This article considers a four-component system for the spatiotemporal interaction between cytotoxic T-lymphocytes (CTLs) and tumor cells given by {[ E_t = Δ E - γ∇· (E ∇α ) + σ h(x) + ρ G/η + T - σ E - μ E T/a+bE^k + ϵ G, x∈Ω , t>0,; α _t = δΔα + θ G - ξα , x∈Ω , t>0,; G_t = μ E T/a+bE^k - ψ G, x∈Ω , t>0,; T_t = ωΔ T + β _1 (1 - β _2 T) T - ϕ E T/a+bE^k + λ G, x∈Ω , t>0, ]. where Ω⊂ℝ^N(N≥ 1) is a bounded domain with smooth boundary, and where γ , σ , ρ , η , μ , ϵ , δ , k, θ , ξ , ω , β _1 , β _2 , ϕ , λ , ψ , a, b are all positive constants. It is demonstrated that, under the conditions k > 1 - 1/N and any ψ >0 , the system possesses a unique global-in-time classical solution. Moreover, if ψ≥ρη+ϵ , then the solution is uniformly bounded.
The current paper is concerned with the following two-species chemotaxis-competition model with weak singular sensitivity and Lotka–Volterra competitive kinetics 0.1 {[ u_t=Δ u-χ _1∇· (u/w^α∇ w)+u(a_1-b_1u-c_1v), t>0, x∈Ω ,; v_t=Δ v-χ _2∇· (v/w^α∇ w)+v(a_2-b_2v-c_2u), t>0, x∈Ω ,; 0=Δ w- w+ u+ v, t>0, x∈Ω ,; ∂ u/∂ν=∂ v/∂ν=∂ w/∂ν=0, t>0, x∈∂Ω ,; u(0,x)=u_0(x), v(0,x)=v_0(x), x∈Ω . ]. Here Ω⊂ℝ^N(N ≥ 2) is a bounded smooth domain, α∈ (0,1) and the parameters χ _i,a_i,b_i,c_i(i=1,2) are all positive constants. When b_1>b_1^*(N) and b_2>b_2^*(N) with N≥ 2, where the value of b_1^*(N) and b_2^*(N) are given explicitly, we establish the global existence and boundedness of positive classical solutions to (0.1) for any given nonnegative initial data u_0, v_0∈ C^0(Ω) with ∫ _Ωu_0(x)dx>0, and ∫ _Ωv_0(x)dx>0. Moreover, we prove that for any globally bounded solution pair (u, v), their sum u+v eventually persists in mass from below, whereas the solution component w is eventually pointwise bounded from below by some positive constants. Based on these results, we derive the asymptotic behavior of the classical solutions of (0.1) in three competition cases, provided that the chemotaxis sensitivity coefficients χ _1 and χ _2 are small relative to max{a_1,a_2} .
This paper focuses on the following quasilinear chemotaxis system with nonlocal term and nonlinear indirect signal production {[ u_t = Δ u - χ∇· (u^m ∇ v_1) + u( a_0 - a_1 u^α + a_2 ∫ _Ω u^βdx) , x ∈Ω , t> 0,; τ v_1t = Δ v_1 - v_1 + v_2^θ _1, x ∈Ω , t> 0,; τ v_2t = Δ v_2 - v_2 + u^θ _2, x ∈Ω , t > 0, ]. subject to the homogeneous Neumann boundary conditions, where τ∈{0, 1} , m ≥ 1 , α≥ 1 , β≥ 1 , χ , a_0 , a_1 , θ _1 , θ _2 are positive constants and a_2 ∈ℝ . For both τ = 0 and τ = 1 , we establish the global boundedness of classical solutions if the parameters satisfy appropriate assumptions. Furthermore, under the conditions α = β and sufficiently large a_1 , exponential convergence towards constant steady state (u_*, v_1_*, v_2_*) is demonstrated. In particular, for the case of nonlocal growth effects (a_2 > 0) , a sufficiently large a_1 is required to avoid the blow–up phenomenon. In comparison, when considering nonlocal damping effects (a_2 < 0) , even if we take a smaller value for α and remove the restrictions on a_1 , the global boundedness of classical solutions still holds. Moreover, it is found that the indirect signal can favorably extend the admissible range of θ _1 in certain spatial dimensions, thereby contributing to the global boundedness of the solutions.
