Life-Span of Semilinear Wave Equations with Scale-invariant Damping: Critical Strauss Exponent Case

Abstract

The blow up problem of the semilinear scale-invariant damping wave equation with critical Strauss type exponent is investigated. The life span is shown to be:n+2 . This result completes our previous study [12] on the sub-Strauss type exponent p < p S (n + µ). Our novelty is to construct the suitable test function from the modified Bessel function. This approach might be also applied to the other type damping wave equations.

Nonexistence of global solutions of wave equations with weak time-dependent damping and combined nonlinearity

“…But, if µ is relatively small, we may conjecture that the influence of u tt will dominate over {µ/(1 + t)}u t , which means that the critical exponent is related to p S (n). See the work [10] by the authors and Wakasa for 0 < µ < (n 2 +n+2)/{2(n+2)}, which was extended to 0 < µ < (n 2 +n+2)/(n+2) by Ikeda and Sobajima [9] and Tu and Lin [15,16]. Unfortunately, till now we are not clear of the boardline of µ, which determines that the critical power of Cauchy problem (1.1) with β = 1 will be Fujita or Strauss.…”

On the nonexistence of global solutions for critical semilinear wave equations with damping in the scattering case

“…In addition, they have derived an upper bound on the lifespan. Tu and Lin [14], [15] have improved the estimates of T ε in [5] recently.…”

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

“…Moreover, the size of µ 1 plays an important role in determining the solution behavior type. In [1,27] it is proved that p F (n) is critical for sufficiently large µ 1 , while for µ 1 < µ * := n 2 +n+2 n+2 in [3,14,9,25,26] several blow-up results are given for p p S (n + µ 1 ). We note that µ * satisfies the identity p F (n) = p S (n + µ * ).…”

“…We note that µ * satisfies the identity p F (n) = p S (n + µ * ). In particular, in [26] a different test function from that of [9] is used in the critical case. Finally, some global (in time) existence results of small data solutions are proved for µ 1 = 2 in [3,2].…”