Global Classical Solvability Research Articles

Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier-Stokes system involving Dirichlet boundary conditions for the signal

The chemotaxis-Navier-Stokes system{nt+u⋅∇n=Δn−∇⋅(nχ(n)∇c),ct+u⋅∇c=Δc−nc,ut+(u⋅∇)u=Δu+∇P+n∇ϕ,∇⋅u=0 is considered in a smoothly bounded domain Ω⊂R2 under the boundary conditions(∇n−nχ(n)∇c)⋅ν=0,c=c⋆,u=0,x∈∂Ω,t>0 with a given nonnegative constant c⋆. It is shown that if χ∈C2([0,∞)) and χ(n)→0 as n→∞, then for all suitably regular initial data, an associated initial value problem possesses a globally defined and bounded classical solution. When ‖n0‖L1(Ω) and ‖c0‖L∞(Ω) are suitably small and c⋆≡0, we further obtain the stabilization of the classical solution.

Global classical solvability and asymptotic behaviors of a parabolic-elliptic Chemotaxis-type system modeling crime activities

This paper is devoted to a two-dimensional parabolic-elliptic system in crime activities. The main difference between this system and the chemotaxis system with logarithmic singular sensitivity and quadratic logistic source is that the source term of this system is a mixtype quadratic term. The local existence of solutions to its intial boundary valve problem associated with homogeneous Neumann boundary conditions was considered by Rodríguez. The present study indicates that for any properly regular initial data there admits global classical solutions. Moreover, the qualitative behaviors of solutions in large time scales are considered as well.

Global classical solvability and stabilization in a two-dimensional chemotaxis–fluid system with sub-logarithmic sensitivity

In this paper, we consider the following system: [Formula: see text] in a smoothly bounded domain [Formula: see text] with [Formula: see text] and a given function [Formula: see text] with [Formula: see text] It is proved that if [Formula: see text] then for appropriately small initial data an associated no-flux/no-flux/Dirichlet initial-boundary value problem is globally solvable in the classical sense, and that if [Formula: see text] then under a different but still suitable smallness restriction of the initial data, a corresponding initial-boundary value problem subject to no-flux/no-flux/Dirichlet boundary conditions admits a unique classical solution which is globally bounded and approaches a constant equilibria [Formula: see text] in [Formula: see text] as [Formula: see text] with [Formula: see text]

Chemotaxis-driven stationary and oscillatory patterns in a diffusive HIV-1 model with CTL immune response and general sensitivity.

In this paper, a reaction-diffusion-chemotaxis HIV-1 model with a cytotoxic T lymphocyte (CTL) immune response and general sensitivity is investigated. We first prove the global classical solvability and L∞-boundedness for the considered model in a bounded domain with arbitrary spatial dimensions, which extends the previous existing results. Then, we apply the global existence result to the case with a linear proliferation immune response and an incidence rate. We study the spatiotemporal dynamics about the three types of spatially homogeneous steady states: infection-free steady state S0, CTL-inactivated infection steady state S1, and CTL-activated infection steady state S∗. Our analyses indicate that S0 is globally asymptotically stable if the basic reproduction number R0 is less than 1; if R0 is between 1 and a threshold, then S1 is globally asymptotically stable. However, if R0 is larger than the threshold, then the chemoattraction and chemorepulsion can destabilize S∗, and thus, a spatiotemporal pattern forms as the chemotactic sensitivity crosses certain critical values. We obtain two kinds of important patterns, which are induced by chemotaxis: stationary Turing pattern and irregular oscillatory pattern. We also find that different chemotactic response functions can affect system's dynamics. Based on some empirical parameter values, numerical simulations are given to illustrate the effectiveness of the theoretical predications.

Cauchy–Dirichlet Problem to Semilinear Multi-Term Fractional Differential Equations

