Blow-up of solutions to nonlinear wave equations in two space dimensions

Agemi, Rentaro

doi:10.1007/bf02567635

Cited by 43 publications

(50 citation statements)

References 6 publications

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“…On the other hand, if 1 < p ≤ p c , then the problem (1.1) and (1.2) does not admit global solutions in general. (See R. Agemi [1], S. Alinhac [3], L. Hörmander [7], A. Hoshiga [9], and F. John [10].) Therefore, we shall call the number p c the critical exponent in the following.…”

Section: Introduction and Statement Of Main Result We Consider The Imentioning

confidence: 99%

Global Small Amplitude Solutions of Nonlinear Hyperbolic Systems with a Critical Exponent under the Null Condition

Hoshiga

Kubo²

2000

SIAM J. Math. Anal.

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Abstract. This paper deals with the Cauchy problems of nonlinear hyperbolic systems in two space dimensions with small data. We assume that the propagation speeds differ from each other and that nonlinearities are cubic. Then it will be shown that if the nonlinearities satisfy the null condition, there exists a global smooth solution. To prove this kind of claim, one usually makes use of the generalized differential operators Ω ij , S, and L i , which will be introduced in section 1. But it is difficult to adopt the operators L i = x i ∂t + t∂x i to our problem, because they do not commute with the d'Alembertian whose propagation speed is not equal to one. We succeed in taking L i away from the proof of our theorem. One can apply our method to a scalar equation; hence L i are needless in this kind of argument.

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Section: Introduction and Statement Of Main Result We Consider The Imentioning

confidence: 99%

Global Small Amplitude Solutions of Nonlinear Hyperbolic Systems with a Critical Exponent under the Null Condition

Hoshiga

Kubo²

2000

SIAM J. Math. Anal.

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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: 95%

“…Let ( f 1 , g1 ) and ( f 2 , g 2 ) satisfy (1.3) and let (u, v) be a weak solution of the system (8.1)satisfying supp(u, v) ⊂ {(x, t) ∈ R N × [0, T ) ; |x| ≤ r 0 + t} for r 0 = sup{|x| ; x ∈ supp ( f 1 , f 2 , g 1 , g 2 )}. If Γ GG (N, p, q) = max{F GG(N, p, q), F GG (N, q, p)} ≥ 0, then T has the following upper bound…”

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confidence: 99%

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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“…= 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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Blow-up of solutions to nonlinear wave equations in two space dimensions

Cited by 43 publications

References 6 publications

Global Small Amplitude Solutions of Nonlinear Hyperbolic Systems with a Critical Exponent under the Null Condition

Global Small Amplitude Solutions of Nonlinear Hyperbolic Systems with a Critical Exponent under the Null Condition

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

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