Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent

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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“…where U 1 is defined by (24). Thanks to (28) we have that U 1 is nonnegative. Then, integrating (34) over [0, t], we get the estimate…”

Section: Lower Bounds For the Spatial Integral Of The Nonlinearitiesmentioning

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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“…Since the critical exponent is p 0 (n + 2) for n = 2 and any n ≥ 3, n odd, we remark for the value = 2 a "wave-like" behavior from the point of view of the critical exponent p in (1.1). Moreover, recently, in several works, namely, Ikeda and Sobajima, Lai et al, and Tu and Lin, [10][11][12][13] it has been studied the blow-up of solutions to (1.1) in the case in which the constant is small. Roughly speaking, in those papers, it is derived p > p 0 (n + ) as a necessary condition for the global (in time) existence of solutions of (1.1), under suitable assumptions on initial data, for 0 < < n 2 +n+2…”

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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“…where Φ is defined by (21). As Φ is an eigenfunction of the Laplace operator and y 2 (t, s; λ, b 1 ), y 2 (t, s; λ, b 2 ) solve L * b1 y = 0 and L * b2 y = 0, respectively, we get that φ and ψ satisfy…”

Section: Remark 42 Even Though Inmentioning

confidence: 99%

“…Before stating the main results of this paper, let us introduce a suitable notion of energy solutions according to [21].…”

Section: Introductionmentioning

confidence: 99%

Blow-up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities

Palmieri

Takamura

2019

Nonlinear Analysis

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In this work we study the blow-up of solutions of a weakly coupled system of damped semilinear wave equations in the scattering case with power nonlinearities. We apply an iteration method to study both the subcritical case and the critical case. In the subcritical case our approach is based on lower bounds for the space averages of the components of local solutions. In the critical case we use the slicing method and a couple of auxiliary functions, recently introduced by Wakasa-Yordanov, to modify the definition of the functionals with the introduction of weight terms. In particular, we find as critical curve for the pair (p, q) of the exponents in the nonlinear terms the same one as for the weakly coupled system of semilinear wave equations with power nonlinearities.

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Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent

Cited by 86 publications

References 24 publications

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

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

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

Blow-up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities

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