In this paper, we analyze the well-posedness of the Cauchy–Dirichlet problem to an integro-differential equation on a multidimensional domain Ω⊂Rn in the unknown u=u(x,t), Dtν0(ϱ0u)−Dtν1(ϱ1u)−L1u−∫0tK(t−s)L2u(x,s)ds=f(x,t)+g(u),0<ν1<ν0<1, where Dtνi are the Caputo fractional derivatives, ϱi=ϱi(x,t) with ϱ0≥μ0>0, and Li are uniform elliptic operators with time-dependent smooth coefficients. The principal feature of this equation is related to the integro-differential operator Dtν0(ϱ0u)−Dtν1(ϱ1u), which (under certain assumption on the coefficients) can be rewritten in the form of a generalized fractional derivative with a non-positive kernel. A particular case of this equation describes oxygen delivery through capillaries to tissue. First, under proper requirements on the given data in the linear model and certain relations between ν0 and ν1, we derive a priori estimates of a solution in Sobolev–Slobodeckii spaces that gives rise to providing the Hölder regularity of the solution. Exploiting these estimates and constructing appropriate approximate solutions, we prove the global strong solvability to the corresponding linear initial-boundary value problem. Finally, obtaining a priori estimates in the fractional Hölder classes and assuming additional conditions on the coefficients ϱ0 and ϱ1 and the nonlinearity g(u), the global one-valued classical solvability to the nonlinear model is claimed with the continuation argument method.

Global existence and stabilization in a two-dimensional chemotaxis-Navier-Stokes system with consumption and production of chemosignals

We are concerned with the initial boundary value problems of chemotaxis-Navier-Stokes systems involving double chemosignals: one is an attractant consumed by the cells themselves and the other is an attractant or a repellent produced by the cells themselves. This work reveals that for any properly regular initial data, if the initial total cells mass is less than some threshold, then the corresponding attraction-atraction Navier-Stokes system admits a unique global classical solution, which, under extra two smallness assumptions on the initial data, is globally bounded and converges at exponential rate to the spatial equilibrium in the large time limit. As byproducts, the global classical solvability and stabilization of the Keller-Segel Navier-Stokes system and the uniform boundedness and large time behavior of the global classical solution of the associated attraction-repulsion Navier-Stokes system are simultaneously achieved.

Stationary and oscillatory patterns of a food chain model with diffusion and predator‐taxis

In this paper, we investigate pattern dynamics in a reaction‐diffusion‐chemotaxis food chain model with predator‐taxis, which extends previous studies of reaction‐diffusion food chain model. By virtue of diffusion semigroup theory, we first prove global classical solvability and boundedness for the considered model over a bounded domain with smooth boundary for arbitrary predator‐taxis sensitivity coefficient. Then the linear stability analysis for the considered model shows that chemotaxis can induce the losing of stability of the unique positive spatially homogeneous steady state via Turing bifurcation and Turing‐spatiotemporal Hopf bifurcation. These bifurcations results in the formation of two kinds of important spatiotemporal patterns: stationary Turing pattern and oscillatory pattern. Simultaneously, the threshold values for Turing bifurcation and Turing‐spatiotemporal Hopf bifurcation are given explicitly. Finally, numerical simulations are performed to illustrate and support our theoretical findings, and some interesting non‐Turing patterns are found in temporal Hopf parameter space by numerical simulation.

Analysis of a chemotaxis-SIS epidemic model with unbounded infection force

A susceptible–infected–susceptible (SIS) model is considered which includes both taxis type cross-diffusive motion of susceptibles and a mass action infection mechanism. An associated no-flux initial–boundary value problem in one-dimensional bounded intervals is shown to admit globally bounded classical solutions for initial data of arbitrary size. For a higher-dimensional counterpart, a result on global classical solvability in the presence of suitably small initial data is derived. This is complemented by a statement on large time stabilization of small-data trajectories toward spatially constant and infection-free equilibria under an assumption on strict positivity of the considered space-dependent recovery rate.

Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension

This manuscript considers the no-flux initial-boundary problem for the migration-consumption system{ut=Δ(uϕ(v)),vt=Δv−uv,(⋆) in a smoothly bounded domain Ω⊂Rn, n≥1, where ϕ suitably generalizes the singular prototype given byϕ(ξ)=ξ−α,ξ>0, with α>0.It is firstly shown that for such diffusion singularities of arbitrary strength, and for any given initial data u0 and v0 from W1,∞(Ω) with u0≥0 and v0>0 in Ω‾, a so-called very weak-strong solution (u,v) with (u,v)|t=0=(u0,v0) can be constructed. Under the additional restrictions that 2≤n≤5 and α>n−26−n, and under an additional assumption on ϕ′, it is furthermore asserted that the flux ∇(uϕ(v)) belongs to some reflexive Lebesgue space, and that (⋆) is satisfied in a standard weak sense.Finally, in the one-dimensional case a statement on global classical solvability is derived under the mere condition that ϕ∈C3((0,∞)) be positive.

