Blow‐up for wave equation with the scale‐invariant damping and combined nonlinearities

In this article, we study the blow‐up of the damped wave equation in the scale‐invariant case and in the presence of two nonlinearities. More precisely, we consider the following equation: utt−Δu+μ1+tut=|ut|p+|u|q,inℝN×[0,∞), with small initial data. For μ 0 is depending on the nonlinearties' p...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2021-01, Vol.44 (1), p.1127-1136
Main Authors: Hamouda, Makram, Hamza, Mohamed  Ali
Format: Article
Language:English
Subjects:
blow‐up
Damping
Invariants
nonlinear wave equations
scale‐invariant damping
Wave equations
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Summary:In this article, we study the blow‐up of the damped wave equation in the scale‐invariant case and in the presence of two nonlinearities. More precisely, we consider the following equation: utt−Δu+μ1+tut=|ut|p+|u|q,inℝN×[0,∞), with small initial data. For μ 0 is depending on the nonlinearties' powers and the space dimension (μ∗ satisfies (q−1)(N+2μ∗−1)p−2=4), we prove that the wave equation, in this case, behaves like the one without dissipation (μ = 0). Our result completes the previous studies in the case where the dissipation is given by μ(1+t)βut;β>1, where, contrary to what we obtain in the present work, the effect of the damping is not significant in the dynamics. Interestingly, in our case, the influence of the damping term μ1+tut is important.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.6817