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Blow-up for semilinear damped wave equations with subcritical exponent in the scattering case. (English) Zbl 1395.35045

In this paper, the authors studied finite time blow-up for solutions to the damped semi-linear wave equation in scattering case (\(\beta>1\) and \(\mu>0\)) with small data \(u_{tt}-\Delta u + \frac{\mu u_{t}}{(1+t)^{\beta}} = |u|^{p}\). When \(\mu=0\), it is related to the Strauss conjecture, and it is well-known that the critical exponent of the problem is the Strauss exponent, \(p_{S}(n)\), which is the positive root of quadratic equation \(\gamma(p,n):= 2 + (n+1)p - (n-1)p^{2} = 0\). Here, as \(\beta > 1\) and \(\mu>0\), we have \((1+t)^{-\beta}\in L^{1}\) and it is natural to expect similar properties for the solutions. As is expected, the authors proved the blow up results for \(1<p< p_{S}(n)\), together with upper bound of the lifespan \[ T \leq C\varepsilon^{\frac{-2p(p-1)}{\gamma(p,n)}} \] for a class of small initial data with size \(\varepsilon\). The idea of proof is to mainly exploit the method of test functions developed in Sideris and Yordanov-Zhang, together with appropriately chosen integration factor.

MSC:

35B44 Blow-up in context of PDEs
35L71 Second-order semilinear hyperbolic equations
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