The Glassey conjecture with radially symmetric data

Hidano, Kunio; Wang, Chengbo; Yokoyama, Kazuyoshi

doi:10.1016/j.matpur.2012.01.007

Cited by 83 publications

(90 citation statements)

References 25 publications

(42 reference statements)

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“…After that Strauss [51] conjectured that the threshold for dividing blowup phenomena in finite time for arbitrary "positive" small initial value and global existence of small solutions is given by An alternative proof of lifespan estimate with critical case p = p S (N) via Gauss's hypergeometric function can be found in Zhou [61] and Zhou-Han [65]. Similar problem can be found for (1.1) with G = |∂ t u| p (see e.g., John [27], Sideris [49], Masuda [43], Schaeffer [48] Rammaha [45], Agemi [1], Hidano-Tsutaya [17] Tzvetkov [55], Zhou [62] and Hidano-Wang-Yokoyama [18]). The complete picture of the blowup phenomena for small solutions can be summarized as follows:…”

Section: Introductionmentioning

confidence: 75%

Blow-up phenomena of semilinear wave equations and their weakly coupled systems

Ikeda

Sobajima

Wakasa

2019

Journal of Differential Equations

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In this paper we consider the wave equations with power type nonlinearities including timederivatives of unknown functions and their weakly coupled systems. We propose a framework of test function method and give a simple proof of the derivation of sharp upper bound of lifespan of solutions to nonlinear wave equations and their systems. We point out that for respective critical case, we use a family of self-similar solution to the standard wave equation including Gauss's hypergeometric functions which are originally introduced by Zhou [59]. However, our framework is much simpler than that. As a consequence, we found new (p, q)-curve for the system ∂ 2 t u − ∆u = |v| q , ∂ 2 t v − ∆v = |∂ t u| p and lifespan estimate for small solutions for new region. (2010): Primary: 35L05, 35L51, 35B44. Mathematics Subject Classification

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Section: Introductionmentioning

confidence: 75%

Blow-up phenomena of semilinear wave equations and their weakly coupled systems

Ikeda

Sobajima

Wakasa

2019

Journal of Differential Equations

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“…= n+1 n−1 . We refer to the classical works [19,48,28,47,45,1,12,55,64,13] for the proof of this conjecture, although up to the knowledge of the author the global existence in the supercritical case for the not radial symmetric case in high dimensions is still open. Recently, in [30] a blow-up result for 1 < p ≤ p Gla (n) has been proved for a semilinear damped wave model in the scattering case, that is, when the time-dependent coefficient of the damping term b(t)u t is nonnegative and summable.…”

Section: Introductionmentioning

confidence: 99%

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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“…Proof. In order to show the validity of (56) and (57) we will employ the definition of weak solution for (12) with a suitable choice of the test functions (φ, ψ) in (52) and (53). If we assume that (u, v) satisfies (8), then, supp u(t, ·), supp v(t, ·) ⊂ B R+t for any t 0.…”

Section: Introduction Of the Functionals For The Critical Casementioning

confidence: 99%

Nonexistence of Global Solutions for a Weakly Coupled System of Semilinear Damped Wave Equations of Derivative Type in the Scattering Case

Palmieri

Takamura

2019

Mediterr. J. Math.

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In this paper we consider the blow-up of solutions to a weakly coupled system of semilinear damped wave equations in the scattering case with nonlinearities of mixed type, namely, in one equation a power nonlinearity and in the other a semilinear term of derivative type. The proof of the blow-up results is based on an iteration argument. As expected, due to the assumptions on the coefficients of the damping terms, we find as critical curve in the pq plane for the pair of exponents (p, q) in the nonlinear terms the same one found by Hidano-Yokoyama and, recently, by Ikeda-Sobajima-Wakasa for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. In the critical and not-damped case we provide a different approach from the test function method applied by Ikeda-Sobajima-Wakasa to prove the blow-up of the solution on the critical curve, improving in some cases the upper bound estimate for the lifespan. More precisely, we combine an iteration argument with the so-called slicing method to show the blow-up dynamic of a weighted version of the functionals used in the subcritical case.

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The Glassey conjecture with radially symmetric data

Cited by 83 publications

References 25 publications

Blow-up phenomena of semilinear wave equations and their weakly coupled systems

Blow-up phenomena of semilinear wave equations and their weakly coupled systems

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

Nonexistence of Global Solutions for a Weakly Coupled System of Semilinear Damped Wave Equations of Derivative Type in the Scattering Case

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