In this paper, we study the asymptotic properties of singular solutions to some nonstandard parabolic equation involving $p(x,t)$-Laplacian and nonlocal anisotropic sources. First, we show the existence and uniqueness of weak solutions by using the Galerkin’s approximations in the anisotropic Orlicz–Sobolev spaces. Second, in order to determine extinction rates and time of solutions, we prove some ordinary differential inequalities with the Sobolev embedding inequalities. Third, we combine priori estimates and the embedding theorems to show the rates and time estimates of blow-up solutions.

中文翻译：

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一类非标准抛物方程奇异解的渐近性质

在本文中，我们研究了某些非标准抛物方程的奇异解的渐近性质 $p（X，Ť）$-拉普拉斯和非本地各向异性源。首先，我们通过在各向异性Orlicz-Sobolev空间中使用Galerkin逼近来证明弱解的存在性和唯一性。其次，为了确定消光率和解的时间，我们用Sobolev嵌入不等式证明了一些常微分不等式。第三，我们结合先验估计和嵌入定理来显示爆破解决方案的速率和时间估计。