This paper is concerned with the well-posedness and asymptotic behavior of stochastic nonlocal PDEs driven by Wong-Zakai type approximation noises on unbounded domains. First, we show the existence and uniqueness of solutions for the approximate equation by applying the Galerkin method. Then, we establish the existence and uniqueness of pullback non-autonomous random attractors for the approximate equation by using the tools and power of the theory of random dynamical systems. When the stochastic system is driven by a linear multiplicative noise, we prove the convergence of solutions of Wong-Zakai approximations and the upper semicontinuity of random attractors of the approximate non-autonomous random system as the size of approximation approaches zero.
Some results on 3D delay Kelvin–Voigt–Brinkman–Forchheimer equations with non-autonomous forcing term and operator type multiplicative white noise are analyzed. First, the existence, uniqueness and backward compactness of pullback random attractors is proved by the backward flattening of solutions and Ascoli-Arzelà theorem. Then we study three types of stability of pullback random attractors: (i) The backward stability of pullback random attractors as the time parameter tends to negative infinity; (ii) The asymptotic autonomous stability of pullback random attractors; (iii) The non-delay stability of pullback random attractors as the delay parameter approaches zero. Since the high regularity of the solution is not easy established, we use the method of the spectrum decomposition to prove the backward asymptotic compactness of the solution operator.
We prove that the L^2 distance between the minimizer of the ℓ ^1 -anisotropic Rudin-Osher-Fatemi (ROF) functional and its minimizer over the space of piecewise constant functions on a rectilinear grid is 𝒪(h^1/2 - q'/2q) , where h is the grid’s mesh size and the datum belongs to L^q , q ≥ 2 . These convergence rates are valid in any dimension d≥ 1 . However, in dimension d = 1 they can be further improved to 𝒪(h^1/2 - 1/2q) . To establish the error bounds, L^q estimates of the ROF minimizer in terms of the datum are critical. Such estimates are particular cases of a universal minimality property of the ROF minimizer derived in the second part of the paper. There it is shown, in both the finite-dimensional and infinite-dimensional settings, that the minimizer simultaneously minimizes a broad class of convex functionals over a neighbourhood of the datum arising in the convex dual of the ROF problem. This extends previous results of similar type about taut strings and the ROF problem.
We consider a wide class of stochastic lattice Selkov systems with three new features: 1) The discrete p-Laplace operator is defined on a high-dimensional unbounded integer set ℤ^d , and has a superlinear growth rate p>2 ; 2) The coupled drift terms are locally Lipschitz from ℓ ^2×ℓ ^2 to ℓ ^2 , and have arbitrary polynomial growth rates; 3) The diffusion terms have time-delay effects, and are locally Lipschitz from ℓ ^2 to ℓ ^2 . The existence of invariant measures of the stochastic systems in the Hilbert space (ℓ ^2×ℓ ^2)× L^2((-ρ ,0),ℓ ^2×ℓ ^2) are established by driving the tightness of a family of probability distributions of the solutions based on the idea of uniform tail-end estimates, the method of high-order moment uniform estimates, the technique of diadic division, and the Arzelà-Ascoli theorem. By improving these uniform estimates for large enough times and bounded initial data, we also establish the tightness of the collection of all invariant measures with respect to the noise intensity and the delay parameter. Then we show that the weak limiting point of any sequence of invariant measures must be an invariant measure of the corresponding limiting system as the noise intensity and the delay parameter tend to zero simultaneously. Our results are new even when the discrete p-Laplacian ( p>2 ) is replaced by the standard discrete Laplacian ( p=2 ), and the arbitrary order growth rate of the drift term reduced to the cubic growth. Our methods can be used for discussing the existence and stability of invariant measures of the systems in the Banach space C([-ρ ,0],ℓ ^2×ℓ ^2) .
