Narrow Results

In this paper, we investigate the Cauchy problem for the $d$-dimensional ($d\ge 2$) incompressible Oldroyd-B model with only fractional dissipation ${(-\mathrm{\Delta })}^{\beta }\tau$, $0\le \beta \le 1$. Applying the pure energy argument rather than spectral analysis method, we obtain the small initial data global existence results in critical Besov spaces, and establish the sharp time-decay rates of the global solutions to such model for the case $0\le \beta \le 1/2$ and the case $1/2\le \beta \le 1$, respectively. In particular, our results include the situation that Oldroyd-B model possesses only stress tensor damping and has no dissipation term.

We consider the one-dimensional compressible Navier–Stokes system for a viscous and heat-conducting ideal polytropic gas when the viscosity $\mu$ and the heat conductivity $\kappa$ depend on the specific volume $v$ and the temperature $\theta$ and are both proportional to $h(v){\theta }^{\alpha }$ for certain non-degenerate smooth function $h$. We prove the existence and uniqueness of a global-in-time non-vacuum solution to its Cauchy problem under certain assumptions on the parameter $\alpha$ and initial data, which imply that the initial data can be large if $\left|\alpha \right|$ is sufficiently small. Such a result appears to be the first global existence result for general adiabatic exponent and large initial data when the viscosity coefficient depends on both the density and the temperature.

Whether or not smooth solutions to the 3D compressible magnetohydrodynamic (MHD) equations without magnetic diffusion are always global in time remains an extremely challenging open problem. No global well-posedness or stability result is currently available for this 3D MHD system in the whole space ${ℝ}^3$ or the periodic box ${𝕋}^3$ even when the initial data is small or near a steady-state solution. This paper presents a global existence and stability result for smooth solutions to this 3D MHD system near any background magnetic field satisfying a Diophantine condition.

The Trojan Y Chromosome (TYC) strategy is a biological control method for controlling invasive populations with an XX–XY sex determinism, wherein a modified organism is introduced into an invasive population to skew the sex ratio over time, causing local extinction. However, the classical three compartment TYC model possesses blow-up solutions, for large initial conditions [Takyi EM, Beauregard MA, Griffin T, Bobo L, Parshad RD, On large and small data blow-up solutions in the Trojan Y Chromosome model, Axioms 11(3):120, 2022]. We investigate model improvements via accounting for a modified logistic term, female mating preference, competition between males and supermales and pair formation. Each one of these models is dynamically explored and is shown to possess global in-time non-negative solutions, in any parameter and initial data regime, and the models are also effective in facilitating extinction of the invasive wild type.

We discuss the existence of the global solution for two types of nonlinear parabolic systems called the Keller–Segel equation and attractive drift–diffusion equation in two space dimensions. We show that the system admits a unique global solution in . The proof is based upon the Brezis–Merle type inequalities of the elliptic and parabolic equations. The proof can be applied to the Cauchy problem which is describing the self-interacting system.

We study the global existence of classical solutions to volume–surface reaction–diffusion systems with control of mass. Such systems appear naturally from modeling evolution of concentrations or densities appearing both in a volume domain and its surface, and therefore have attracted considerable attention. Due to the characteristic volume–surface coupling, global existence of solutions to general systems is a challenging issue. In particular, the duality method, which is powerful in dealing with mass conserved systems in domains, is not applicable on its own. In this paper, we introduce a new family of $L^p$-energy functions and combine them with a suitable duality method for volume–surface systems, to ultimately obtain global existence of classical solutions under a general assumption called the intermediate sum condition. For systems that conserve mass, but do not satisfy this condition, global solutions are shown under a quasi-uniform condition, that is, under the assumption that the diffusion coefficients are close to each other. In the case of mass dissipation, we also show that the solution is bounded uniformly in time by studying the system on each time-space cylinder of unit size, and showing that the solution is sup-norm bounded independently of the cylinder. Applications of our results include global existence and boundedness for systems arising from membrane protein clustering or activation of Cdc42 in cell polarization.

This paper establishes the existence-and-uniqueness theorems of the global solution to a class of stochastic delay differential equations only with the local Lipschitz condition but does not consider the linear growth condition and studies stochastic ultimate boundedness and the moment average in time. Here we allow the coefficients of these equations to be polynomial or their growths are controlled by polynomial speed. To illustrate our idea clearly, we consider a scalar equation and a special n-dimensional equation carefully.

The objective of this paper is to prove the existence of a global solution to a certain stochastic partial differential equation subject to the $L^2$-norm being constrained. The corresponding evolution equation can be seen as the projection of the unconstrained problem onto the tangent space of the unit sphere $\mathcal{ℳ}$ in a Hilbert space $L^2$.

Global existence results have been obtained by Serre and Grassin–Serre for smooth solutions to the Euler equations of a perfect gas, provided the initial data belong to suitable spaces, the initial sound speed is small, and the initial velocity forces particles to spread out. We work in two space dimensions and start with initial data which are rotation invariant around 0 and of the type considered by Serre and Grassin–Serre. We then consider slightly perturbed initial data which are also rotation invariant around 0 and jump across a given circle centered at 0, in such a way that there is a solution with these perturbed initial data which presents two centered waves (in radial coordinates) and one contact discontinuity for small positive time. We show that this solution is global in positive time and keeps the same structure.

We prove the unique existence of time-periodic solutions to general hyperbolic equations with periodic external forces autonomous or nonautonomous over a domain bounded by two parallel planes, provided that all the characteristics with respect to the direction normal to the planes have the same sign. It is also shown that global-in-time solutions to initial-boundary value problems coincide with the solutions to corresponding time-periodic problems after a finite time. We devote one section to the reformulation of several realistic problems and see our results have wide applicability.

We consider a fragmentation-coagulation equation with growth, where the nonlinear coagulation term, introduced in O. Arino and R. Rudnicki [2], is designed to model processes in which only a part of particles in the aggregates is capable of coalescence. We introduce various growth models, describing both biological and inorganic processes, and discuss their effect on the generation of the linear growth-fragmentation semigroup. Once its existence has been established, the solution of the full nonlinear problem follows by showing that the coagulation term is globally Lipschitz.

This article is concerned with the behavior of global solutions of a parabolic partial differential equation with a power nonlinearity. Depending on parameter values and initial data, various behavior of global solutions can be observed such as grow-up, convergence, quasi-convergence and non-convergence. We give precise descriptions of the relationship between the spatial decay rate of initial data and the large-time behavior of solutions.