Global Existence Time Research Articles

Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix

Abstract The aim of this work is to study the global existence in time of solutions for the tridiagonal system of reaction-diffusion by order m m . Our techniques of proof are based on compact semigroup methods and some L 1 {L}^{1} -estimates. We show that global solutions exist. Our investigation can be applied for a wide class of nonlinear terms of reaction.

Time-global existence of generalized BV flow via the Allen–Cahn equation

We show that a mean curvature flow obtained as the limit of the Allen–Cahn equation is not only a Brakke flow but also a generalized BV flow proposed by Stuvard and Tonegawa (2024).

Generalized result on the global existence of positive solutions for a parabolic reaction diffusion model with a full diffusion matrix

"In this paper, we study the global existence in time of solutions for a parabolic reaction di usion model with a full matrix of di usion coe cients on a bounded domain. The technique used is based on compact semigroup methods and some estimates. Our objective is to show, under appropriate hypotheses, that the proposed model has a global solution with a large choice of nonlinearities."

Positive solutions to the prey–predator equations with dormancy of predators

AbstractThe time-global unique classical positive solutions to the reaction–diffusion equations for prey–predator models with dormancy of predators are constructed. The feature appears on the nonlinear terms of Holling type $\rm I\!I$ functional response. The crucial step is to establish time-local positive classical solutions by using a new approximation associated with time-evolution operators. Although the system does not equip usual comparison principle for solutions to partial differential equation, a priori bounds are derived by enclosing and renormalising arguments of solutions to the corresponding ordinary differential equations. Furthermore, time-global existence, invariant regions and asymptotic behaviours of solutions follow from such a priori bounds.

Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation

We consider chemotaxis-Navier–Stokes systems with logistic proliferation and signal consumption of the form for parameter choices kappa ge 0 and mu >0. Herein, we moreover impose a nonnegative and time-constant prescribed concentration c_star in C^2({overline{Omega }}) for the signal chemical on the boundary of the domain Omega subset {mathbb {R}}^{mathcal {N}} with {mathcal {N}}in {2,3}. After first extending the previously known result on time-global existence of weak solutions for the Stokes variant to the full Navier–Stokes setting, we proceed with an investigation of eventual regularity properties in the slightly more restrictive setting of c_star being also constant in space. We show that sufficiently strong logistic influence, in the sense that for omega >0 and mu _0>0 there is some eta =eta (omega ,mu _0,c_star )>0 with the property that whenever μ0≤μandκmin{μ,μN+66+ω}<η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mu _0\\le \\mu \\quad \ ext {and}\\quad \\frac{\\kappa }{\\min \\{\\mu ,\\mu ^{\\frac{{\\mathcal {N}}+6}{6}+\\omega }\\}}<\\eta \\end{aligned}$$\\end{document}are satisfied the global weak solution eventually becomes a smooth and classical solution with waiting time depending on omega ,mu _0,eta ,c_star and the initial data.

Kobayashi–Warren–Carter type systems with nonhomogeneous Dirichlet boundary data for crystalline orientation

In this paper we study the Dirichlet problem for the Kobayashi–Warren–Carter system. This system of parabolic PDE’s models the grain boundary motion in a polycrystal with a prescribed orientation at the boundary of the domain. We obtain global existence in time of energy-dissipative solutions. The regularity of the solutions as well as the energy-dissipation property permit us to derive the steady-state problem as the asymptotic in time limit of the system. We finally study the ω-limit set of the solutions; we completely characterize it in the one dimensional case, showing, in particular that orientations in the ω-limit set belong to the space of SBV functions. In the two dimensional case, we give sufficient conditions for existence of radial symmetric piecewise constant solutions.

Inhomogeneous Dirichlet boundary value problem for nonlinear Schr\xf6dinger equations in the upper half-space

We consider the inhomogeneous Dirichlet-boundary value problem for nonlinear Schrödinger equations with a power nonlinearity in general space dimension nge 3. We present some results on global existence in time and uniquness of small solutions to integral equations associated to the original problem.

English

The Cauchy problem of the Cahn-Hilliard equation with inertial term in multi space dimension is considered. Based on detailed analysis of Green’s function, using fixed-point theorem, we get the global existence in time of classical solution with large initial data. Furthermore, we get Lp decay rate of the solution.

