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Universal bound for global solution of nonlinear heat equation

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Abstract

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.

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Authors and Affiliations

  1. Laboratory of Mathematical Analysis and Applications (LMAA-LR11-ES11), Faculty of Mathematical, Physical and Natural Sciences of Tunis, University of Tunis El-Manar, 2092, Tunis, Tunisia

    Nedra Belhaj Rhouma & Mouldi Seddik

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  1. Nedra Belhaj Rhouma
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  2. Mouldi Seddik
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Correspondence to Mouldi Seddik.

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Belhaj Rhouma, N., Seddik, M. Universal bound for global solution of nonlinear heat equation. Positivity 24, 837–854 (2020). https://doi.org/10.1007/s11117-019-00712-1

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