Unique Classical Solution Research Articles

Global Hilbert expansion for some nonrelativistic kinetic equations

AbstractThe Vlasov–Maxwell–Landau (VML) system and the Vlasov–Maxwell–Boltzmann (VMB) system are fundamental models in dilute collisional plasmas. In this paper, we are concerned with the hydrodynamic limits of both the VML and the noncutoff VMB systems in the entire space. Our primary objective is to rigorously prove that, within the framework of Hilbert expansion, the unique classical solution of the VML or noncutoff VMB system converges globally over time to the smooth global solution of the Euler–Maxwell system as the Knudsen number approaches zero. The core of our analysis hinges on deriving novel interplay energy estimates for the solutions of these two systems, concerning both a local Maxwellian and a global Maxwellian, respectively. Our findings address a problem in the hydrodynamic limit for Landau‐type equations and noncutoff Boltzmann‐type equations with a magnetic field. Furthermore, the approach developed in this paper can be seamlessly extended to assess the validity of the Hilbert expansion for other types of kinetic equations.

Stochastic Control Problems with State Reflections Arising from Relaxed Benchmark Tracking

This paper studies stochastic control problems motivated by optimal consumption with wealth benchmark tracking. The benchmark process is modeled by a combination of a geometric Brownian motion and a running maximum process, indicating its increasing trend in the long run. We consider a relaxed tracking formulation such that the wealth compensated by the injected capital always dominates the benchmark process. The stochastic control problem is to maximize the expected utility of consumption deducted by the cost of the capital injection under the dynamic floor constraint. By introducing two auxiliary state processes with reflections, an equivalent auxiliary control problem is formulated and studied, which leads to the Hamilton-Jacobi-Bellman equation with two Neumann boundary conditions. We establish the existence of a unique classical solution to the dual partial differential equation using some novel probabilistic representations involving the local time of some dual processes together with a tailor-made decomposition-homogenization technique. The proof of the verification theorem on the optimal feedback control can be carried out by some stochastic flow analysis and technical estimations of the optimal control. Funding: L. Bo and Y. Huang are supported by the National Natural Science Foundation of China [Grant 12471451], the Natural Science Basic Research Program of Shaanxi [Grant 2023-JC-JQ-05], the Shaanxi Fundamental Science Research Project for Mathematics and Physics [Grant 23JSZ010], and Fundamental Research Funds for the Central Universities [Grant 20199235177]. X. Yu is supported by the Hong Kong RGC General Research Fund (GRF) [Grant 15304122] and the Research Centre for Quantitative Finance at the Hong Kong Polytechnic University [Grant P0042708].

Well-posedness of the Tricomi Problem for the Multidimensional Lavrent'ev-Bitsadze Equation

Numerous applications in physics and engineering involve models with partial differential equations of mixed type. The theory of boundary-value problems for such equations in two dimensions has been well studied. However, the key problem of well-posedness of mixed problems for such equations in multidimensional bounded domains remains currently unsolved. This paper establishes a mixed domain in which the solution of the Tricomi problem for the multidimensional Lavrent'ev-Bitsadze equation has a unique classical solution.

Optimal controls for forward-backward stochastic differential equations: Time-inconsistency and time-consistent solutions

This paper is concerned with an optimal control problem for a forward-backward stochastic differential equation (FBSDE, for short) with a recursive cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). It is found that such an optimal control problem is time-inconsistent in general, even if the cost functional is reduced to a classical Bolza type one as in Peng [47], Lim–Zhou [38], and Yong [72]. Therefore, instead of finding a global optimal control (which is time-inconsistent), we will look for a time-consistent and locally optimal equilibrium strategy, which can be constructed via the solution of an associated equilibrium Hamilton–Jacobi–Bellman (HJB, for short) equation. A verification theorem for the local optimality of the equilibrium strategy is proved by means of the generalized Feynman–Kac formula for BSVIEs and some stability estimates of the representation parabolic partial differential equations (PDEs, for short). Under certain conditions, it is proved that the equilibrium HJB equation, which is a nonlocal PDE, admits a unique classical solution. As special cases and applications, the linear-quadratic problems, a mean-variance model, a social planner problem with heterogeneous Epstein–Zin utilities, and a Stackelberg game are briefly investigated. It turns out that our framework can cover not only the optimal control problems for FBSDEs studied in [47,38,72], and so on, but also the problems of the general discounting and some nonlinear appearance of conditional expectations for the terminal state, studied in Yong [73,75] and Björk–Khapko–Murgoci [6].

