The aim of this paper is to study some properties of positive solutions to the nonlinear diffusion equation$$\begin{aligned} \frac{\partial u(x,t)}{\partial t} = \Delta _p u(x,t) + c(x)f(u(x,t)), \;\; (x,t) \in \Omega \times (0,\infty ). \end{aligned}$$Assuming that f is of a bistable type with stable constant steady states 0 and \(c_0 >0\), we show, that there exist a universal, a priori upper bound for all positive solutions of the previous equation. Moreover, we prove the convergence of these solutions to the constant \(c_0\) as t tends to \(+\,\infty \). Some examples where our results can be applied are provided.

中文翻译：

---

非线性热方程整体解的通用界

本文的目的是研究非线性扩散方程$$ \ begin {aligned} \ frac {\ partial u（x，t）} {\ partial t} = \ Delta _p u（x， t）+ c（x）f（u（x，t）），\; \; [x，t）\ in \ Omega \ times（0，\ infty）\ end {aligned} $$假设f是具有稳定恒定稳态0和\（c_0> 0 \）的双稳态类型，我们证明对于前一个所有正解，存在一个通用的先验上限方程。此外，随着t趋于\（+ ,, \ infty \），我们证明了这些解对常数\（c_0 \）的收敛性。提供了一些可以应用我们的结果的示例。