Global classical solvability in a food metric chemotaxis system under zero boundary conditions at infinity

We consider the nutrient–chemotaxis model, derived by a food metric, on the real line. The model consists of the equation for the nutrient density of ordinary differential equation type and that for the microorganism density of parabolic type, in which the diffusion coefficient is singular at the zero nutrient density. We establish a global existence result of bounded classical solutions for a large class of initial data under zero boundary conditions at infinity. To this end, we propose a transformation related to the food metric that transforms the original system into a parabolic–hyperbolic system with linear diffusion. We obtain a proper global well-posedness result for the transformed system using standard estimates, including the heat kernel estimates. We convert this to the solution of the original equation via an inverse transformation.

Global classical solvability and stabilization in a reaction–diffusion intraguild predation model with chemotaxis

Global classical solvability and stabilization in a reaction–diffusion intraguild predation model with chemotaxis

Global classical solvability in a three‐dimensional haptotaxis system modeling oncolytic virotherapy

In this paper, we concern with the haptotaxis system modeling oncolytic virotherapy. We present the global classical solution under appropriate regularity assumptions on the initial data. Our results generalize some well‐known results in the literature.

Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis–haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more)

We propose and study a class of parabolic-ordinary differential equation models involving chemotaxis and haptotaxis of a species following signals indirectly produced by another, non-motile one. The setting is motivated by cancer invasion mediated by interactions with the tumour microenvironment, but has much wider applicability, being able to comprise descriptions of biologically quite different problems. As a main mathematical feature constituting a core difference to both classical Keller–Segel chemotaxis systems and Chaplain–Lolas type chemotaxis–haptotaxis systems, the considered model accounts for certain types of indirect signal production mechanisms. The main results assert unique global classical solvability under suitably mild assumptions on the system parameter functions in associated spatially two-dimensional initial-boundary value problems. In particular, this rigorously confirms that at least in two-dimensional settings, the considered indirectness in signal production induces a significant blow-up suppressing tendency also in taxis systems substantially more general than some particular examples for which corresponding effects have recently been observed.

Solving a fully parabolic chemotaxis system with periodic asymptotic behavior using Generalized Finite Difference Method

This work studies a parabolic-parabolic chemotactic PDE's system which describes the evolution of a biological population “U” and a chemical substance “V”, using a Generalized Finite Difference Method, in a two dimensional bounded domain with regular boundary. In a previous paper [12], the authors asserted global classical solvability and periodic asymptotic behavior for the continuous system in 2D. In this continuous work, a rigorous proof of the global classical solvability to the discretization of the model proposed in [12] is presented in two dimensional space. Numerical experiments concerning the convergence in space and in time, and long-time simulations are presented in order to illustrate the accuracy, efficiency and robustness of the developed numerical algorithms.

Boundedness in a two-dimensional Keller–Segel–Navier–Stokes system involving a rapidly diffusing repulsive signal

This paper is concerned with the Keller–Segel–Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{lcll} n_t + u\cdot \nabla n &{}=&{} \Delta n + \nabla \cdot (n\nabla c), \qquad &{} x\in \Omega , \ t>0, \\ u\cdot \nabla c &{}=&{} \Delta c -c+n, \qquad &{} x\in \Omega , \ t>0, \\ u_t + (u\cdot \nabla ) u &{}=&{} \Delta u + \nabla P + n\nabla \Phi , \qquad \nabla \cdot u=0, \qquad &{} x\in \Omega , \ t>0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$with a given smooth gravitational potential $$\Phi $$. It is shown that for all suitably regular initial data, a corresponding no-flux/no-flux/Dirichlet initial boundary value problem posed in a smoothly bounded planar domain admits a uniquely determined global classical solution (n, c, u, P) which has the additional property that n remains uniformly bounded. This partially goes beyond a recent result asserting global classical solvability, but without any boundedness information, in a related slightly more complex variant of ($$\star $$) accounting for parabolic evolution of the quantity c. In particular, the obtained outcome therefore provides further evidence indicating that the considered fluid interaction does not substantially reduce a certain explosion-avoiding character of the Keller–Segel-type chemorepulsion mechanisms, as known to form an essential feature of corresponding fluid-free analogues. The reasoning at its core relies on the use of a quasi-Lyapunov inequality which operates at regularity levels that seem rather unusual in this and related contexts, but which in the considered two-dimensional setting can be seen to serve as a starting point sufficient for a bootstrap-type series of arguments, finally providing global boundedness.

