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Blow up of solutions to semilinear wave equations. (English) Zbl 1032.35137
Summary: This work shows the absence of global solutions to the equation \[ u_{tt} = \Delta u + p^{-k} u ^m, \] in the Minkowski space \(\mathbb {M}_0=\mathbb {R}\times\mathbb {R}^N\), where \( m > 1\), \((N-1)m < N+1\), and \(p \) is a conformal factor approaching zero at infinity. Using a modification of the method of conformal compactification, we prove that any solution develops a singularity at a finite time.
MSC:
35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35L15 Initial value problems for second-order hyperbolic equations
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