Wave equation on certain noncompact symmetric spaces

Abstract

In this paper, we prove sharp pointwise kernel estimates and dispersive properties for the linear wave equation on noncompact Riemannian symmetric spaces G/K of any rank with G complex. As a consequence, we deduce Strichartz inequalities for a large family of admissible pairs and prove global well-posedness results for the corresponding semilinear equation with low regularity data as on hyperbolic spaces. 2020 Mathematics Subject Classification. 22E30, 35J10, 35P25, 43A85, 43A90. Key words and phrases. noncompact symmetric space of higher rank, semilinear wave equation, dispersive property, Strichartz inequality, global well-posedness.1 The symbol , let us recall, means precisely that there exists a constant 0 < C < +∞ such that, wherep is the dual exponent of p, defined by the formula 1 p + 1 p = 1, and similarly forq .

Smoothing properties of dispersive equations on non-compact symmetric spaces

“…These properties are consequences of Herz's principle (see [Her70]) with bi-Kinvariant functions. The inequality (3.3) follows by [Cow97], see also [APV11;Zha21]. The weighted version (3.4) was previously stated in [Kai14, Corollary 2.6] for g ∈ L 2 (X, x σ dx) with σ > ν.…”

Stochastic heat equations with logarithmic nonlinearity

“…k > 0, with the metric in geodesic polar coordinates g H n = dr 2 + (sinh r) 2 dω 2 , for (r, ω) ∈ (0, ∞) × S n−1 , see Section 2. For general spatial dimensions n ≥ 2, we refer the reader to [2], [14], [11], [12], [1] for shifted and nonshifted wave equations and also [15], [16], [21], [22], [3] for similar results on related spaces. All these results show that the critical power is p c (n) = 1, or we can say that there is no critical powers on hyperbolic spaces with power-type nonlinearities.…”