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We study the behavior as t tends to infinity of the solutions, \(u(x,t)\) , to the heat equation that are bounded in the half-space \(\mathbb {R}^n \times (0,\infty )\) . As t tends to infinity, such solutions either converge to a constant or fail to converge for every x; the outcome depends on the initial data. Examples of convergence and divergence are characterized in this article.

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Behavior of Bounded Solutions, \(u(x,t)\) , to the Heat Equation in \(\mathbb {R}^n \times (0,\infty )\) as \(t \to \infty \)

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  • 发表日期：

    2026年

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  • Wolodymyr R. Madych

摘要

We study the behavior as t tends to infinity of the solutions, \(u(x,t)\) , to the heat equation that are bounded in the half-space \(\mathbb {R}^n \times (0,\infty )\) . As t tends to infinity, such solutions either converge to a constant or fail to converge for every x; the outcome depends on the initial data. Examples of convergence and divergence are characterized in this article.