This paper deals with qualitative properties of solutions in a parabolic equation with nonstandard growth conditions and singular medium void. For the sublinear source, we prove that all solutions are global. For the linear source, the solutions are global provided that the measure of domain is small. For the superlinear source, the solutions blow up at infinite time given the initial data belonging to the unstable set, and blow up in finite time with positive or negative energy, respectively. The exponential decay rates of global solutions are demonstrated as well and the bounds of blow-up time are determined for all dimensions of space domain. At last, we discuss the blow-up rates of solutions.

本文研究具有非标准增长条件和奇异介质空隙的抛物方程的解的定性性质。对于亚线性源，我们证明所有解都是全局的。对于线性源，如果域的度量较小，则解决方案是全局的。对于超线性源，在给出属于不稳定集的初始数据的情况下，解在无限时间内爆炸，并在有限时间内分别以正或负能量爆炸。还证明了整体解的指数衰减率，并确定了空间域所有维度的爆炸时间界限。最后，我们讨论了解决方案的爆炸率。