Principal Eigenvalue Research Articles

Dynamics of a nonlocal dispersal in‐host viral model with humoral immunity

AbstractThis work investigates the existence and asymptotic behavior of solutions to a nonlocal dispersal in‐host viral model with humoral immunity. The model features both spatial movement of virus and humoral immunity response to the virus. This paper gives the principal eigenvalue () of the nonlocal dispersal problem, and it is verified to be a critical value that determines the infection dynamics. The uninfected equilibrium is unique and stable for . For , the equilibrium is not stable, and infected with/without B cells response equilibrium emerges. This paper also establishes the dynamics of infected equilibria. Numerical simulations are carried out to demonstrate the analytical results of this study.

Bifurcation results for a non-local elliptic equation with a nonlinear boundary condition

In this paper we study a logistic elliptic equation with a nonlinear boundary condition and a nonlocal term. By employing the local and global bifurcation theories, a priori bounds for elliptic equations, blow-up techniques for parabolic equations and the properties of the principal eigenvalues, we establish some existence results for compact and unbounded connected components of the positive solutions set of the nonlocal problem.

Hopf Bifurcation in a Delayed Equation with Diffusion Driven by Carrying Capacity

In this paper, a delayed reaction–diffusion equation with carrying capacity-driven diffusion is investigated. The stability of the positive equilibrium solutions and the existence of the Hopf bifurcation of the equation are considered by studying the principal eigenvalue of an associated elliptic operator. The properties of the bifurcating periodic solutions are also obtained by using the normal form theory and the center manifold reduction. Furthermore, some representative numerical simulations are provided to illustrate the main theoretical results.

AVO-Friendly Velocity Analysis Based on the High-Resolution PCA-Weighted Semblance

Velocity analysis using the semblance spectrum can provide an effective velocity model for advanced seismic imaging technology, in which the picking accuracy of velocity analysis is significantly affected by the resolution of the semblance spectrum. However, the peak broadening of the conventional semblance spectrum leads to picking uncertainty, and it cannot deal with the amplitude-variation-with-offset (AVO) phenomenon. The well-known AB semblance can process the AVO anomalies, but it has a lower resolution compared with conventional semblance. To improve the resolution of the AB semblance spectrum, we propose a new weighted AB semblance based on principal component analysis (PCA). The principal components or eigenvalues of seismic events are highly sensitive to the components with spatial coherence. Thus, we utilized the principal components of the normal moveout (NMO)-corrected seismic events with different scanning velocities to construct a weighting function. The new function not only has a high resolution for velocity scanning, but it is also a friendly method for the AVO phenomenon. Numerical experiments with the synthetic and field seismic data sets proved that the new method significantly improves resolution and can provide more accurate picked velocities compared with conventional methods.

Dynamical analysis of a multi-group SIR epidemic model with nonlocal diffusion and nonlinear incidence rate

In this paper, a multi-group SIR epidemic model with nonlocal diffusion and nonlinear incidence rate in spatially heterogeneous environment is proposed. The well-posedness, including the existence, positivity and boundedness of solutions, is achieved. The basic reproduction number R0 is defined, and the existence of principal eigenvalue is studied. Further, the threshold criteria on the global dynamics of the model are established in terms of R0. That is, when R0<1, the disease-free steady state is globally asymptotically stable, when R0>1, the disease is uniformly persistent. Particularly, a single-group degenerated SIR epidemic model is also studied. We give some properties on the principle eigenvalue and prove that there exists a compact global attractor for the solution semiflow. Furthermore, when the basic reproduction number R̄0>1, we obtain that the endemic steady state is unique to exist and globally asymptotically stable.

Mixed boundary value problems for fully nonlinear degenerate or singular equations

We prove existence, uniqueness and regularity results for mixed boundary value problems associated with fully nonlinear, possibly singular or degenerate elliptic equations. Our main result is a global Hölder estimate for solutions, obtained by means of the comparison principle and the construction of ad hoc barriers. The global Hölder estimate immediately yields a compactness result in the space of solutions, which could be applied in the study of principal eigenvalues and principal eigenfunctions of mixed boundary value problems.

Nonzero-Sum Risk-Sensitive Continuous-Time Stochastic Games with Ergodic Costs

We study nonzero-sum stochastic games for continuous time Markov decision processes on a denumerable state space with risk-sensitive ergodic cost criterion. Transition rates and cost rates are allowed to be unbounded. Under a Lyapunov type stability assumption, we show that the corresponding system of coupled HJB equations admits a solution which leads to the existence of a Nash equilibrium in stationary strategies. We establish this using an approach involving principal eigenvalues associated with the HJB equations. Furthermore, exploiting appropriate stochastic representation of principal eigenfunctions, we completely characterize Nash equilibria in the space of stationary Markov strategies.

