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

Abstract

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.

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

“…Even though the existence of global in time small data solutions in the supercritical case is an open problem, some partial results for the single semilinear equation (2) in the case δ = 1 (cf. [28,29]) suggest the likelihood and plausibility of this conjecture.…”

“…Let us begin with the base case j = 1 in (33) and (34). Plugging (22) in (29) and shrinking the domain of integration, we find for t T 0…”

“…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.…”

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.…”