Let the objective function f depends on the target variable x along with a nuisance variable s : f(υ) = f(x,s) . Consider the task of identifying the marginal solution x^*= arginf _xinf _s f(x,s) . This paper discusses three related problems. The plug-in approach, widely used, e.g., in inverse problems, suggests using a preliminary guess (pilot) s and apply the solution of the partial optimization x = arginf _x f(x,s) . The main question to address within this approach is the required quality of the pilot, ensuring the prescribed accuracy of x . The popular alternating optimization approach suggests the following procedure: given a starting guess x_0 , for t ≥ 1 , define s_t = arginf _s f(x_t-1,s) , and then x_t = arginf _x f(x,s_t) . The main question here is the set of conditions ensuring a convergence of x_t to x^* . Finally, the paper discusses an interesting connection between marginal optimization and sup-norm estimation. The basic idea is to consider one component of the variable υ as a target and the rest as nuisance. In all cases, we provide accurate closed-form results under realistic assumptions. The results are illustrated by an example for Bradley–Terry–Luce model of ranking from pairwise comparisons.
This paper rigorously analyzes a Keller–Segel–Navier–Stokes system with indirect signal production and subquadratic logistic degradation: 0.1 {[ n_t + u·∇ n = Δ n - χ∇· ( n ∇ v )+ρ n-μ n^α , x ∈Ω , t> 0,; v_t + u·∇ v = Δ v - v + w, x ∈Ω , t> 0,; w_t + u·∇ w = Δ w - w + n, x ∈Ω , t> 0,; u_t + (u·∇ )u + ∇ P = Δu + n ∇ϕ , x ∈Ω , t> 0,; ∇·u = 0, x ∈Ω , t > 0. ]. On a bounded smooth domain Ω⊂ℝ^3 with no-flux for n, v, w and no-slip for u, parameters are: ρ∈ℝ , μ > 0 , χ > 0 (chemotactic sensitivity), ϕ∈ W^2,∞(Ω ) , and α (logistic exponent) critically affecting dynamics. A known bottleneck for weak degradation ( α < 4/3 ) is the need for an L^4/3 estimate of n to obtain fluid energy bounds. While α≥ 4/3 yields solutions directly [10, 11, 43, 63, 68], the case α < 4/3 remains open in 3D. To address this, we construct a quasi-energy inequality: 0.2 ∫ _Ω n^2/3 + A ∫ _Ω |∇ v|^2 + B ∫ _Ω |u|^2, with large constants A, B. For α > 5/4 , we combine analysis of v’s regularity with the w-equation to obtain an L^q estimate ( q > 3/2 ) for w, leading via Moser iteration to an L^∞ bound for v. Under small χ , we prove uniform boundedness of this quasi-energy, overcoming classical barriers and providing new tools for such systems.
We study a class of abstract evolution equations driven by singular memory laws that combine a local time derivative with a nonlocal convolution term. This framework includes, as particular or limiting cases, the first-order abstract Cauchy problem, abstract Basset equations, multi-term and distributed-order fractional models, and purely nonlocal diffusion equations. Our starting point is the constitutive memory law itself, rather than a Volterra kernel prescribed in advance. We show that each singular memory law in this class determines a unique canonical locally integrable kernel, and that the original problem is therefore equivalent to an abstract Volterra equation governed by this kernel and by an associated Laplace symbol. We then establish transfer principles from the original memory law to the canonical kernel, obtaining consequences for complete monotonicity, regularity, and sectoriality that lead to generation criteria for the corresponding resolvent families. Finally, we analyze singular limits connecting the local and purely nonlocal regimes and discuss canonical examples arising from fractional, Basset-type, and distributed-order models. In this way, the paper provides a unified operator-theoretic framework for a broad family of evolution equations with memory.