Wave equations associated with Liouville-type problems: global existence in time and blow-up criteria

We are concerned with wave equations associated with some Liouville-type problems on compact surfaces, focusing on sinh-Gordon equation and general Toda systems. Our aim is on one side to develop the analysis for wave equations associated with the latter problems and second, to substantially refine the analysis initiated in Chanillo and Yung (Adv Math 235:187–207, 2013) concerning the mean field equation. In particular, by exploiting the variational analysis recently derived for Liouville-type problems we prove global existence in time for the subcritical case and we give general blow-up criteria for the supercritical and critical case. The strategy is mainly based on fixed point arguments and improved versions of the Moser–Trudinger inequality.

Asymptotic behavior of solutions for the 2D micropolar equations in Sobolev–Gevrey spaces

This work guarantees the existence of a positive instant t = T and a unique solution ( u , w ) ∈ [ C ( [ 0 , T ] ; H a , σ s ( R 2 ) ) ] 3 (with a &gt; 0, σ &gt; 1, s &gt; 0 and s ≠ 1) for the micropolar equations. Furthermore, we consider the global existence in time of this solution in order to prove the following decay rates: lim t → ∞ t s 2 ‖ ( u , w ) ( t ) ‖ H ˙ a , σ s ( R 2 ) 2 = lim t → ∞ t s + 1 2 ‖ w ( t ) ‖ H ˙ a , σ s ( R 2 ) 2 = lim t → ∞ ‖ ( u , w ) ( t ) ‖ H a , σ λ ( R 2 ) = 0 , ∀ λ ⩽ s . These limits are established by applying the estimate ‖ F − 1 ( e T | · | ( u ˆ , w ˆ ) ( t ) ) ‖ H s ( R 2 ) ⩽ [ 1 + 2 M 2 ] 1 2 , ∀ t ⩾ T , where T relies only on s , μ , ν and M (the inequality above is also demonstrated in this paper). Here M is a bound for ‖ ( u , w ) ( t ) ‖ H s ( R 2 ) (for all t ⩾ 0) which results from the limits lim t → ∞ t s 2 ‖ ( u , w ) ( t ) ‖ H ˙ s ( R 2 ) = lim t → ∞ ‖ ( u , w ) ( t ) ‖ L 2 ( R 2 ) = 0.

Existence of local and global solution for a spatio-temporal predator-prey model

In this paper we prove the existence and uniqueness of weak solutions for a kind of Lotka–Volterra system, by using successive linearization techniques. This approach has the advantage to treat two equations separately in each iteration step. Under suitable initial conditions, we construct an invariant region to show the global existence in time of solutions for the system. By means of Sobolev embeddings and regularity results, we find estimates for predator and prey populations in adequate norms. In order to demonstrate the convergence properties of the introduced method, several numerical examples are given.

Time global existence and finite time blow-up criterion for solutions to the Keller-Segel system coupled with the Navier-Stokes fluid

We will deal with the chemotaxis model under the effect of the Navier-Stokes fluid, i.e., the incompressible viscous fluid. We shall show the existence of a local mild solution for large initial data and a global mild solution for small initial data in the scale invariant class demonstrating that n0∈L1(R2) and u0∈Lσ2(R2). Our method is based on the perturbation of linearization together with the Lp−Lq-estimates of the heat semigroup. As a by-product of our method, we shall prove the smoothing effect and uniqueness of our mild solution. In addition, we shall show a blow-up criterion which almost covers the well-known threshold number 8π of the size ‖n0‖L1(R2) under the rest state of the fluid motion. Furthermore, the blow-up rate will be also discussed.

Cauchy problem for the Kuznetsov equation

We consider the Cauchy problem for a model of non-linear acoustic, named the Kuznetsov equation, describing a sound propagation in thermo-viscous elastic media. For the viscous case, it is a weakly quasi-linear strongly damped wave equation, for which we prove the global existence in time of regular solutions for sufficiently small initial data, the size of which is specified, and give the corresponding energy estimates. In the inviscid case, we update the known results of John for quasi-linear wave equations, obtaining the well-posedness results for less regular initial data. We obtain, using a priori estimates and a Klainerman inequality, the estimations of the maximal existence time, depending on the space dimension, which are optimal, thanks to the blow-up results of Alinhac. Alinhac's blow-up results are also confirmed by a $L^2$-stability estimate, obtained between a regular and a less regular solutions.

