Georgiev, Vladimir; Heiming, Charlotte; Kubo, Hideo Supercritical semilinear wave equation with non-negative potential. (English) Zbl 0994.35091 Commun. Partial Differ. Equations 26, No. 11-12, 2267-2303 (2001). The authors of this interesting paper investigate the Cauchy problem for the hyperbolic equation \(\partial_t^2u+Au=F_p(u)\), \(u(0,x)=u_0(x)\), \(\partial_tu(0,x)=u_1(x)\), where \(x\in \mathbb R^3\) and \(F_p(u)\) behaves like \(|u|^p\) for some \(p>1\). Here \(A\) is a self-adjoint non-negative operator in \(L(\mathbb R^3)\). It is proved a weighted \(L^{\infty }\) estimate for the solution to the considered problem with a smooth positive time independent potential. The proof is based on application of generalized Fourier transform for the perturbed Laplace operator and a finite dependence domain argument. This estimate is applied to prove the existence of global small data solution to supercritical semilinear wave equations \(\partial_t^2u-\triangle u+V(x)u=F_p(u)\) in \([0,\infty)\times \mathbb R^3\) (\(p>p_0=1+\sqrt{2}\)) with a smooth non-negative potential \(V(x)\) with compact support. Reviewer: Dimitar A.Kolev (Sofia) Cited in 1 ReviewCited in 7 Documents MSC: 35L70 Second-order nonlinear hyperbolic equations 35B45 A priori estimates in context of PDEs 35L15 Initial value problems for second-order hyperbolic equations 35A08 Fundamental solutions to PDEs Keywords:weighted \(L^\infty\) estimates; decay estimates; generalized Fourier transform; global small data solution PDFBibTeX XMLCite \textit{V. Georgiev} et al., Commun. Partial Differ. Equations 26, No. 11--12, 2267--2303 (2001; Zbl 0994.35091) Full Text: DOI