Stabilization in a chemotaxis-consumption model involving Robin-type boundary conditions

The chemotaxis-consumption model{ut=∇⋅((u+1)α∇u)−∇⋅(u(u+1)β−1∇v),x∈Ω,t>00=Δv−uv,x∈Ω,t>0 is considered in a smooth bounded domain Ω⊂Rn under the boundary conditions (u+1)α∂νu=u(u+1)β−1∂νv and ∂νv=(γ−v)g on ∂Ω, with parameters α,β∈R, γ>0 and the nonnegative function g∈C1+ω(Ω¯) for some ω∈(0,1). If β−α≤1, it is proved that there exists a global bounded classical solution (u,v) for suitably regular initial data u0. Furthermore, for the given mass m=∫Ωu∞>0, the corresponding stationary system admits a unique classical solution (u∞,v∞), appearing as a large-time limit in the sense that if γ>0 is small enough, then‖u(⋅,t)−u∞‖L∞(Ω)+‖v(⋅,t)−v∞‖L∞(Ω)→0 as t→∞, with ∫Ωu∞=∫Ωu0=m>0.

Local existence and global boundedness for a chemotaxis system with gradient dependent flux limitation

In this paper, we investigate the following Keller–Segel system with flux limitation [Formula: see text] under no-flux boundary conditions in a ball [Formula: see text]), where [Formula: see text] is positive and [Formula: see text] with [Formula: see text]. It is proved that the problem (⋆) possesses a unique classical solution that can be extended in time up to a maximal [Formula: see text]. Moreover, it is shown that the above solution is global and bounded when either [Formula: see text] if [Formula: see text] and [Formula: see text], or [Formula: see text] if [Formula: see text] and [Formula: see text]. We point out that when [Formula: see text], our result is consistent with that of [N. Bellomo and M. Winkler, Comm. Partial Differential Equations 42 (2017) 436–473].

The classical solvability for a one-dimensional nonlinear thermoelasticity system with the far field degeneracy

Abstract We study the local classical solvability of the Cauchy problem to the equations of one-dimensional nonlinear thermoelasticity. The governing model is a coupled system of a nonlinear hyperbolic equation for the displacement and a parabolic equation for the temperature of the elastic material. We allow the hyperbolic equation to degenerate at spacial infinity, which results that the coefficients of the coupled system are not uniformly bounded and then the previous methods for the strictly hyperbolic-parabolic coupled systems are invalid. We introduce a suitable weighted norm to establish the local existence and uniqueness of classical solutions by the contraction mapping principle. The existence time T of the solution is independent of the spatial variable.

Global boundedness and asymptotic behavior in a double haptotaxis model for oncolytic virotherapy

This work studies a haptotactic cross-diffusion system modeling oncolytic virotherapy{ut=Δu−∇⋅(u∇v)−ρuz,vt=−(u+w)v−κv,wt=DwΔw−ξw∇⋅(w∇v)−w+ρuz,zt=DzΔz−z−ρuz+βw, in a smooth bounded domain Ω⊂R2, with parameters Dw>0, Dz>0, ξw≥0, ρ≥0, κ>0 and β≥0. This system describes the interaction among uninfected and infected cancer cells, extracellular matrix and oncolytic viruses. It is rigorously proved that an associated no-flux initial-boundary value problem has a unique global classical solution which is bounded in (L∞(Ω))4. Moreover, it is shown that this bounded solution can approach a spatially constant equilibrium in the large time limit under the additional assumption that 0≤β<1.

The non-homogeneous Cauchy problem of semigroup of linear operators

This paper presents the results of ω-order reversing partial contraction mapping generated by the non-homogeneous Cauchy problem. We investigated several concepts of solution for non-homogeneous Cauchy problem where an analysis was arrived out on the relationship between C' strong and respectively C0-solution. We showed that Z is the infinitesimal generator of a C0-semigroup of contractions. Finally, we obtained that the problem has a unique classical solution and f, a member of L1(a,b; X) is continuous on (a,b).

On the motion of a body with a cavity filled with magnetohydrodynamic fluid

We study the dynamics of a coupled system, formed by a rigid body with a cavity entirely filled with magnetohydrodynamic compressible fluid. Our aim is to derive the global existence of the unique classical solutions and weak solutions to this system. Moreover, we show the weak-strong uniqueness principle which means that a weak solution coincides with a strong solution on the time existence of a strong solution, provided they emanate from the same initial data.

