This paper deals with the existence and non-xistence of the solutions to the Cauchy problem of a class of quasilinear parabolic equations with a gradient term. We establish Fujita-type blowup theorems and determine the critical Fujita exponent in terms of spatial dimension, the asymptotic behavior of the coefficients of the gradient term at infinity, the exponents of spatial positions in the coefficients of the time-derivative term, and the source term. In particular, we classify the critical case as the blowup case.

中文翻译：

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带梯度项的拟线性抛物方程柯西问题解的存在与不存在

本文研究一类具有梯度项的拟线性抛物方程的柯西问题解的存在与不存在。我们建立了 Fujita 型膨胀定理，并确定了空间维数方面的临界 Fujita 指数、无穷远梯度项系数的渐近行为、时间导数项系数中空间位置的指数以及源学期。特别是，我们将临界案例归类为爆炸案例。