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Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential. (English) Zbl 1030.35024

Consider the equations \[ i\partial_{t}u + \Delta u - {a\over |x|^{2}}u = 0, \qquad u(0,x) = f(x), \tag{1} \]
\[ \partial_{t}^{2}u + \Delta u - {a\over |x|^{2}}u = 0, \qquad u(0,x) = f(x),\;\partial_{t}u(0,x) = g(x),\tag{2} \] where \(\Delta\) denotes the \(n\)-dimensional Laplacian and \(a\) is a real number. Let \(n\geq 2\), \(\lambda(n) = {n-2\over 2}\) and for \(a\geq -\lambda(n)^{2}\) set \(\nu_{d}= \sqrt{(\lambda(n) +d)^{2} + a}\).
In the case that \(u\) solves (1) the authors prove the following estimate for \(f\in L^{2}(\mathbb R^{n})\) \[ \|u\|_{L^{p}_{t}(L^{q}_{x})} \leq C\|f\|_{L^{2}} \] where \(p\geq 2\) and \(q\) such that \({2\over p} + {n\over q} = {n\over 2}\) with \((n,p) \neq (2,2)\). The proof is based on a weighted \(L^{p}\)-estimate for the operator \(P_{a} = -\Delta + {a\over |x|^{2}}\).
If \(u\) solves the wave equation (2) it is proven that for Cauchy data \((f,g) \in {\dot H}^{1/2} \times {\dot H}^{-1/2}\) \[ \|(-\Delta)^{\sigma/2}u\|_{L_{t}^{p}(L_{x}^{q})} \leq C(\|f\|_{{\dot H}^{1/2}} + \|g\|_{{\dot H}^{-1/2}}) \] holds, where \(p\geq 2\) and \(q\) such that \({2\over p} + {n-1\over q} = {n-1 \over 2}\) and \(a+\lambda^{2}>0\), \(\sigma = {1\over p} +{n\over q} - {n-1\over 2}\). This result is based on a generalized Morawetz estimate again for \(P_{a}\).
The last section contains estimates of \((-\Delta)^{s/2}u\) in suitable norms. This is applied to show that the nonlinear initial value problem for \(\partial_{t}^{2}u + P_{a}u = \pm |u|^{\kappa}\), for \(\kappa \geq {n+3 \over n-1}\) with appropriate Cauchy data has a unique global solution.

MSC:

35G10 Initial value problems for linear higher-order PDEs
35J10 Schrödinger operator, Schrödinger equation
35L05 Wave equation
35L15 Initial value problems for second-order hyperbolic equations
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References:

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