Volume 8, Issue 4
Lp-Lq Estimates for a Linear Perturbed Klein-Gordon Equation

Chunlai Mu

DOI:

J. Part. Diff. Eq., 8 (1995), pp. 341-350.

Published online: 1995-08

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  • Abstract

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 > 1 where a is the piecewise-linear function of 1/p and 1/q.

  • Keywords

L^p-L^q estimates Klein-Gordon equation perturbed potential

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@Article{JPDE-8-341, author = {}, title = {Lp-Lq Estimates for a Linear Perturbed Klein-Gordon Equation}, journal = {Journal of Partial Differential Equations}, year = {1995}, volume = {8}, number = {4}, pages = {341--350}, abstract = { 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 > 1 where a is the piecewise-linear function of 1/p and 1/q.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5666.html} }
TY - JOUR T1 - Lp-Lq Estimates for a Linear Perturbed Klein-Gordon Equation 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 > 1 where a is the piecewise-linear function of 1/p and 1/q.
Chunlai Mu . (2019). Lp-Lq Estimates for a Linear Perturbed Klein-Gordon Equation. Journal of Partial Differential Equations. 8 (4). 341-350. doi:
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