The present paper is on the existence and behaviour of solutions for a class of semilinear parabolic equations, defined on a bounded smooth domain and assuming a nonlinearity asymptotically linear at infinity. The behavior of the solutions when the initial data varies in the phase space is analyzed. Global solutions are obtained, which may be bounded or blow-up in infinite time (grow-up). The main tools are the comparison principle and variational methods. In particular, the Nehari manifold is used to separate the phase space into regions of initial data where uniform boundedness or grow-up behavior of the semiflow may occur. Additionally, some attention is paid to initial data at high energy level.

中文翻译：

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饱和介质中半线性抛物线问题的爆破有界解

本文是关于一类半线性抛物方程的解的存在性和性质，该半抛物型方程定义在有界光滑域上，并假定无穷大处的非线性渐近线性。分析了当初始数据在相空间中变化时解的行为。获得了全局解，该解可能在无限时间内被限制或爆炸（增长）。主要工具是比较原理和变分方法。特别是，使用Nehari流形将相空间分成初始数据区域，在该区域中可能会出现均匀的有界或半流动的长大行为。另外，在高能级时要注意初始数据。