In this paper, a doubly nonlinear parabolic problem involving nonstandard growth conditions under homogeneous Dirichlet boundary conditions is studied. To be a little precise, first, some energy estimates and inequalities are exploited to obtain energy decay estimates of the solution. Then, it is proved that the rest field u(x, t) = 0 is asymptotically stable in terms of natural energy associated with the solution. Second, we give two new blow-up criteria that one of them is obtained to remove the exponent restriction by using an extended form of the concavity method and the other one is a blow-up result of weak solutions with arbitrarily high initial energy. Moreover, an extinction result is also showed by establishing a new auxiliary lemma. The present results of this paper complement and improve a recent paper investigated by Liu et al. [J. Math. Phys. 59, 121504 (2018)].

中文翻译：

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一类具有各向异性非标准生长条件的非线性抛物线方程

本文研究了在齐次狄利克雷边界条件下涉及非标准生长条件的双非线性抛物线问题。准确地说，首先，利用一些能量估计和不等式来获得解的能量衰减估计。然后，证明了剩余场 u(x, t) = 0 在与解相关的自然能量方面是渐近稳定的。其次，我们给出了两个新的blow-up标准，其中一个是通过使用凹度方法的扩展形式来消除指数限制的，另一个是具有任意高初始能量的弱解的blow-up结果。此外，还通过建立新的辅助引理来显示灭绝结果。本文的当前结果补充和改进了 Liu 等人最近研究的一篇论文。[J. 数学。物理。59, 121504 (2018)]。