Blow-up phenomena of semilinear wave equations and their weakly coupled systems

Abstract

In this paper we consider the wave equations with power type nonlinearities including timederivatives of unknown functions and their weakly coupled systems. We propose a framework of test function method and give a simple proof of the derivation of sharp upper bound of lifespan of solutions to nonlinear wave equations and their systems. We point out that for respective critical case, we use a family of self-similar solution to the standard wave equation including Gauss's hypergeometric functions which are originally introduced by Zhou [59]. However, our framework is much simpler than that. As a consequence, we found new (p, q)-curve for the system ∂ 2 t u − ∆u = |v| q , ∂ 2 t v − ∆v = |∂ t u| p and lifespan estimate for small solutions for new region. (2010): Primary: 35L05, 35L51, 35B44. Mathematics Subject Classification

Global Existence Results for a Semilinear Wave Equation with Scale-Invariant Damping and Mass in Odd Space Dimension

“…where the kernel functions E, K 0 , K 1 are defined by (14), (15) and (16), respectively. Due to the sign assumption for u 0 it follows that U 0 is a nonnegative function.…”

“…Let us point out that we assumed throughout the paper that the parameters µ, ν 2 satisfy δ ≥ 0 for the single semilinear equation (respectively, µ 1 , µ 2 , ν 2 1 , ν 2 2 satisfy δ 1 , δ 2 ≥ 0 for the weakly coupled system). This assumptions are made in order to guarantee that the kernel functions defined by (14), (15) and (16) are nonnegative functions. Moreover, in the blow-up argument we estimate from below the hypergeometric functions by positive constants.…”

“…The non-exitence of global in time solutions to (6) (which corresponds to (5) in the case µ 1 = µ 2 = 0 and ν 2 1 = ν 2 2 = 0) has been studied in [8,60], while the existence part has been proved in the three dimensional and radial case in [24]. Recently, in [15,Section 8] the upper bound for the lifespan has been derived. Summarizing the main results of these works we can sea that max{Λ(n, p, q), Λ(n, q, p)} = 0,…”

Nonexistence of Global Solutions for a Weakly Coupled System of Semilinear Damped Wave Equations of Derivative Type in the Scattering Case

“…the cases G 1 (v, ∂ t v) = |v| p , G 2 (u, ∂ t u) = |u| q and G 1 (v, ∂ t v) = |∂ t v| p , G 2 (u, ∂ t u) = |∂ t u| q have been studied in [5,3,4,2,23,22,8,24] and in [6,52,21,16], respectively. While in the case of power nonlinearities (that is, for G 1 (v, ∂ t v) = |v| p , G 2 (u, ∂ t u) = |u| q ) the critical curve is given by…”

“…even though the global existence part has been studied so far only in the three dimensional and radial symmetric case. Recently, the case with mixed nonlinear terms G 1 (v, ∂ t v) = |v| q , G 2 (u, ∂ t u) = |∂ t u| p has been investigated for (4) in [13,16]. In this paper we shall prove that the for same range of exponents p, q > 1 as in [16] a blow-up result can be proved in the subcritiacal case even when we add as lower order terms in the linear part damping terms with time-dependent and scattering producing coefficients (see [49,50,51] for this classification of a damping term with time-dependent coefficient for wave models).…”