No 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.

We concern the blow up problem to the scale invariant damping wave equations with sub-Strauss exponent. This problem has been studied by Lai, Takamura and Wakasa ([5]) and Ikeda and Sobajima [4] recently. In present paper, we extend the blowup exponent from p F (n) ≤ p < p S (n + 2µ) to 1 < p < p S (n + µ) without small restriction on µ. Moreover, the upper bound of lifespan is derived with uniformly estimate T (ε) ≤ Cε −2p(p−1)/γ (p,n+2µ) . This result extends the blowup result of semilinear wave equation and shows the wave-like behavior of scale invariant damping wave equation's solution even with large µ > 1.

In this work, we investigate the influence of general damping and potential terms on the blow-up and lifespan estimates for energy solutions to power-type semilinear wave equations. The space-dependent damping and potential functions are assumed to be critical or short range, spherically symmetric perturbation. The blow up results and the upper bound of lifespan estimates are obtained by the so-called test function method. The key ingredient is to construct special positive solutions to the linear dual problem with the desired asymptotic behavior, which is reduced, in turn, to constructing solutions to certain elliptic "eigenvalue" problems.

No abstract