Abstract
This paper is twofold. The first part aims to study the long-time asymptotic behavior of solutions to the heat equation on Riemannian symmetric spaces G/K of noncompact type and of general rank. We show that any solution to the heat equation with bi-K-invariant L1 initial data behaves asymptotically as the mass times the fundamental solution, and provide a counterexample in the non bi-K-invariant case. These answer problems recently raised by J.L. Vázquez. In the second part, we investigate the long-time asymptotic behavior of solutions to the heat equation associated with the so-called distinguished Laplacian on G/K. Interestingly, we observe in this case phenomena which are similar to the Euclidean setting, namely L1 asymptotic convergence with no bi-K-invariance condition and strong L∞ convergence.
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