Global existence of small data solutions for wave models with sub-exponential propagation speed. (English) Zbl 1331.35236
Summary: In the paper [the authors, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 121, 82–100 (2015; Zbl 1320.35215)] we explained that it is reasonable to divide the study of global existence of small data solutions to semi-linear classical damped wave models with time-dependent speed of propagation, time-dependent dissipation and power nonlinearity into two cases. The super-exponential case was treated in [loc. cit.]. The present paper is devoted to the sub-exponential case. Both cases arise from a different influence of the interplay of time-dependent coefficients on the critical exponent. Here we only sketch differences to the approach for the super-exponential case. These differences appear in handling the nonlinearity. The corresponding Matsumura type estimates for a family of linear Cauchy problems depending on a parameter coincide in both cases (see [the authors, in: Fourier analysis. Pseudo-differential operators, time-frequency analysis and partial differential equation. Based on the presentations at the international conference, Aalto University. Cham: Birkhäuser/Springer. 9–45 (2014; Zbl 1332.35038); loc. cit.]).
MSC:
35L71 | Second-order semilinear hyperbolic equations |
35L15 | Initial value problems for second-order hyperbolic equations |
35B40 | Asymptotic behavior of solutions to PDEs |
Keywords:
power nonlinearity; Matsumura type estimates; decay behavior; time-dependent speed of propagation; time-dependent dissipationReferences:
[1] | Bui, Tang Bao Ngoc, Semi-linear waves with time-dependent speed and dissipation (2014), TU Bergakademie Freiberg, (Ph.D. thesis) · Zbl 1332.35038 |
[2] | Bui, Tang Bao Ngoc; Reissig, M., The interplay between time-dependent speed of propagation and dissipation in wave models, (Ruzhansky, M.; Turunen, V., Fourier Analysis. Fourier Analysis, Trends in Mathematics (2014), Birkhäuser), 9-45 · Zbl 1332.35038 |
[3] | Bui, Tang Bao Ngoc; Reissig, M., Global existence of small data solutions for wave models with super-exponential propagation speed, Nonlinear Anal., 121, 82-100 (2015) · Zbl 1320.35215 |
[4] | D’Abbicco, M.; Lucente, S.; Reissig, M., Semi-linear wave equations with effective damping, Chin. Ann. Math. Ser. B, 34, 3, 345-380 (2013) · Zbl 1278.35150 |
[5] | Lin, J.; Nishihara, K.; Zhai, J., Critical exponent for the semilinear wave equation with time-dependent damping, Discrete Contin. Dyn. Syst., 32, 12, 4307-4320 (2012) · Zbl 1255.35058 |
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