In this paper, the existence, uniqueness, and positivity of solutions, as well as the asymptotic behavior through a finite fractal dimensional global attractor for a general Advection-Diffusion-Reaction (ADR) equation, are investigated. Our findings are innovative, as we employ semigroups and global attractors theories to achieve these results. Also, an analytical solution of a two-dimensional Advection-Diffusion Equation is presented. And finally, two Explicit Finite Difference schemes are used to simulate solutions in the two- and three-dimensional cases. The numerical simulations are conducted with predefined initial and Dirichlet boundary conditions.

翻译：本文研究了一般平流-扩散-反应(ADR)方程解的存在性、唯一性、正性以及通过有限分形维全局吸引子体现的渐近行为。我们的研究具有创新性，利用半群和全局吸引子理论得出了这些结果。此外，本文还给出了二维平流-扩散-反应方程的解析解。最后，采用两种显式有限差分格式对二维和三维情形进行数值模拟。数值模拟在预设初始条件和Dirichlet边界条件下进行。