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The lifespan of solutions of semilinear wave equations with the scale-invariant damping in one space dimension. (English) Zbl 1463.35358

Initial value problem for semilinear wave equations with the scale-invariant damping \[v_{tt}-\Delta v+{\mu}{(1+t)^{-1}}v_t=|v|^p,\,\,\,v=v(x,t),\,\,\,(x,t)\in R^n\times[0,\infty),\] \[v(x,0)=\varepsilon f(x),\,\,\,v_t(x,0)=\varepsilon g(x),\,\,\,x\in R^n\] is considered where \(p>1\), \(\mu>0\), \((f,g)\in H^1(R^n)\times L^2(R^n)\), \(p>1\), \(\varepsilon>0\) is a small parameter. Provided that the total integral of the sum of initial position and speed vanishes, the authors study a new type of the lifespan estimates which is closely related to the non-damped case in shifted space dimensions.

MSC:

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