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Supercritical semilinear wave equation with non-negative potential. (English) Zbl 0994.35091

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.

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
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