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On the stability of solutions for a family of parabolic equations with time delay on \({\mathbb {R}}^n\)

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Abstract

In this paper we consider the following initial value problem

$$\begin{aligned} \left\{ \begin{array}{ll} (\partial _t - \Delta _x + w\cdot \nabla _x +\alpha I) u(t,x) - \beta u(t-\tau ,x) = 0, &{} (t,x)\in {\mathbb {R}}_+\times {\mathbb {R}}^n \\ u(t,x) = \phi (t,x)\in C([-\tau ,0],L^1({\mathbb {R}}^n)) &{} \end{array} \right. \end{aligned}$$

We describe the ranges of the parameters \(\alpha , \beta ,\tau \) that guarantee that every solution u(t, x) vanishes as \(t\rightarrow \infty \).

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Acknowledgements

The author would like to express sincerest gratitude to his supervisors: Dr Ming Mei, for pointing out the problem considered in this paper and reading the manuscript, and Dr Galia Dafni, for her valuable suggestions, guidance and comments. The author would like to thank Professor Glenn Webb and an anonymous referee for reading the manuscript and their valuable feedbacks.

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  1. Department of Mathematics and Statistics, Concordia University, 1455 De Maisonneuve Blvd. W, Montreal, QC, H3G 1M8, Canada

    Almaz Butaev

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  1. Almaz Butaev
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Correspondence to Almaz Butaev.

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Butaev, A. On the stability of solutions for a family of parabolic equations with time delay on \({\mathbb {R}}^n\). Complex Anal Synerg 5, 4 (2019). https://doi.org/10.1007/s40627-019-0029-1

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  • DOI: https://doi.org/10.1007/s40627-019-0029-1

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