TY - JOUR
T1 - Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma
AU - Lai, Ning An
AU - Schiavone, Nico Michele
AU - Takamura, Hiroyuki
N1 - Funding Information:
The first author is supported by Natural Science Foundation of Zhejiang Province ( LY18A010008 ) and National Natural Science Foundation of China ( 11771194 ). The second author is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica ( INdAM ). The third author is partially supported by the Grant-in-Aid for Scientific Research (B) (No. 18H01132 ), Japan Society for the Promotion of Science .
Funding Information:
The first author is supported by Natural Science Foundation of Zhejiang Province (LY18A010008) and National Natural Science Foundation of China (11771194). The second author is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit? e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The third author is partially supported by the Grant-in-Aid for Scientific Research (B) (No. 18H01132), Japan Society for the Promotion of Science.
Publisher Copyright:
© 2020 The Authors
PY - 2020/12/5
Y1 - 2020/12/5
N2 - In this work we consider several semilinear damped wave equations with “subcritical” nonlinearities, focusing on studying lifespan estimates for energy solutions. Our main concern is on equations with scale-invariant damping and mass. By imposing different assumptions on the initial data, we prove lifespan estimates from above, distinguishing between “wave-like” and “heat-like” behaviours. Furthermore, we conjecture logarithmic improvements for the estimates on the “transition surfaces” separating the two behaviours. As a direct consequence, we reorganize the blow-up results and lifespan estimates for the massless case, and we obtain in particular improved lifespan estimates for the one dimensional case, compared to the known results. We also study semilinear wave equations with scattering damping and negative mass term, finding that if the decay rate of the mass term equals to 2, the lifespan estimate coincides with the one in a special case of scale-invariant damped equation. The main tool employed in the proof is a Kato's type lemma, established by iteration argument.
AB - In this work we consider several semilinear damped wave equations with “subcritical” nonlinearities, focusing on studying lifespan estimates for energy solutions. Our main concern is on equations with scale-invariant damping and mass. By imposing different assumptions on the initial data, we prove lifespan estimates from above, distinguishing between “wave-like” and “heat-like” behaviours. Furthermore, we conjecture logarithmic improvements for the estimates on the “transition surfaces” separating the two behaviours. As a direct consequence, we reorganize the blow-up results and lifespan estimates for the massless case, and we obtain in particular improved lifespan estimates for the one dimensional case, compared to the known results. We also study semilinear wave equations with scattering damping and negative mass term, finding that if the decay rate of the mass term equals to 2, the lifespan estimate coincides with the one in a special case of scale-invariant damped equation. The main tool employed in the proof is a Kato's type lemma, established by iteration argument.
KW - Blow-up
KW - Lifespan
KW - Scale-invariant damping
KW - Semilinear wave equation
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U2 - 10.1016/j.jde.2020.08.020
DO - 10.1016/j.jde.2020.08.020
M3 - Article
AN - SCOPUS:85091390446
VL - 269
SP - 11575
EP - 11620
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
IS - 12
ER -