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This article studies the nonequilibrium–diffusion limit of the compressible Euler model arising in radiation hydrodynamics in ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$ with the general initial data. Combining the moment method with Hilbert expansion, we show that the radiative intensity can be approximated by the sum of interior solution and initial layer. We also show that the solution satisfied by the density, temperature, and velocity can be approximated by the interior solutions. Our results can be considered as an extension from 핋3 in arXiv.2312.15208 by Ju, Li, and Zhang to ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$. In contrast to arXiv.2312.15208, we get the exact convergence rates by studying the error system derived from the primitive system, the zeroth‐order to the second‐order about the radiative intensity, and the zeroth‐order about the hydrodynamics. [ABSTRACT FROM AUTHOR]

This article studies the asymptotic analysis of a coupled model arising in radiative transfer in $ \mathbb{T}^{3} $ with one-speed velocity. The model consists of a kinetic equation satisfied by the specific intensity of radiation coupled to a nonhomogeneous heat equation satisfied by the material temperature. We show that the radiative intensity can be approximated by the sum of the interior solution and initial layer, while the temperature can be approximated by the interior solution for the general initial data. [ABSTRACT FROM AUTHOR]

The Kolmogorov–Petrovsky–Piskunov (KPP) partial differential equation (PDE) is solved in this article using the moving mesh finite difference technique (MMFDM) in conjunction with physics‐informed neural networks (PINNs). We construct a time‐dependent mesh to obtain approximate solutions for the KPP problem. The temporal derivative is discretized using a backward Euler, while the spatial derivatives are discretized using a central implicit difference scheme. Depending on the error measure, several moving mesh partial differential equations (MMPDEs) are employed along the arc‐length and curvature mesh density functions (MDF). The proposed strategy has been suggested to yield remarkably precise and consistent results. To find the approximate solution, we additionally employ physics‐informed neural networks (PINNs) to compare the outcomes of the adaptive moving mesh approach. It has been observed that solutions obtained using the moving mesh method (MMM) are sufficiently accurate, and the absolute error is also much lower than the PINNs. [ABSTRACT FROM AUTHOR]

In this paper, we consider the properties of solutions for the following fractional parabolic equation \begin{document}$ \begin{equation*} \frac{\partial u}{\partial t}(x, t)+(-\Delta)^su(x, t) = f(u(x, t)), \ (x, t)\in\mathbb{R}^N\times\mathbb{R}. \end{equation*} $\end{document} Without assuming any decay behavior of $ u $ near infinity, we first establish a narrow region principle in unbounded domains, then employing the direct method of moving planes, we derive the monotonicity, antisymmetry and non-existence of the solutions. In most previous articles, to carry out the direct method of moving planes in the whole space, they often needed to assume that the solution tends to zero near infinity or that the solution is divided by a function to make the new solution satisfy such an asymptotic decay. Here we develop a new approach–estimating the singular integrals defining $ (-\Delta)^s $ and the derivative of the solution with respect to time along a sequence of approximate extreme points.We believe that the new method employed here will be very helpful to study a class of parabolic equations involving nonlinear nonlocal operators. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamic behaviors of a fractional order predator–prey system, in which the prey population has three effects: Allee effect, fear effect, and shelter effect. First, we prove in detail the positivity, existence, uniqueness, and boundedness of the solutions of the model from the perspective of mathematical analysis. Second, the stability of the system is considered by analyzing the stability of all equilibria and possible bifurcations of the system. It is proved that the system undergoes Hopf bifurcation with respect to four important parameters at the positive equilibrium point. Third, through stability analysis of the system, we find that: (i) as long as the initial density of the prey population is small enough, it will enter the attraction region of an extinction equilibrium point, making the system population at risk of extinction; (ii) we can eliminate the limit-cycle to make the system achieve stable coexistence by appropriately increasing the fear level or refuge rate, or reducing the prey natality or the order of fractional order systems; (iii) fractional order system is more stable than integer order systems, when the system has periodic solution, the two species can coexist stably by increasing the fear level or refuge rate appropriately. The threshold of fear level and refuge rate in fractional order systems is smaller than that in integer order systems. Finally, the rationality of the research results is verified by numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. The primary aim is to study the impact of small advection terms and heterogeneous environment, which is on two species' dynamics via a free boundary. The function $ m(x) $ represents heterogeneous environment, and it can satisfy positive everywhere condition or changeable sign condition. Firstly, on one hand, we provide long time behaviors of the solution in vanishing case when $ m(x) $ satisfies both conditions above; on the other hand, long time behaviors of the solution in spreading case are got when $ m(x) $ satisfies positive everywhere condition. Secondly, a spreading-vanishing dichotomy and several sufficient conditions through the initial data and the moving parameters are obtained to determine whether spreading or vanishing of two species happens when $ m(x) $ satisfies both conditions above. Furthermore, we derive estimates of spreading speed of the free boundary when $ m(x) $ satisfies positive everywhere condition and two species spreading occurs. [ABSTRACT FROM AUTHOR]

