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\(L^ p\) estimates for the wave equation with a potential. (English) Zbl 0795.35059

\(L^ p\) estimates for the wave equation \[ \partial_ t^ 2 u-\Delta u+V(x) u=0 \] with the initial conditions \(u(0,x)= f_ 1(x)\), \(u_ t(0,x)= f_ 2(x)\), \(x\in\mathbb{R}^ n\) are considered. Under some conditions on the functions \(f_ 1(x)\), \(f_ 2(x)\), \(V(x)\) and the constants \(p\), \(p'\), \(n\), the inequality \[ \| u\|_{p'}\leq Ct^{-d} \Bigl[\| f_ 1\|_ p+ \| \nabla f_ 1\|_ p+ \| f_ 2\|_ p\Bigr], \] is proved with some constants \(C\) and \(d\). The following cases are considered: Global decay with weight, weighted estimates for sufficiently small potentials, estimate for large frequencies, estimate for bounded frequencies, space-time estimates.

MSC:

35L05 Wave equation
35B40 Asymptotic behavior of solutions to PDEs
35Q40 PDEs in connection with quantum mechanics
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