A rearrangement minimization problem related to a nonlinear parametric boundary value problem

This paper deals with a rearrangement minimization problem, which is associated with a nonlinear parametric boundary value problem. When the parameter is positive and less than the principal eigenvalue of the p-Laplacian type operator, we obtain that the nonlinear parametric boundary value problem has a unique solution. We then establish the solvability of the rearrangement minimization problem. Finally, based on Stampacchia truncation method, we establish the regularity property of the solution to the nonlinear boundary value problem, and then we investigate the symmetric property of the solution to the rearrangement minimization problem when the domain is a ball at the origin.

A Convergent Interacting Particle Method and Computation of KPP Front Speeds in Chaotic Flows

In this paper, we study the propagation speeds of reaction-diffusion-advection (RDA) fronts in time-periodic cellular and chaotic flows with Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We first apply the variational principle to reduce the computation of KPP front speeds to a principal eigenvalue problem of a linear advection-diffusion operator with space-time periodic coefficients on a periodic domain. To this end, we develop efficient Lagrangian particle methods to compute the principal eigenvalue through the Feynman-Kac formula. By estimating the convergence rate of Feynman-Kac semigroups and the operator splitting methods for approximating the linear advection-diffusion solution operators, we obtain convergence analysis for the proposed numerical methods. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method in computing KPP front speeds in time-periodic cellular and chaotic flows, especially the time-dependent Arnold-Beltrami-Childress (ABC) flow and time-dependent Kolmogorov flow in three-dimensional space.

First-passage times to anisotropic partially reactive targets

We investigate restricted diffusion in a bounded domain towards a small partially reactive target in three- and higher-dimensional spaces. We propose a simple explicit approximation for the principal eigenvalue of the Laplace operator with mixed Robin-Neumann boundary conditions. This approximation involves the harmonic capacity and the surface area of the target, the volume of the confining domain, the diffusion coefficient, and the reactivity. The accuracy of the approximation is checked by using a finite-elements method. The proposed approximation determines also the mean first-reaction time, the long-time decay of the survival probability, and the overall reaction rate on that target. We identify the relevant lengthscale of the target, which determines its trapping capacity, and we investigate its relation to the target shape. In particular, we study the effect of target anisotropy on the principal eigenvalue by computing the harmonic capacity of prolate and oblate spheroids in various space dimensions. Some implications of these results in chemical physics and biophysics are briefly discussed.

Ambrosetti–Prodi problems for the Robin [formula omitted]-Laplacian

The classical Ambrosetti–Prodi problem considers perturbations of the linear Dirichlet Laplace operator by a nonlinear reaction whose derivative jumps over the principal eigenvalue of the operator. In this paper, we develop a related analysis for parametric problems driven by the nonlinear Robin (p,q)-Laplace operator (sum of a p-Laplacian and a q-Laplacian). Under hypotheses that cover both the (p−1)-linear and the (p−1)-superlinear case, we prove an optimal existence, multiplicity, and non-existence result, which is global in the parameter λ>0.

Diffusion Tensor Imaging of a Median Nerve by Magnetic Resonance: A Pilot Study.

The magnetic resonance Diffusion Tensor Imaging (DTI) is a powerful extension of Diffusion Weighted Imaging (DWI) utilizing multiple bipolar gradients, allowing for the evaluation of the microstructural environment of the highly anisotropic tissues. DTI was predominantly used for the assessment of the central nervous system (CNS), but with the advancement in magnetic resonance (MR) hardware and software, it has now become possible to image the peripheral nerves which were difficult to evaluate previously because of their small caliber. This study focuses on the assessment of the human median peripheral nerve ex vivo by DTI microscopy at 9.4 T magnetic field which allowed the evaluation of diffusion eigenvalues, the mean diffusivity and the fractional anisotropy at 35 μm in-plane resolution. The resolution was sufficient for clear depiction of all nerve anatomical structures and therefore further image analysis allowed the obtaining of average values for DT parameters in nerve fascicles (intrafascicular region and perineurium) as well as in the surrounding epineurium. The results confirmed the highest fractional anisotropy of 0.33 and principal diffusion eigenvalue of 1.0 × 10−9 m2/s in the intrafascicular region, somewhat lower values of 0.27 and 0.95 × 10−9 m2/s in the perineurium region and close to isotropic with very slow diffusion (0.15 and 0.05 × 10−9 m2/s) in the epineurium region.

The number of positive solutions for Schrödinger–Poisson system under the effect of eigenvalue

We investigate the eigenvalue problem for the Schrödinger–Poisson system as follows: where is nonnegative and is a potential well with bottom A bounded Cerami sequence is established by using the truncation technique, decomposition of Hilbert space and mountain pass theorem. Then exploiting the property of steep potential well, under some proper assumptions on , we conclude that one positive solution exists for while two positive solutions exist for when where is the principal eigenvalue of in with weight function and is the corresponding principal eigenfunction. Moreover, an interesting phenomenon is that when , one positive solution is always present near for unaffected by the non‐local term, however large its norm might be.

