In this paper we prove existence of nonnegative solutions to parabolic Cauchy–Dirichlet problems with (eventually) singular superlinear gradient terms. The model equation is

where \(\Omega \) is an open bounded subset of \({{\,\mathrm{{\mathbb {R}}}\,}}^N\) with \(N>2\), \(0<T<+\infty \), \(1<p<N\), and \(q<p\) is superlinear. The functions \(g,\,h\) are continuous and possibly satisfying \(g(0) = +\infty \) and/or \(h(0)= +\infty \), with different rates. Finally, f is nonnegative and it belongs to a suitable Lebesgue space. We investigate the relation among the superlinear threshold of q, the regularity of the initial datum and the forcing term, and the decay rates of \(g,\,h\) at infinity.

在本文中，我们证明了（最终）具有奇异超线性梯度项的抛物型Cauchy-Dirichlet问题非负解的存在。模型方程为

其中\（\ Omega \）是\（{{\，\ mathrm {{\\ mathbb {R}}} \，}} ^ N \）的开放界子集，其中\（N> 2 \），\（0 <T <+ \ infty \），\（1 <p <N \）和\（q <p \）是超线性的。函数\（g，\，h \）是连续的，并可能以不同的速率满足\（g（0）= + \ infty \）和/或\（h（0）= + \ infty \）。最后，f为非负数，并且属于适当的Lebesgue空间。我们研究了q的超线性阈值，初始基准面和强迫项的规律性以及无穷大\（g，\，h \）的衰减率之间的关系。