Global solutions of continuous coagulation–fragmentation equations with unbounded coefficients

In this paper we prove the existence of global classical solutions to continuous coagulation–fragmentation equations with unbounded coefficients under the sole assumption that the coagulation rate is dominated by a power of the fragmentation rate, thus improving upon a number of recent results by not requiring any polynomial growth bound for either rate. This is achieved by proving a new result on the analyticity of the fragmentation semigroup and then using its regularizing properties to prove the local and then, under a stronger assumption, the global classical solvability of the coagulation–fragmentation equation considered as a semilinear perturbation of the linear fragmentation equation. Furthermore, we show that weak solutions of the coagulation–fragmentation equation, obtained by the weak compactness method, coincide with the classical local in time solutions provided the latter exist.

Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy

This work studies a haptotaxis system proposed as a model for oncolytic virotherapy, accounting for interaction between uninfected cancer cells, infected cancer cells, extracellular matrix (ECM) and oncolytic virus. In addition to random movement, both uninfected and infected tumor cells migrate haptotactically toward higher ECM densities; moreover, besides degrading the non-diffusible ECM upon contact the two cancer cell populations are subject to an infection-induced transition mechanism driven by virus particles which are released by infected cancer cells, and which assault the uninfected part of the tumor.The main results assert global classical solvability in an associated initial-boundary value problem posed in one- or two-dimensional domains with any given suitably regular initial data. This is achieved by discovering a quasi-Lyapunov functional structure that allows to appropriately cope with the presence of nonlinear zero-order interaction terms which apparently form the most significant additional mathematical challenge of the considered system in comparison to previously studied haptotaxis models.

Global classical solvability and generic infinite-time blow-up in quasilinear Keller–Segel systems with bounded sensitivities

The chemotaxis system(⋆){ut=∇⋅(D(u,v)∇u)−∇⋅(S(u,v)∇v),vt=Δv−v+u, is considered under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn, n≥2, along with initial conditions involving suitably regular and nonnegative data.It is firstly asserted that if the positive smooth function D decays at most algebraically with respect to u, then for any smooth nonnegative and bounded S fulfilling a further mild assumption especially satisfied when S≡S(u) with S(0)=0, (⋆) possesses a globally defined classical solution.If Ω is a ball, then under appropriate assumptions on D and S generalizing the prototypical choices in(⋆⋆)D(u,v)=(u+1)m−1andS(u,v)=u(u+1)σ−1,u≥0,v≥0, with m∈R and σ∈R such that(⋆⋆⋆)m−n−2n<σ≤0, the phenomenon of infinite-time blow-up is next shown to occur for all initial data within a set B of functions which inter alia is found to be dense in the set of all radially symmetric and suitably regular positive functions on Ω‾.Up to equality in (⋆⋆⋆) thereby covering the largest possible range of nonpositive σ for the appearance of unbounded solutions, this extends previous findings on blow-up in infinite time which in the context of (⋆⋆) were limited to a smaller parameter region, and which were restricted to mere existence results without information on the richness of B.

Solvability of linear boundary value problems for subdiffusion equations with memory

For $\nu \in (0,1)$, the nonautonomous integro-differential equation \[ \mathbf {D}_{t}^{\nu }u-\mathcal {L}_{1}u-\int _{0}^{t}\mathcal {K}_{1}(t-s)\mathcal {L}_{2}u(\cdot ,s)\,ds =f(x,t) \] is considered here, where $\mathbf {D}_{t}^{\nu }$ is the Caputo fractional derivative and $\mathcal {L}_{1}$ and $\mathcal {L}_{2}$ are uniformly elliptic operators with smooth coefficients dependent on time. The global classical solvability of the associated initial-boundary value problems is addressed.

Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier–Stokes system modeling coral fertilization

The interplay of chemotaxis, convection and reaction terms is studied in the particular framework of a refined model for coral broadcast spawning, consisting of three equations describing the population densities of unfertilized sperms and eggs and the concentration of a chemical released by the latter, coupled to the incompressible Navier–Stokes equations.Under mild assumptions on the initial data, global existence of classical solutions to an associated initial-boundary value problem in bounded planar domains is established. Moreover, all these solutions are shown to approach a spatially homogeneous equilibrium in the large time limit.