Boundedness in a chemotaxis-May–Nowak model with exposed state

This paper is concerned with the Neumann initial–boundary value problem of the chemotaxis-May–Nowak model with exposed state for virus dynamics. It is proved that the problem admits a unique global classical solution which is uniformly bounded for all sufficiently smooth initial data in smoothly bounded domains Ω⊂Rn, n≤3.

Dynamic Analysis of a Ratio-Dependent Food Chain Model with Prey-Taxis

In this paper, we consider a food chain model with ratio-dependent functional response and prey-taxis. We first investigate the global existence and boundedness of the unique positive classical solutions of the system in a bounded domain with smooth boundary and Neumann boundary conditions. Then, we analyze the local stability of the system and the existence of Hopf bifurcation. In addition, we prove the global asymptotic stability of steady states under some conditions by constructing a Lyapunov functional, and investigate convergence rates. Finally, we present several numerical simulations to illustrate the results.

Numerical analysis of the SIS infectious disease model with spatial heterogeneity

PurposeThe susceptible-infectious-susceptible (SIS) infectious disease models without spatial heterogeneity have limited applications, and the numerical simulation without considering models’ global existence and uniqueness of classical solutions might converge to an impractical solution. This paper aims to develop a robust and reliable numerical approach to the SIS epidemic model with spatial heterogeneity, which characterizes the horizontal and vertical transmission of the disease.Design/methodology/approachThis study used stability analysis methods from nonlinear dynamics to evaluate the stability of SIS epidemic models. Additionally, the authors applied numerical solution methods from diffusion equations and heat conduction equations in fluid mechanics to infectious disease transmission models with spatial heterogeneity, which can guarantee a robustly stable and highly reliable numerical process. The findings revealed that this interdisciplinary approach not only provides a more comprehensive understanding of the propagation patterns of infectious diseases across various spatial environments but also offers new application directions in the fields of fluid mechanics and heat flow. The results of this study are highly significant for developing effective control strategies against infectious diseases while offering new ideas and methods for related fields of research.FindingsThrough theoretical analysis and numerical simulation, the distribution of infected persons in heterogeneous environments is closely related to the location parameters. The finding is suitable for clinical use.Originality/valueThe theoretical analysis of the stability theorem and the threshold dynamics guarantee robust stability and fast convergence of the numerical solution. It opens up a new window for a robust and reliable numerical study.

On small-data solution of the chemotaxis–SIS epidemic system with bilinear incidence rate

This paper deals with the initial–boundary value problem for chemotaxis susceptible-infected-susceptible epidemic system with bilinear incidence rate ut=d1Δu+χ∇⋅(u∇v)−β(x)uv+γ(x)v,x∈Ω,t>0,vt=d2Δv+β(x)uv−γ(x)v,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn (n≥1), where d1>0, d2>0, χ∈R, 0<β∈C1(Ω¯) and 0<γ∈C1(Ω¯). It is proved that if (u(x,0),v(x,0)) in L∞(Ω)×L1(Ω) is suitably small and ‖v(x,0)‖L∞(Ω)+‖∇v(x,0)‖Lq(Ω)≤K for each K>0 and q>n, then the problem possesses a unique global classical solution which is bounded in Ω×(0,∞). Moreover, we prove that the solution exponentially stabilizes to the disease-free equilibrium N|Ω|,0 with N=∫Ω(u(x,0)+v(x,0)) in L∞(Ω) as t→∞.

The Cauchy Problem for the General Telegraph Equation with Variable Coefficients under the Cauchy Conditions on a Curved Line in the Plane

The Riemann method is used to prove the global correctness theorem to Cauchy problem for a general telegraph equation with variable coefficients under Cauchy conditions on a curved line in the plane. The global correctness theorem consists of an explicit Riemann formula for a unique and stable classical solution and a Hadamard correctness criterion for this Cauchy problem. From the formulation of the Cauchy problem, the definition of its classical solutions and the established smoothness criterion of the right-hand side of the equation, its correctness criterion is derived. These results are obtained by Lomovtsev’s new implicit characteristics method which uses only two differential characteristics equations and twelve inversion identities of six implicit mappings. If the righthand side of general telegraph equation depends only on one of two independent variables, then it is necessary and sufficient that it be continuous with respect to this variable. If the right-hand side of this equation depends on two variables and is continuous, then in its integral smoothness requirements it is necessary and sufficient the continuity in one and continuous differentiability in the other variable. The correctness criterion represents the necessary and sufficient smoothness requirements of the right-hand side of the equation and the Cauchy data. From the established global correctness theorem, the well-known Riemann formulas for classical solutions and correctness criteria to Cauchy problems for the general and model telegraph equations in the upper half-plane are derived. In the works of other authors, there is no necessary (minimally sufficient) smoothness on the right-hand sides of the hyperbolic equations of real Cauchy problems for the set of classical (twice continuously differentiable) solutions.

