We give a comprehensive study of the analytic properties and long-time behavior of solutions of a reaction-diffusion system in a bounded domain in the case where the nonlinearity satisfies the standard monotonicity assumption. We pay the main attention to the supercritical case, where the nonlinearity is not subordinated to the linear part of the equation trying to put as small as possible amount of extra restrictions on this nonlinearity. The properties of such systems in the supercritical case may be very different in comparison with the standard case of subordinated nonlinearities. We examine the global existence and uniqueness of weak and strong solutions, various types of smoothing properties, asymptotic compactness and the existence of global and exponential attractors.

在非线性满足标准单调性假设的情况下，我们全面研究了有界域中反应扩散系统解的解析性质和长时间行为。我们主要关注超临界情况，其中非线性不从属于方程的线性部分，试图对这种非线性施加尽可能小的额外限制。与从属非线性的标准情况相比，这种系统在超临界情况下的特性可能非常不同。我们研究了弱解和强解的全局存在性和唯一性、各种类型的平滑特性、渐近紧致性以及全局和指数吸引子的存在。