A competition between Fujita and Strauss type exponents for blow-up of semi-linear wave equations with scale-invariant damping and mass

Math Methods in App Sciences

2019

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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.

1 < p \leq \max \{p_{Fuj} (n + \frac{μ - 1}{2} - \frac{\sqrt{δ}}{2}), p_{0} (n + μ)\},

Section: Concluding Remarks and Open Problemsmentioning

confidence: 98%

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

See 1 more Smart Citation

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

Math Methods in App Sciences

2019

Self Cite

“…Therefore, (1) is "parabolic-like" from the point of view of the critical exponent for "large" δ. On the other hand, in [22] it has been proved a blow-up result for δ ∈ (0, 1] provided that 1 < p max p S (n + µ 1 ), p F n + µ1−1− √ δ 2 with the exception of the critical case p = p S (n + µ 1 ) in dimension n = 1. In the preceding condition p S (r) denotes the so-called Strauss exponent, that is, the positive root of the quadratic equation γ(p, r) := 2 + (r + 1)p − (r − 1)p 2 = 0 for r > 1.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Lifespan of semilinear wave equation with scale invariant dissipation and mass and sub-Strauss power nonlinearity

Journal of Mathematical Analysis and Applications

Tu²

2019

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In this paper, we study the blow-up of solutions for semilinear wave equations with scale invariant dissipation and mass in the case in which the model is somehow "wave-like". A Strauss type critical exponent is determined as the upper bound for the exponent in the nonlinearity in the main theorems. Two blow-up results are obtained for the sub-critical case and for the critical case, respectively. In both cases, an upper bound lifespan estimate is given.

“…homogeneous linear equation. According to [3,40,5,4,39,22,30,27,21,13,31,37,38,28,29,6,35,16,20] for δ 0 the model in (2) is somehow an intermediate model between the semilinear free wave equation and the semilinear classical damped equation, whose critical exponent is p Fuj (n + µ−1 2 − √ δ 2 ) for δ ≥ (n + 1) 2 and seems reasonably to be p 0 (n + µ) for small and nonnegative values of delta, where p Fuj (n) and p 0 (n) denote the Fujita exponent and the Strauss exponent, respectively.…”

Section: Introductionmentioning

confidence: 99%

A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale‐invariant lower order terms

Math Methods in App Sciences

2020

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In this note two blow-up results are proved for a weakly coupled system of semilinear wave equations with distinct scale-invariant lower order terms both in the subcritical case and in the critical case, when the damping and the mass terms make both equations in some sense "wave-like". In the proof of the subcritical case an iteration argument is used. This approach is based on a coupled system of nonlinear ordinary integral inequalities and lower bound estimates for the spatial integral of the nonlinearities. In the critical case we employ a test function type method, that has been developed recently by Ikeda-Sobajima-Wakasa and relies strongly on a family of certain self-similar solutions of the adjoint linear equation. Therefore, as critical curve in the p -q plane of the exponents of the power nonlinearities for this weakly coupled system we conjecture a shift of the critical curve for the corresponding weakly coupled system of semilinear wave equations.