A note on the blowup of scale invariant damping wave equation with sub-Strauss exponent

Tu, Ziheng; Lin, Jiayun

doi:10.48550/arxiv.1709.00866

Cited by 28 publications

(35 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Comparing the lifespan estimates in Theorem 1 with the known results summarized in the above table "Blow-up in finite time for β = 1", we remark that the heat-like estimates for n ≥ 1 were already proved by Wakasugi [49], whereas the wave-like ones for n ≥ 2 by Tu and Lin [44]. The wave-like estimates for n = 1 were almost obtained by Ikeda and Sobajima [11] for p F (n) ≤ p < p S (n + µ), with a loss in the exponent given by a constant δ > 0.…”

Section: Authorssupporting

confidence: 53%

See 1 more Smart Citation

Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma

Lai,

Schiavone,

Takamura

2020

Preprint

View full text Add to dashboard Cite

In this paper we study several semilinear damped wave equations with "subcritical" nonlinearities, focusing on demonstrating lifespan estimates for energy solutions. Our main concern is on equations with scale-invariant damping and mass. Under different assumptions imposed on the initial data, lifespan estimates from above are clearly showed. The key fact is that we find "transition surfaces", which distinguish lifespan estimates between "wave-like" and "heat-like" behaviours. Moreover we conjecture that the lifespan estimates on the "transition surfaces" can be logarithmically improved. As direct consequences, we reorganize the blow-up results and lifespan estimates for the massless case in which the "transition surfaces" degenerate to "transition curves". Furthermore, we obtain improved lifespan estimates in one space dimension, comparing to the known results.We also study semilinear wave equations with the scattering damping and negative mass term, and find that if the decay rate of the mass term equals to 2, the lifespan estimate is the same as one special case of the equations with the scaleinvariant damping and positive mass.The main strategy of the proof consists of a Kato's type lemma in integral form, which is established by iteration argument.We recall that the critical exponent p crit of (1.1) is the smallest exponent p crit > 1 such that, if p > p crit , there exists a unique global energy solution to the problem, whereas if 1 < p ≤ p crit the solution blows up in finite time. In the latter case, one is also interested in finding estimates for the lifespan T ε , which is the maximal existence time of the solution, depending on the parameter ε.Our principal model is the one in (1.1), for which we obtain Theorem 2 and Theorem 4, according to the different conditions imposed on the initial data. As straightforward consequences, we also obtain Theorem 1 and Theorem 3 for the massless case, i.e. the model with µ 2 = 0. The lifespan estimate in dimension n = 1 in this case is improved, comparing to the known results. Moreover, we continue the study of semilinear wave equations with scattering damping, negative mass term and power nonlinearity, introduced by the authors in [20,21].The paper is organized in this way: in the rest of the Introduction, we will sketch the background of the problems under consideration and we will exhibit our results, which will be proved in Section 3, exploiting, as main tool, a Kato's type lemma in integral form presented in Section 2.

show abstract

Section: Authorssupporting

confidence: 53%

“…Hence our improvements are given by the wave-like estimates if n = 1 and by the logarithmic gain T ε φ 0 (ε) if n = µ = 1 and 1 < p ≤ 2. Moreover, about the wave-like estimates for n ≥ 2, in [44] the initial data are supposed to be non-negative, whereas our conditions on the initial data are less restrictive.…”

Section: Authorsmentioning

confidence: 99%

Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma

Lai,

Schiavone,

Takamura

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…it is conjectured that the critical power is p = p c (n + µ). There are some partial results on blow-up and lifespan estimate, see [7,9,10,11,6,20] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Weighted $L^2-L^2$ estimate for wave equation and its applications

Lai

2018

Preprint

View full text Add to dashboard Cite

In this work we establish a weighted L 2 − L 2 estimate for inhomogeneous wave equation in 3-D, by introducing a Morawetz multiplier with weight of powerand then integrating on the light cones and t slice. With this weighted L 2 − L 2 estimate in hand, we may give a new proof of global existence for small data Cauchy problem of semilinear wave equation with supercritical power in 3-D. What is more, by combining the Huygens' principle for wave equations in 3-D, the global existence for semilinear wave equation with scale invariant damping in 3-D is established.

show abstract

“…Existence, blow-up and asymptotic behavior in time of solutions for the problem (1.8) have been extensively studied (see [2,31,12,7,9,27,28] for example). We only recall results closely related to the present study (µ = 2).…”

mentioning

confidence: 99%

“…where θ = θ(x, t) ≥ 0 is a positive solution on R n ×[0, T ). See [12,7,9,27,28] for related works with general positive µ.…”

mentioning

confidence: 99%

Critical exponent for the wave equation with a time-dependent scale invariant damping and a cubic convolution

Ikeda

Tanaka

Wakasa

2020

Preprint

View full text Add to dashboard Cite

In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping 2 1+t ∂ t v and a cubic convolution (|x|) is an unknown function to the problem on R 3 × [0, T ). Here T denotes a maximal existence time of v.The first aim of the present paper is to prove unique global existence of the solution to the problem and asymptotic behavior of the solution in the supercritical case γ ∈ (0, 3), and show a lower estimate of the lifespan in the critical or subcritical case γ ∈ − 1 2 , 0 . The

show abstract

A note on the blowup of scale invariant damping wave equation with sub-Strauss exponent

Cited by 28 publications

References 11 publications

Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma

Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma

Weighted $L^2-L^2$ estimate for wave equation and its applications

Critical exponent for the wave equation with a time-dependent scale invariant damping and a cubic convolution

Contact Info

Product

Resources

About