TY - JOUR
T1 - <em>L<sup>p</sup></em>-<em>L<sup>q</sup></em> Estimates for a Linear Perturbed Klein-Gordon Equation
AU - Chunlai Mu 
JO - Journal of Partial Differential Equations
VL - 4
SP - 341
EP - 350
PY - 1995
DA - 1995/08
SN - 8
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jpde/5666.html
KW - L^p-L^q estimates
KW - Klein-Gordon equation
KW - perturbed potential
AB -  We consider L^p-L^q estimates for the solution u(t,x) to tbe following perturbed Klein-Gordon equation ∂_{tt}u - Δu + u + V(x)u = 0 \qquad x∈ R^n, n ≥ 3 u(x,0) = 0, ∂_tu(x,0) = f(x) We assume that the potential V(x) and the initial data f(x) are compact, and V(x) is sufficiently small, then the solution u(t,x) of the above problem satisfies ||u(t)||_q ≤ Ct^{-a}||f||_p for t &gt; 1 where a is the piecewise-linear function of 1/p and 1/q.