## Tsukuba Journal of Mathematics

### Nonlinear wave equation with potential

Sandra Lucente

#### Abstract

We study the Cauchy problem for $u_{tt}-\Delta u+V(x)|u|^{p-1}u=0$ with $x\in R^{n}$. The function $V(x)$ is positive and regular. The exponent $p$ is subcritical or critical. By the aid of Shatah-Struwe technique (cf.[7]), we prove the existence of the global classical solution with suitable hypotheses on $V(x):V(x)>0,3\leq n\leq 7$ or $V(x)=|x|^{2}$, $n=3$. To approach this second case we cannot follow directly the argument used in [7]: we need and prove weighted nonlinear estimates in Besov spaces.

#### Article information

Source
Tsukuba J. Math., Volume 24, Number 1 (2000), 81-107.

Dates
First available in Project Euclid: 30 May 2017

https://projecteuclid.org/euclid.tkbjm/1496164047

Digital Object Identifier
doi:10.21099/tkbjm/1496164047

Mathematical Reviews number (MathSciNet)
MR1791332

Zentralblatt MATH identifier
0979.35103

#### Citation

Lucente, Sandra. Nonlinear wave equation with potential. Tsukuba J. Math. 24 (2000), no. 1, 81--107. doi:10.21099/tkbjm/1496164047. https://projecteuclid.org/euclid.tkbjm/1496164047