Harnack inequality and principal eigentheory for general infinity Laplacian operators with gradient in $$\mathbb {R}^N$$ and applications

Under the lack of variational structure and nondegeneracy, we investigate three notions of generalized principal eigenvalue for a general infinity Laplacian operator with gradient and homogeneous term. A Harnack inequality is proved to support our analysis. This is a continuation of our first work (Biswas and Vo in Liouville theorems for infinity Laplacian with gradient and KPP type equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. https://doi.org/10.2422/2036-2145.202105_050) and a contribution in the development of the theory of generalized principal eigenvalue beside the works (Berestycki et al. in Commun Pure Appl Math 47(1):47–92, 1994; Berestycki and Rossi in JEMS 8:195–215, 2006; Berestycki and Rossi in Commun Pure Appl Math 68(6):1014–1065, 2015; Berestycki et al. in J Math Pures Appl 103:1276–1293, 2015; Nguyen and Vo in Calc Var Partial Differ Equ 58(3):102 2019). We use these notions to characterize the validity of maximum principle and study the existence, nonexistence and uniqueness of positive solutions of Fisher-KPP type equations in the whole space. The sliding method is intrinsically improved for infinity Laplacian to solve the problem. The results are related to the Liouville type results, which will be meticulously explained.

On a second order eigenvalue problem and its application

A linear eigenvalue problem governed by a second order differential equation with separate and general boundary conditions is considered and a new monotonicity result on the principal eigenvalue with respect to the coefficient of the advection term is established. The main approach is based on the functional proposed by Liu and Lou [16] and a key finding lies in the nice properties of the associated Fréchet operator when confined at suitable points and function spaces. As an application, this monotonicity result is used to study a class of competitive parabolic systems and the so-called “competitive exclusion principle” is observed in a larger parameter region than several existing works, which is a nontrivial improvement.

Dynamics of an HIV infection model with virus diffusion and latently infected cell activation

Effective combination therapy usually reduces the plasma viral load of HIV to below the detection limit, but it cannot eradicate the virus. The latently infected cell activation is considered to be the main obstacle to completely eradicating HIV infection. In this paper, we consider an HIV infection model with latently infected cell activation, virus diffusion and spatial heterogeneity under Neumann boundary condition. The basic reproduction ratio is characterized by the principal eigenvalue of the related elliptic eigenvalue problem. Besides, by constructing Lyapunov functionals and using Green’s first identity, the global threshold dynamics of the system are completely established. Numerical simulations are carried out to illustrate the theoretical results, in particular, the influence of virus diffusion rate on the basic reproduction ratio is addressed.

Sharp estimates of the generalized principal eigenvalue for superlinear viscous Hamilton–Jacobi equations with inward drift

This paper is concerned with the ergodic problem for viscous Hamilton–Jacobi equations with superlinear Hamiltonian, inward-pointing drift, and positive potential function which vanishes at infinity. Assuming some radial symmetry of the drift vector field and the potential function outside a large ball, we obtain sharp estimates of the generalized principal eigenvalue with respect to a perturbation of the potential function. We also specify the necessary and sufficient condition so that the spectral function contains a plateau.

A New Monotonicity for Principal Eigenvalues with Applications to Time-Periodic Patch Models

A new monotonocity of principal eigenvalues in time-periodic patch environments is established. As an application, a patch model for two competing species in spatio-temporally varying environments is investigated. When two species are identical except for their relaxation time, the species with the shorter relaxation time will always drive the other one to extinction. When two species are identical except for their diffusion rates, our results suggest that the faster diffusing species could be favored for some intermediate range of relaxation time, while the slower diffusing species will be favored for both short and long relaxation time. In general, short relaxation time and slow diffusion rate tend to help species gain advantage in competition.

Global Limit Theorem for Parabolic Equations with a Potential

We obtain the asymptotics, as $t + |x| \rightarrow \infty$, of the fundamental solution to the heat equation with a compactly supported potential. It is assumed that the corresponding stationary operator has at least one positive eigenvalue. Two regions with different types of behavior are distinguished: inside a certain conical surface in the $(t,x)$ space, the asymptotics is determined by the principal eigenvalue and the corresponding eigenfunction; outside of the conical surface, the main term of the asymptotics is a product of a bounded function and the fundamental solution of the unperturbed operator, with the contribution from the potential becoming negligible if $|x|/t \rightarrow \infty$. A formula for the global asymptotics, as $t + |x| \rightarrow \infty$, of the solution in the entire half-space $t > 0$ is provided. In probabilistic terms, the result describes the asymptotics of the density of particles in a branching diffusion with compactly supported branching and killing potentials.

Principal eigenvalue, maximum principles and linear stability for nonlocal diffusion equations in metric measure spaces

We study principal eigenvalues and maximum principles for stationary nonlocal operators in spaces of integrable functions defined on general metric measure spaces under minimal assumptions on the kernels. Several characterizations of the principal eigenvalue are given as well as several conditions guaranteeing existence. Characterization of the (strong) maximum principle is also given. For evolution problems we prove the strong maximum principle and characterize stability in terms of the sign of the principal eigenvalue. We recover, extend and improve all previously known results, obtained for smooth open sets in euclidean space under continuity assumptions on the data.