Sharp lifespan estimates for the weakly coupled system of semilinear damped wave equations in the critical case

Chen, Wenhui; Dao, Tuan Anh

doi:10.1007/s00208-021-02335-y

Cited by 5 publications

(3 citation statements)

References 34 publications

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“…We inherit the method of dealing with fractional Laplace operator in paper [5], then apply it to derive the blow-up result and the upper bound estimates for lifespan simultaneously. Among other things, the next novelty is that we have succeeded in getting its lower bound to establish sharp estimates for lifespan of solutions to (1) by developing the method of used in the paper [1]. This means that our results come to fill out the gap related to lifespan estimates appearing in [13].…”

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confidence: 89%

“…With ℓ " 1, 2, ¨¨¨, k, we denote by p ℓ ą 1, the power exponents of nonlinear terms, moreover, the dissipative terms B t u ℓ and p´∆q σ B t u ℓ stand for the frictional dissipation (or external damping) and visco-elastic dissipation (or strong damping), respectively. The corresponding linear equation of (1) we have in mind is # u tt `p´∆q σ u `ut `p´∆q σ u t " 0, x P R n , t ě 0 up0, xq " u 0 pxq, u t p0, xq " u 1 pxq,…”

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confidence: 99%

“…Thanks to the conditions ( 10) and ( 11), one recognizes that ε ℓ ą 0 for all ℓ " 1, 2, ¨¨¨, k ´1, which represent some loss of decay of solutions to (1)…”

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confidence: 99%

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On the Cauchy problem for semi-linear σ -evolution equations with time-dependent damping

Dao

Aslan

2023

Preprint

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In this paper, we would like to consider the Cauchy problem for semi-linear σ-evolution equations with time-dependent damping for any σ≥1. Motivated strongly by the classification of damping terms in the paper34, the first main goal of the present work is to make some generalizations from σ=1 to σ>1 and simultaneously to investigate decay estimates for solutions to the corresponding linear equations in the so-called effective damping cases. For the next main goals, we are going not only to prove the global well-posedness property of small data solutions but also to indicate blow-up results for solutions to the semi-linear problem. In this concern, the novelty which should be recognized is that the application of a modified test function combined with a judicious choice of test functions gives blow-up phenomena and upper bound estimates for lifespan in both the subcritical case and the critical case, where σ is assumed to be any fractional number. Finally, lower bound estimates for lifespan in some spatial dimensions are also established to find out their sharp results.

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confidence: 89%

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confidence: 99%