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

Abstract

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.

A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case

“…In these papers, the time -dependent multiplier is bounded by positive constants from above and from below and it is used to study semilinear damped wave models with time -dependent coefficients for the damping terms in the scattering producing case. On the other hand, the case of unbounded time -dependent multipliers is considered for semilinear wave models with scaleinvariant damping and mass terms in [24,22,35,31,36].…”

Weakly coupled system of semilinear wave equations with distinct scale-invariant terms in the linear part

“…Since for the single equation (1.7) we expect p 0 (n + µ) to be the critical exponent for small and nonnegative values of δ (cf. [37,47,44,45,48]), it is clear that the result from Theorem 2.1 cannot be sharp in this case.…”

“…The value of δ has a strong influence on some properties of solutions to (1.7) and to the corresponding homogeneous linear equation. According to [4,57,6,5,56,37,46,43,27,17,47,54,55,44,45,7,48,21,26] for δ 0 the model in (1.7) is somehow an intermediate model between the semilinear free wave equation and the semilinear classical damped equation, whose critical exponent is p Fuj (n + α − 1) for δ ≥ (n + 1) 2 , where α is defined analogously as in (1.3), and seems reasonably to be p 0 (n + µ) for small values of delta. In this paper we will deal with the system (1.1) and we will investigate how the interaction between the powers p, q in the nonlinearities provides either the global in time existence of the solution or the blow-up in finite time.…”

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

“…In the critical case of blow-up phenomena for semilinear wave equations with scale-invariant damping and mass, it is important to have a precise description of the behavior of solutions to the adjoint equation to the corresponding linear homogeneous equation. According to this purpose, in this section we will introduce a family of self-similar solutions to this equation, that can be represented by using Gauss hypergeometric functions (see also [45,46,12,13,15,35]). In particular, we refer to [35,Section 4] for the proofs of results which are not proved here.…”

“…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.…”