Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation, II

Palmieri, Alessandro; Reissig, Michael

doi:10.1002/mana.201700144

Cited by 65 publications

(64 citation statements)

References 10 publications

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“…and, hence, as j → ∞ in (39) we see that the lower bound for U (z) is not finite. Therefore, in order to guarantee the existence of U (z), it must hold the converse inequality for z.…”

Section: Iteration Argument: Critical Casementioning

confidence: 84%

Global Existence Results for a Semilinear Wave Equation with Scale-Invariant Damping and Mass in Odd Space Dimension

Palmieri

2019

Trends in Mathematics

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In this note, we prove blow-up results for semilinear wave models with damping and mass in the scale-invariant case and with nonlinear terms of derivative type. We consider the single equation and the weakly coupled system. In the first case we get a blow-up result for exponents below a certain shift of the Glassey exponent. For the weakly coupled system we find as critical curve a shift of the corresponding curve for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. Our approach follows the one for the respective classical wave equation by Zhou Yi. In particular, an explicit integral representation formula for a solution of the corresponding linear scale-invariant wave equation, which is derived by using Yagdjian's integral transform approach, is employed in the blow-up argument. While in the case of the single equation we may use a comparison argument, for the weakly coupled system an iteration argument is applied. Keywords Blow-up, Glassey exponent, Nonlinearity of derivative type, Time-dependent and scale-invariant lower order terms, Integral representation formula, Upper bound estimates of the lifespan AMS Classification (2010) Primary: 35B44, 35L71; Secondary: 35B33, 35C15

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“…and, hence, as j → ∞ in (39) we see that the lower bound for U (z) is not finite. Therefore, in order to guarantee the existence of U (z), it must hold the converse inequality for z.…”

Section: Iteration Argument: Critical Casementioning

confidence: 84%

Global Existence Results for a Semilinear Wave Equation with Scale-Invariant Damping and Mass in Odd Space Dimension

Palmieri

2019

Trends in Mathematics

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“…For further considerations on how the quantity describes the interplay between the damping term 1+t u t and the mass term 2 (1+t) 2 u one can see. 14 Recently, (1.3) has been studied in D'Abbicco and Palmieri, Nunes do Nascimento et al, Palmieri, and Palmieri and Reissig [15][16][17][18][19][20] under different assumptions on .…”

Section: Introductionmentioning

confidence: 99%

A global existence result for a semilinear scale‐invariant wave equation in even dimension

Palmieri

2019

Math Methods in App Sciences

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In the present article a, semilinear scale‐invariant wave equation with damping and mass is considered. The global (in time) existence of radial symmetric solutions in even spatial dimension n is proved by using weighted L∞ − L∞ estimates, under the assumption that the multiplicative constants, which appear in the coefficients of damping and of mass terms, fulfill an interplay condition, which yields somehow a “wave‐like” model. In particular, combining this existence result with a recently proved blow‐up result, a suitable shift of Strauss exponent is proved to be the critical exponent for the considered model. Moreover, the still open part of a conjecture done by D'Abbicco‐Lucente‐Reissig is proved to be true in the massless case.

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“…( As in the paper, 25 we also prove the optimality of the condition (C1) for p k . One choice for the parameters q 1 , q 2 and r 1 , … , r 6 is the following: q 1 = 3(p k − 1), q 2 = 6, r 1 = 6, r 2 = 3, r 3 = 3(p k − 1), r 4 = 6(p k − 1) 3(p k − 1) − 2 , r 5 = 3(p k − 1), r 6 = 6.…”

Section: Casementioning

confidence: 74%

“…and data belonging to  0 m,1 with m ∈ [ 1, 6∕5), where exactly one exponent is not above the exponent p c (m, ). Without loss of generality we choose 1 < p k 1 < p c (m, ) and p k 2 , p k 3 > p c (m, ), where the exponent p c (m, ) is defined by (25). Let us choose the evolution space (35) with the norm (39).…”

Section: Philosophy Of Our Approachmentioning

confidence: 99%

Weakly coupled systems of semilinear elastic waves with different damping mechanisms in 3D

Chen

2018

Math Methods in App Sciences

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We consider the following Cauchy problem for weakly coupled systems of semilinear damped elastic waves with a power source nonlinearity in three dimensions: Utt−a2ΔU−b2−a2∇divU+(−Δ)θUt=F(U),(t,x)∈(0,∞)×double-struckR3, where U=Ufalse(t,xfalse)=()Ufalse(1false)false(t,xfalse),Ufalse(2false)false(t,xfalse),Ufalse(3false)false(t,xfalse)normalT with b2 > a2 > 0 and θ ∈ [0,1]. Our interests are some qualitative properties of solutions to the corresponding linear model with vanishing right‐hand side and the influence of the value of θ on the exponents p1,p2,p3 in Ffalse(Ufalse)=()false|Ufalse(3false)|p1,false|Ufalse(1false)|p2,false|Ufalse(2false)|p3normalT to get results for the global (in time) existence of small data solutions.

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Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation, II

Cited by 65 publications

References 10 publications

Global Existence Results for a Semilinear Wave Equation with Scale-Invariant Damping and Mass in Odd Space Dimension

Global Existence Results for a Semilinear Wave Equation with Scale-Invariant Damping and Mass in Odd Space Dimension

A global existence result for a semilinear scale‐invariant wave equation in even dimension

Weakly coupled systems of semilinear elastic waves with different damping mechanisms in 3D

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