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# Global Existence Results for a Semilinear Wave Equation with Scale-Invariant Damping and Mass in Odd Space Dimension

## Abstract

We consider a semilinear wave equation with scale-invariant damping and mass and power nonlinearity. For this model we prove some global (in time) existence results in odd spatial dimension *n*, under the assumption that the multiplicative constants *μ* and *ν*^{2}, which appear in the coefficients of the damping and of the mass terms, respectively, satisfy an interplay condition which makes the model somehow “wave-like”. Combining these global existence results with a recently proved blow-up result, we will find as critical exponent for the considered model the largest between suitable shifts of the Strauss exponent and of Fujita exponent, respectively. Besides, the competition among these two kind of exponents shows how the interrelationship between *μ* and *ν*^{2} determines the possible transition from a “hyperbolic-like” to a “parabolic-like” model. Nevertheless, in the case *n* ≥ 3 we will restrict our considerations to the radial symmetric case.

## Notes

### Acknowledgements

The author thanks his supervisor Michael Reissig, who introduced him first to the study of the model considered in this work. Furthermore, the author thanks Marcello D’Abbicco, who provided a crucial hint to overcome a difficulty in the one dimensional case.

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