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Time decay estimates for a perturbed wave equation. (English) Zbl 0754.35078
Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1991, No.XIII, 10 p. (1991).
We consider global time estimates for solutions \(u(t,x)\) on \(\mathbb{R}\times\mathbb{R}^ n\), \(n\geq 3\), to problems of the form \[ (\square+V(x))u=0, \qquad u(0,x)=0,\quad \partial_ t u(0,x)=f(x). \] Theorem 1. Let \(V\in C_ 0^ \infty(\mathbb{R}^ n)\), and let \(f\in L^ p(\mathbb{R}^ n)\), \(n\geq 3\), have compact support. If either \(\| V\|_{(n+1)/2}\) is sufficiently small or \(V\geq 0\), then \(\| u(t)\|_{p'}\leq C_ p t^{-d}\| f\|_ p\).

35L05 Wave equation
35P05 General topics in linear spectral theory for PDEs
35Q35 PDEs in connection with fluid mechanics
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