Search

In this paper we consider the wave equations with power type nonlinearities including time-derivatives of unknown functions and their weakly coupled systems. We propose a framework of test function methods and give a simple proof of the derivation of sharp upper bounds for lifespan of solutions to nonlinear wave equations and their systems. We point out that for respective critical cases, we use a family of self-similar solutions to the linear wave equation including Gauss's hypergeometric functions, which are originally introduced by Zhou [59]. We emphasize that our framework does not require the pointwise positivity of the initial data even in the high dimensional case N ≥ 4. Moreover, we find a new (p , q) -curve for the system ∂ t 2 u − Δ u = | v | q , ∂ t 2 v − Δ v = | ∂ t u | p with lifespan estimates for small solutions in a new region. [ABSTRACT FROM AUTHOR]

We consider the initial value problem for a semi-linear wave equation with a time-dependent effective damping term. The interest is the behavior of lifespan of solutions in view of the asymptotic profile of the coefficient of damping as time tends to infinity. A simple case of effective damping and a threshold case in the sense of overdamping were discussed in Ikeda and Wakasugi (2015) and Ikeda and Inui (2019), respectively. We discuss here general damping terms under a certain assumption that is discussed. The result of this paper is the sharp lifespan estimates of blowup solutions including the typical cases treated in previous studies. The proof of the upper bound of lifespan is a modification of the test-function method by Ikeda–Sobajima (2019) and the one of lower bound is based on the technique of scaling variables introduced by Gallay and Raugel (1998) for the one-dimensional case and Wakasugi (2017) for the higher-dimensional case. [ABSTRACT FROM AUTHOR]

We study a kind of nonlinear wave equations with damping and potential, whose coefficients are both critical in the sense of the scaling and depend only on the spatial variables. Based on the earlier works, one may think there are two kinds of blow-up phenomenons when the exponent of the nonlinear term is small. It also means there are two kinds of law to determine the critical exponent. In this paper, we obtain a blow-up result and get the estimate of the upper bound of the lifespan in critical and sub-critical cases. All of the results support such a conjecture, although for now, the existence part is still open. [ABSTRACT FROM AUTHOR]