This paper is concerned with an initial-boundary value problem for a doubly haptotactic oncolytic virotherapy model. Thus far, existing results on the existence and boundedness of solutions to oncolytic virus models with doubly haptotaxis and extracellular matrix remodeling have been largely confined to the two-dimensional setting with linear diffusion. In particular, in three space dimensions, the doubly haptotactic system with linear diffusion requires additional coefficient restrictions and small initial data to guarantee the solvability of classical solutions. The present work aims to remove the constraints of small initial data and narrow parameter regimes in the three-dimensional setting. To this end, we investigate a three-dimensional doubly haptotactic oncolytic virus model in which linear diffusion is replaced by porous medium diffusion. Precisely, we focus on the following model {[ u_t=Δ u^m-∇· (u∇ v)+μ _u u(1-u)-uz,; v_t=-(u+w)v+μ _v v(1-v),; w_t=Δ w^l-∇· (w∇ v)-w+uz,; z_t=Δ z-z-uz+β w, ]. in a smoothly bounded domain. For some slow diffusion case m>1 and l>6/5 , this problem admits a globally defined weak solution for any large initial values. Unlike the approach based on the exponential transformation used in the linear diffusion case, we introduce a new technique that builds on our previous work to handle the solvability issues posed by the porous medium diffusion. The technique relies on delicate foundational energy estimates and matching the estimates of u, v and w.
This study aims to generalize the concepts of quasi-efficiency and quasi-proper efficiency for variable ordering structures within the framework of vector optimization. In addition to unifying the different concepts of approximate solution defined in the literature, this new concept is useful in problems related to the analysis of non-dominated approximate solutions and minimal approximate solutions. Furthermore, some of the main properties and related theorems are established and, scalarization results and a practical example are presented to demonstrate its efficiency and applicability. Finally, using the new notions of being quasi-non-dominated and quasi-minimal, a generalized subdifferential is defined in vector optimization with variable ordering structures.
We study convex optimization problems on Hadamard manifolds and propose a projection based variant of the proximal point algorithm, called the Busemann hybrid projection-proximal point algorithm. The method replaces Euclidean hyperplanes by horospheres defined through Busemann functions and uses the associated projection geometry to build an intrinsic update rule. The projection step is available in closed form and avoids tangent space linearization. The method allows inexactness in the subproblem solution under a relative error level strictly below one. We establish a Fejér type descent property, prove global convergence, and derive a sublinear complexity bound. We also show that, in the exact case, the method reduces to the classical Riemannian proximal point algorithm. The results highlight the role of Busemann based support inequalities and subdifferentials in optimization on spaces of nonpositive curvature.
We consider the ensemble controllability problem for a linear time-invariant system ẋ(t,θ )=A(θ )x(t,θ )+B(θ )u(t) , where A and B are continuous matrices with respect to the parameter θ , which belongs to some compact set Θ⊂ℝ . Given any continuous initial state datum θ↦ x^0(θ ) and any continuous target state θ↦ x^1(θ ) , we investigate the numerical computation of a θ -independent open loop control u such that x^0 is steered, in a given time T>0 , at a distance ε >0 of x^1 in the uniform norm (with respect to the parameter). We approach the problem both theoretically and numerically. Using the Fenchel–Rockafellar duality, we first prove the existence and uniqueness of the ensemble control of a minimal L^2 norm. The numerical recovery of the optimal control is obtained by solving the dual problem, which consists in the unconstrained minimization of a non-differentiable functional in the space of Radon measures.