A Keller–Segel system involving mixed local and nonlocal operators with logistic term on ℝN

In this paper, we investigate a parabolic–elliptic Keller–Segel system with a logistic term on , involving mixed local and nonlocal operators with . At first, we prove that the semigroup generated by is an analytical semigroup and uses blow‐up arguments in combination with the classical Liouville‐type theorem to demonstrate the regularity of weak solutions of the parabolic equation with the mixed operators. Next, the local existence and uniqueness of classical solutions are established by applying the semigroup theory and regularity results in case of . Moreover, we obtain the global existence and boundedness of classical solutions under some conditions on given initial values and the asymptotic behavior of the global solutions with strictly positive initial conditions.

Improved decay estimates and C2-asymptotic stability of solutions to the Einstein-scalar field system in spherical symmetry

We investigate the asymptotic stability of solutions to the characteristic initial value problem for the Einstein (massless) scalar field system with a positive cosmological constant. We prescribe spherically symmetric initial data on a future null cone with a wider range of decaying profiles than previously considered. New estimates are then derived in order to prove that, for small data, the system has a unique global classical solution. We also show that the solution decays exponentially in (Bondi) time and that the radial decay is essentially polynomial, although containing logarithmic factors in some special cases. This improved asymptotic analysis allows us to show that, under appropriate and natural decaying conditions on the initial data, the future asymptotic solution is differentiable, up to and including spatial null-infinity, and approaches the de Sitter solution, uniformly, in a neighborhood of infinity. Moreover, we analyze the decay of derivatives of the solution up to second order showing the (uniform) [Formula: see text]-asymptotic stability of the de Sitter attractor in this setting. This corresponds to a surprisingly strong realization of the cosmic no-hair conjecture.

Spatiotemporal Dynamics and Bifurcation Analysis of a Generalized Two-Prey One-Predator System with Diffusion and Double Prey-Taxes

The presence of a predator can force the mediation of a coexistence state in three-species ordinary differential equation model, where two competing species are preyed on by a common predator. To understand how the addition of diffusion and prey-taxis affects predator-mediated coexistence in such an ecological system, we consider a general two-competing-prey and one-predator model with double prey-taxes under Neumann boundary conditions. We first show that there is a unique global classical solution to this model with ratio-dependent and nonratio-dependent predator functional responses. Then, we demonstrate the emergence of the so-called stationary patterns. Finally, in detail, we give some sufficient conditions for the existence, nonexistence, and stability of nonconstant positive steady states and time-periodic positive solutions. Surprisingly, we find that the combination of a repulsive prey-taxis and an attractive prey-taxis can also induce the emergence of pattern formations. The theoretical results imply that double prey-taxes play an extremely important part in biological control.

On the Correction Method of Test Solutions to the New Model Wave Equation with Two Variable Rates in First Quarter of the Plane

A correction method of test solutions into classical solutions to a new inhomogeneous model wave equation with variable two-rates in the first quarter of the plane was developed to derive the minimum smoothness requirement of its right-hand side. In the case of different rates, it is not possible to derive the minimum smoothness of right-hand side of the inhomogeneous model wave equation in the first quarter of the plane without correcting both test generalized and test classical solutions. In this paper, classical solutions to the inhomogeneous two-rate model wave equation and the smoothness criterion of their right-hand side are obtained. We need them to find explicit unique and stable classical solutions and Hadamard correctness criteria to mixed (initial-boundary) problems for this wave equation using developed by the author of the ”implicit characteristics method”.

Global boundedness in a chemotaxis system with signal-dependent motility and indirect signal consumption

This paper deals with the following chemotaxis system ut=Δuγv,x∈Ω,t>0,vt=Δv−vw,x∈Ω,t>0,wt=−δw+u,x∈Ω,t>0,under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn (n≤3), where the motility function γ∈C30,+∞ is positive on [0,∞). For all suitably regular initial data, then the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded. The purpose of this work is to remove the smallness assumption on ‖v0‖L∞(Ω) in three dimension (Xu et al., 0000).