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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 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]