This study is devoted to control problems governed by a second-order semilinear evolution equation with a memory kernel κ∈L^1(0,∞ ) satisfying ∫ _0^∞κ (t)dt<1 and the function ϰ (t):=∫ _t^∞κ (s)ds is of positive type. In this process, we first discuss the well-posedness of the linear evolution equation with memory by introducing the concept of a resolvent family. We then examine several fundamental properties of the resolvent family ℛ(· ) and the associated family 𝒫(· ) that are crucial for the development of our main results. Subsequently, we study a linear–quadratic regulator problem in order to derive an optimal control leading to the approximate controllability of the state and its derivative for the linear control system. Furthermore, we determine sufficient conditions for the existence of a mild solution of the semilinear control system, corresponding to a given control u∈L^2(J;𝕌) , by using the Schaefer fixed point theorem. We also investigate controllability of a semilinear system in a Hilbert space by constructing a single control that ensures the weak approximate controllability of both the state and its derivative. Our main results are determined by combining the fixed point techniques with the approximation solvability method and weak topology. Finally, the theoretical results are applied to analyze the weak approximate controllability problem for a wave equation with a weakly singular kernel of the form κ (t)=e^-att^ν -1/Γ (ν ), a>1, 0<ν <1 .
A steady-state Oseen problem for incompressible fluid with nonsmooth unilateral and frictional type boundary conditions is studied. The problem serves as a new mathematical model for the gas–liquid two-phase fluid problem in shale gas development. First the physical setting of a two-phase returning fluid model is described. Then, the weak formulation which consists of a coupled system of two elliptic variational–hemivariational inequalities and an elliptic equation is delivered. The main existence theorem is proved by solving three auxiliary problems and applying a Schauder–Tychonoff fixed-point argument. Finally, a partial pressure-recovery result is provided.
In this paper, we establish the strong order 1/2 of convergence in the averaging principle for multiscale forward–backward stochastic differential equations with Lipschitz coefficients. Moreover, this convergence order is shown to be optimal through an example.
This paper aims to study the model of Klein–Gordon–Schrödinger equations with a memory term and locally distributed damping : {[ iψ '+ Δψ + iα (|ψ |^2 +1)ψ =- ϕψ in Ω× (0, ∞ ),; ϕ” - Δϕ +∫ _0^t g(t-τ ) div[a(x) ∇ϕ (τ )] d τ +b(x)ϕ '= |ψ |^2θχ _ω in Ω× (0, ∞ ),; ]. where Ω is a bounded domain of ℝ^n with n≤ 3 and smooth boundary ∂Ω =Γ . Here, α and θ are positive constants. In this work, χ _ω represents the cutoff function of ω . Assuming that a and b are non-negative functions such that a(x) + b(x) ≥δ > 0 in Ω , the exponential decay rate is demonstrated for each regular solution of the above system.
This paper is devoted to the study of the existence and uniqueness of solutions for a class of evolutionary variational–hemivariational inequalities involving generalized history-dependent operators. The main contributions of this work are twofold. First, the notion of generalized history-dependent operators is introduced, which allows a wide range of problems with differential–integral operators to be unified under the framework of generalized history-dependent operator problems. Second, unlike most existing studies on evolutionary variational–hemivariational inequalities, where the so-called “smallness condition” is commonly imposed, we eliminate this assumption and establish the existence and uniqueness of solutions and their well-posedness through a concise mathematical analysis in Hilbert spaces.
This paper studies multi-objective infinite-horizon discounted continuous-time Markov decision processes (CTMDPs) with distinct discount rates across the objectives. We start the analysis through the weighted sum method. A counterexample demonstrates the absence of Pareto-optimal stationary policies, contrasting with the classic discounted CTMDP results. Inspired by the idea for finite-horizon scenarios, we derive the Hamilton-Jacobi-Bellman equation with time as the differential variable for the infinite-horizon weighted sum optimization. To solve the new equation, we introduce a novel dynamic programming operator along with a carefully constructed initial function. Our value iteration method establishes the existence of weighted-optimal Markov deterministic policies, thereby ensuring the existence of Pareto-optimal policies. Moreover, we prove weak Pareto sufficiency of weighted-optimal policies by the occupation measure technique. Finally, we characterize the structure of ε -optimal policies—they are ultimately stationary under specific conditions, and provide an algorithm to calculate ε -optimal policies for finite-state cases.