In this work, a prey–predator system with Holling type II response function including a Michaelis–Menten type capture and fear effect is put forward to be studied. Firstly, the existence and stability of equilibria of the system are discussed. Then, by considering the harvesting coefficient as bifurcation parameter, the occurrence of Hopf bifurcation at the positive equilibrium point and the existence of limit cycle emerging through Hopf bifurcation are proved. Furthermore, through the analysis of fear effect and capture item, we find that: (i) the fear effect can either stabilize the system by excluding periodic solutions or destroy the stability of the system and produce periodic oscillation behavior; (ii) increasing the level of fear can reduce the final number of predators, but not lead to extinction; (iii) the harvesting coefficient also has significant influence on the persistence of the predator. Finally, numerical simulations are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

The paper is devoted to investigating a semilinear parabolic equation with a nonlinear gradient source term: ut = uxx + xm|ux|p, t > 0, 0 < x < 1,where p > m + 2, m ≥ 0. Zhang and Hu [Discrete Contin. Dyn. Syst. 26 (2010) 767-779] showed that finite time gradient blowup occurs at the boundary and the accurate blowup rate is also obtained for super-critical boundary value. Throughout this paper, we present a complete large time behavior of a classical solution : is global and converges to the unique stationary solution in norm for subcritical boundary value, and blows up in infinite time for critical boundary value. Gradient growup rate is also established by the method of matched asymptotic expansions. In addition, gradient estimate of solutions is obtained by the Bernstein-type arguments. [ABSTRACT FROM AUTHOR]

In this paper we consider a free boundary tumor model with angiogenesis. The model consists of a reaction-diffusion equation describing the concentration of nutrients σσ and an elliptic equation describing the distribution of the internal pressure p. The vasculature supplies nutrients to the tumor, so that ∂σ/∂n + β(σ − σ) = 0 holds on the boundary, where a positive constant β is the rate of nutrient supply to the tumor and ˉσ is the nutrient concentration outside the tumor. The tumor cells proliferate at a rate μ. If 0 < ˜σ < σ, where ˜σ is a threshold concentration for proliferating, then there exists a unique radially symmetric stationary solution (σS(r),pS(r),RS). In this paper, we found a function μ∗ = μ∗ (RS) such that if μ < μ∗ then the radially symmetric stationary solution is linearly stable with respect to non-radially symmetric perturbations, whereas if μ > μ∗ then the radially symmetric stationary solution is linearly unstable. [ABSTRACT FROM AUTHOR]

In this paper, we consider a free boundary tumor model with a periodic supply of external nutrients, so that the nutrient concentration σ satisfies σ = ϕ(t) on the boundary, where ϕ(t) is a positive periodic function with period T. A parameter μ in the model is proportional to the "aggressiveness" of the tumor. If 0<σ˜ 0 with respect to all radially symmetric perturbations. We prove that under nonradially symmetric perturbations, there exists a number μ∗ such that if 0 < μ < μ∗, then the T‐periodic solution is linearly stable, whereas if μ > μ∗, then the T‐periodic solution is linearly unstable. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a two-species weak competition system of reaction-diffusion-advection with double free boundaries that represent the expanding front in a one-dimensional habitat, where a combination of random movement and advection is adopted by two competing species. The main goal is to understand the effect of small advection environment and dynamics of the two species through double free boundaries. We provide a spreading-vanishing dichotomy, which means that both of the two species either spread to the entire space successfully and survive in the new environment as time goes to infinity, or vanish and become extinct in the long run. Furthermore, if the spreading or vanishing of the two species occurs, some sufficient conditions via the initial data are established. When spreading of the two species happens, the long time behavior of solutions and estimates of spreading speed of both free boundaries are obtained. [ABSTRACT FROM AUTHOR]

Throughout this paper, we investigate the blowup set for the semilinear reaction‐diffusion system ut=Δu+f(u,v),x∈Ω,t>0,vt=Δv+g(u,v),x∈Ω,t>0,u(x,t)=v(x,t)=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω,where Ω=BR:={x∈Rn;|x|

This paper is focused on the gradient blowup rate for the following general semilinear parabolic equation withbeing a smoothly bounded domain inandbeing a smooth function. Under some suitable assumptions on, we establish the gradient blowup rate lower bound by the rescaling method for any. While the upper bound which is established by the comparison method can be obtained only for. At last, we provide some specific examples to verify our results. [ABSTRACT FROM PUBLISHER]