, Bendong Lou
Summary: The reaction diffusion equation with generalized bistable nonlinearity shows that the solution tends to vanish as time goes to infinity, when the nonlinearity decreases rapidly for large u.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Cole Graham
Summary: We investigated integro-differential Fisher-KPP equations with nonlocal diffusion. For typical equations, we established the logarithmic Bramson delay for solutions with step-like initial data. Furthermore, certain strongly asymmetric diffusions exhibited more exotic behavior.
COMMUNICATIONS IN MATHEMATICAL SCIENCES
(2022)
Article
Mathematics
T. J. Christiansen, K. Datchev
Summary: In this paper, we study the analytic estimates of the Dirichlet Laplacian near the continuous spectrum on unbounded domains, as well as the consequences for wave decay and resonance-free regions in domains with infinite cylindrical ends. Our results also extend to examples beyond the star-shaped case, such as scattering by a strictly convex obstacle inside a straight planar waveguide.
MATHEMATICAL RESEARCH LETTERS
(2022)
Article
Mathematics, Applied
Felix del Teso, Jorgen Endal, Juan Luis Vazquez
Summary: The study investigates the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in Double-struck capital R-N. We prove the existence of a continuous and bounded selfsimilar solution with a free boundary at the change-of-phase level. The study also provides well-posedness and basic properties of very weak solutions for general bounded data in several dimensions, and explores limits and connections with other diffusion problems.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Lorenzo Brasco, David Gomez-Castro, Juan Luis Vazquez
Summary: The study aims to characterize the homogeneous fractional Sobolev-Slobodecki.i spaces D-s,D-p(R-n) and their embeddings, showing isomorphisms to suitable function spaces or space of equivalence classes of functions. The Morrey-Campanato inequality is presented as a main tool in the analysis.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Gabriele Grillo, Giulia Meglioli, Fabio Punzo
Summary: This study investigates reaction-diffusion equations in various geometric settings, proving global existence and boundedness of solutions in specific cases, as well as discovering instances where solutions exhibit blow up in infinite time. The methods rely on functional analysis and are applicable to diverse situations, with a focus on weighted reaction-diffusion equations in the Euclidean setting.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2021)
Article
Mathematics, Applied
Gabriele Grillo, Giulia Meglioli, Fabio Punzo
Summary: The discussion primarily revolves around the global existence of solutions to the porous medium equation with a power-like reaction term on Riemannian manifolds. Utilizing functional analytic methods, it is proven that under certain conditions, global solutions do exist.
JOURNAL OF EVOLUTION EQUATIONS
(2021)
Article
Mathematics
Gabriele Grillo, Matteo Muratori, Fabio Punzo
Summary: The study investigates the well-posedness of the fast diffusion equation on noncompact Riemannian manifolds and establishes uniqueness conditions for different types of initial data. The crucial role of curvature and stochastic completeness in determining the uniqueness of solutions is highlighted, along with the proof of nonexistence of nontrivial distributional subsolutions to certain semilinear elliptic equations.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics, Applied
Kleber Carrapatoso, Jean Dolbeault, Frederic Herau, Stephane Mischler, Clement Mouhot
Summary: We prove functional inequalities on vector fields u : R-d -> R-d when R-d is equipped with a bounded measure e(-phi) dx that satisfies a Poincare inequality, and study associated self-adjoint operators. The weighted Korn inequality compares the differential matrix Du, once projected orthogonally to certain finite-dimensional spaces, with its symmetric part D(s)u and, in an improved form of the inequality, an additional term del phi.u. We also consider Poincare-Korn inequalities for estimating a projection of u by D(s)u and zeroth-order versions of these inequalities obtained using the Witten-Laplace operator. The constants depend on geometric properties of the potential phi and the estimates are quantitative and constructive. These inequalities are motivated by kinetic theory and related with the Korn inequality (1906) in mechanics, which compares Du and D(s)u on a bounded domain.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2022)