Space dependent adhesion forces mediated by transient elastic linkages: New convergence and global existence results

In the first part of this work we show the convergence with respect to an asymptotic parameter ε of a delayed heat equation. It represents a mathematical extension of works considered previously by the authors [14–16]. Namely, this is the first result involving delay operators approximating protein linkages coupled with a spatial elliptic second order operator. For the sake of simplicity we choose the Laplace operator, although more general results could be derived. The main arguments are (i) new energy estimates and (ii) a stability result extended from the previous work to this more involved context. They allow to prove convergence of the delay operator to a friction term together with the Laplace operator in the same asymptotic regime considered without the space dependence in [14]. In a second part we extend fixed-point results for the fully non-linear model introduced in [16] and prove global existence in time. This shows that the blow-up scenario observed previously does not occur. Since the latter result was interpreted as a rupture of adhesion forces, we discuss the possibility of bond breaking both from the analytic and numerical point of view.

The variational formulation of the fully parabolic Keller–Segel system with degenerate diffusion

We prove the time-global existence of solutions of the degenerate Keller–Segel system in higher dimensions, under the assumption that the mass of the first component is below a certain critical value. What we deal with is the full parabolic–parabolic system rather than the simplified parabolic–elliptic system. Our approach is to formulate the problem as a gradient flow on the Wasserstein space. We first consider a time-discretized problem, in which the values of the solution are determined iteratively by solving a certain minimizing problem at each time step. Here we use a new minimizing scheme at each time level, which gives the time-discretized solutions favorable regularity properties. As a consequence, it becomes relatively easy to prove that the time-discretized solutions converge to a weak solution of the original system as the time step size tends to zero.

Chemotactic systems in the presence of conflicts: A new functional inequality

The evolution of a chemotactic system involving a population of cells attracted to self-produced chemicals is described by the Keller–Segel system. In dimension 2, this system demonstrates a balance between the spreading effect of diffusion and the concentration due to self-attraction. As a result, there exists a critical “mass” (i.e. total cell's population) above which the solution of this system collapses in a finite time, while below this critical mass there is global existence in time. In particular, subcritical mass leads under certain additional conditions to the existence of steady states, corresponding to the solution of an elliptic Liouville equation. The existence of this critical mass is related to a functional inequality known as the Moser–Trudinger inequality.An extension of the Keller–Segel model to several cells populations was considered before in the literature. Here we review some of these results and, in particular, consider the case of conflict between two populations, that is, when population one attracts population two, while, at the same time, population two repels population one. This assumption leads to a new functional inequality which generalizes the Moser–Trudinger inequality. As an application of this inequality we derive sufficient conditions for the existence of steady states corresponding to solutions of an elliptic Liouville system.

Generalized result with a unified proof of global existence to a class of reaction‐diffusion systems

We prove in this paper a generalized result with a unified proof of global existence in time of classical solutions to a class of a reaction diffusion system with triangular diffusion matrix on a bounded domain in . The system in question is ut=aΔu − f(x,t,u,v), vt=cΔu + dΔv + ρf(x,t,u,v), , t &gt; 0 with f(x,t,0,η) = 0 and f(x,t,ξ,η)≤Kφ(ξ)eση, for all x∈Ω, t &gt; 0, ξ≥0, η≥0; where ρ, K and σ are real positive constants. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Global weak solutions to a model of micropolar fluids with Maxwell–Cattaneo heat transfer law

This work is devoted to the study of a system describing the dynamics of a heated micropolar fluid under the action of an applied magnetic field. The heat transfer is subjected to Maxwell–Cattaneo law instead of the usual Fourier law. Thanks to regularization and compactness methods the global existence in time of weak solutions with finite energy has been proved.

Global existence for reaction–diffusion systems with nonlinear diffusion and control of mass

We prove here global existence in time of weak solutions for some reaction–diffusion systems with natural structure conditions on the nonlinear reactive terms which provide positivity of the solutions and uniform control of the total mass. The diffusion operators are nonlinear, in particular operators of the porous media type ui↦−diΔuimi. Global existence is proved under the assumption that the reactive terms are bounded in L1. This extends previous similar results obtained in the semilinear case when the diffusion operators are linear of type ui↦−diΔui.

Nonlinear Evolution Equations and Its Application to a Tumour Invasion Model

We consider nonlinear evolution equations with logistic term satisfying initial Neumann-boundary condition and show global existence in time of solutions to the problem in arbitrary space dimension by using the method of energy. Applying the result to a mathematical model of tumour invasion, we discuss the property of the rigorous solution to the model. Finally we will show the time depending relationship and interaction between tumour cells, the surrounding tissue and matrix degradation enzymes in the model by computer simulations. It is seen that our mathematical result of the existence and asymptotic behaviour of solutions verifies our simulations, which also confirm the mathematical result visibly.