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Strichartz estimates for wave equations with charge transfer Hamiltonians. (English) Zbl 1484.35092

Memoirs of the American Mathematical Society 1339. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-4974-2/pbk; 978-1-4704-6807-1/ebook). v, 84 p. (2021).
In this paper, the author proves Strichartz estimates (both regular and reversed) and (local) energy estimates for scattering states to the wave equation with a charge transfer Hamiltonian in \(\mathbb{R}^3\): \((\partial_t^2-\Delta+V_1(x)+V_2(x-vt)) u=0\) (\(t>0\)). Here, \(|v|<1\), the potentials are assumed to decay fast with rate \(\alpha>3\), and zero is neither an eigenvalue nor a resonance of the corresponding operator. The following Strichartz estimates (as well as the generalized version in polar coordinates) are obtained: \(\|u\|_{L^p_t L^q_x}+ \|u\|_{L^2_t L^\infty_{|x|}L^\lambda_{\omega}}\le C \|\partial_{t,x} u(0)\|_{L^2}\) with \(p>2\), \(\lambda<\infty\) and \(1/p+3/q=1/2\). The energy estimate and the local energy decay estimate are also established: \(\|\partial_{t,x} u\|_{L^\infty_t L^2_x} +\|(1+|x-\mu t|)^{-1/2-\epsilon}\partial_{t,x} u\|_{L^2_t L^2_x}\le C \|\partial_{t,x} u(0)\|_{L^2}\), when \(|\mu|<1\) and \(\epsilon>0\). In addition, the author obtains the endpoint reversed Strichartz estimates \(\|u(x,t)\|_{L^\infty_xL^2_t} + \|u(x+vt,t)\|_{L^\infty_xL^2_t} \le C \|\partial_{t,x} u(0)\|_{L^2}\). In order to study nonlinear multisoltion systems, the author also proved the inhomogeneous generalizations of Strichartz estimates and local energy decay estimates. As an application of these results, the author shows that scattering states indeed scatter to solutions to the free wave equation. These estimates for those linear models are also of crucial importance for problems related to interactions of potentials and solitons, for example, in the author’s companion paper [Commun. Math. Phys. 364, No. 1, 45–82 (2018; Zbl 1420.35287)].

MSC:

35B45 A priori estimates in context of PDEs
35L05 Wave equation
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35L15 Initial value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 1420.35287
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References:

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