Article
Physics, Mathematical
Emeric Bouin, Jean Dolbeault, Laurent Lafleche
Summary: This paper investigates the large time behavior of kinetic equations without confinement, focusing on collision operators with fat tailed local equilibria and their anomalous diffusion limit. The study develops an L-2-hypocoercivity approach at the kinetic level to establish a decay rate compatible with the fractional diffusion limit.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
Jose A. Carrillo, David Gomez-Castro, Juan Luis Vazquez
Summary: This paper studies radial solutions of an aggregation-diffusion model that combines nonlinear fast diffusion with a convection term driven by the gradient of a potential. Depending on the exponent of fast diffusion and the potential, the steady state is given by the sum of an explicit integrable function, plus a Dirac delta at the origin containing the rest of the mass of the initial datum, showing a splitting phenomenon.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2022)
Article
Mathematics, Applied
Juan Luis Vazquez
Summary: The study establishes the existence, uniqueness, and quantitative estimates for solutions to the fractional nonlinear diffusion equation, focusing on a certain range of exponents. By obtaining weighted global integral estimates, the existence of solutions for a class of large data is proved, and the dichotomy positivity versus extinction for nonnegative solutions at any given time is established. The analysis includes the conditions for extinction in finite time.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Article
Mathematics
Hardy Chan, David Gomez-Castro, Juan Luis Vazquez
Summary: In this paper, we investigate the existence and uniqueness of solutions for the parabolic problem with singular data, as well as the impact of initial data and forcing term. When the boundary data is zero, the results coincide with the standard fractional heat semigroup. Furthermore, we explore the spectral theory of the fractional heat semigroup and obtain bounds on the fractional heat kernel.
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
(2022)
Article
Mathematics, Applied
Gabriele Grillo, Giulia Meglioli, Fabio Punzo
Summary: This study investigates the Fujita phenomenon for reaction-diffusion evolution equations on manifolds similar to hyperbolic space, showing that a different version of the phenomenon occurs with a different critical exponent. The results indicate that on a larger class of manifolds, solutions to equations with power nonlinearities are global in time for sufficiently small data.
ANNALI DI MATEMATICA PURA ED APPLICATA
(2023)
Article
Mathematics, Applied
Jean Dolbeault, An Zhang
Summary: The carre du champ method is a powerful technique for proving interpolation inequalities with explicit constants in presence of a non-trivial metric on a manifold. The method applies to some classical Gagliardo-Nirenberg-Sobolev inequalities on the sphere, with optimal constants. Very nonlinear regimes close to the critical Sobolev exponent can be covered using nonlinear parabolic flows of porous medium or fast diffusion type.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2022)
Correction
Mathematics
Jean Dolbeault, Maria J. Esteban, Eric Sere
Summary: This corrigendum addresses some overlooked closability issues in [1].
JOURNAL OF FUNCTIONAL ANALYSIS
(2023)
Article
Mathematics, Applied
Filomena Feo, Juan Luis Vazquez, Bruno Volzone
Summary: We prove the existence of self-similar fundamental solutions (SSF) of the anisotropic porous medium equation in the suitable fast diffusion range. Each of such SSF solutions is uniquely determined by its mass. We also obtain the asymptotic behaviour of all finite mass solutions in terms of the family of self-similar fundamental solutions.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2023)
Article
Mathematics, Applied
J. A. Carrillo, D. Gomez-Castro, J. L. Vazquez
Summary: This paper discusses the density solutions for gradient flow equations of the form u(t) = del center dot (gamma(u) backward difference N(u)), and compares the results for linear mobility and concave mobility cases. It is found that solutions with compactly supported initial data remain compactly supported and generate moving free boundaries. For the linear mobility case, a special solution in the form of a disk vortex is present, while for the concave mobility case, viscosity solutions exist in the whole space and display a fat tail at infinity.
ADVANCES IN NONLINEAR ANALYSIS
(2022)
