A competition between Fujita and Strauss type exponents for blow-up of semi-linear wave equations with scale-invariant damping and mass |
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Authors: | Alessandro Palmieri Michael Reissig |
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Affiliation: | Institute of Applied Analysis, Faculty for Mathematics and Computer Science, Technical University Bergakademie Freiberg, Prüferstraße 9, 09596, Freiberg, Germany |
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Abstract: | We obtain a blow-up result for solutions to a semi-linear wave equation with scale-invariant dissipation and mass and power non-linearity, in the case in which the model has a “wave like” behavior. We perform a change of variables that transforms our starting equation in a strictly hyperbolic semi-linear wave equation with time-dependent speed of propagation. Applying Kato's lemma we prove a blow-up result for solutions to the transformed equation under some assumptions on the initial data. The limit case, that is, when the exponent p is exactly equal to the upper bound of the range of admissible values of p yielding blow-up needs special considerations. In this critical case an explicit integral representation formula for solutions of the corresponding linear Cauchy problem in 1d is derived. Finally, carrying out the inverse change of variables we get a non-existence result for global (in time) solutions to the original model. |
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Keywords: | primary 35B44 35L05 secondary 33C15 35L15 35L71 Semi-linear wave equation Time-dependent speed of propagation Power non-linearity Blow-up Critical case Integral representation formula |
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