The Strichartz estimates for the damped wave equation and the behavior of solutions for the energy critical nonlinear equation

Abstract

For the linear damped wave equation (DW), the L p -L q type estimates have been well studied. Recently, Watanabe [32] showed the Strichartz estimates for DW when d = 2, 3. In the present paper, we give Strichartz estimates for DW in higher dimensions. Moreover, by applying the estimates, we give the local well-posedness of the energy critical nonlinear damped waveEspecially, we show the small data global existence for NLDW. In addition, we investigate the behavior of the solutions to NLDW. Namely, we give a decay result for solutions with finite Strichartz norm and a blow-up result for solutions with negative Nehari functional.where d ∈ N, (φ 0 , φ 1 ) is given, and φ is an unknown complex valued function.

Unconditional well-posedness for the energy-critical nonlinear damped wave equation

“…It follows from standard way and thus we omit the details. In [5], we also did not show the inhomogeneous Strichartz estimate in the wave-endpoint case (q, r ) = (2, 2(d−1) d−3 ) for d ≥ 4. It is possible to show the endpoint Strichartz estimates by applying the argument of Keel and Tao [10] as follows.…”

“…In the previous paper [5] (see also [22]), we showed the local well-posedness of (NLDW) when 3 ≤ d ≤ 5 and we gave the global behavior of the solutions. In the present paper, we will show the local well-posedness of (NLDW) when d ≥ 6.…”

“…In the previous paper [5], we did not have the Besov-type estimates. It follows from standard way and thus we omit the details.…”

“…in the sense of tempered distributions for every t ∈ I . First, we show the local well-posedness when d ≥ 6, which was not treated in [5].…”

“…On the other hand, the derivative losses γ and γ come from the high frequency part, which behaves like the wave equation with exponential time decay. See the previous work [5] for more detail.…”