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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 $$\| u(t)\|_{p'}\leq C_ p t^{-d}\| f\|_ p$$.

##### MSC:
 35L05 Wave equation 35P05 General topics in linear spectral theory for PDEs 35Q35 PDEs in connection with fluid mechanics
##### Keywords:
global time estimates
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