Blow-up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities

Abstract

In this work we study the blow-up of solutions of a weakly coupled system of damped semilinear wave equations in the scattering case with power nonlinearities. We apply an iteration method to study both the subcritical case and the critical case. In the subcritical case our approach is based on lower bounds for the space averages of the components of local solutions. In the critical case we use the slicing method and a couple of auxiliary functions, recently introduced by Wakasa-Yordanov, to modify the definition of the functionals with the introduction of weight terms. In particular, we find as critical curve for the pair (p, q) of the exponents in the nonlinear terms the same one as for the weakly coupled system of semilinear wave equations with power nonlinearities.

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

“…In particular, in the critical case we employ the so-called slicing method in order to deal with logarithmic factors. For further details on the slicing method see [2], where this method was introduced for the first time or [51,52,57,41,42,43] where the slicing method is used in critical cases in order to manage factors of logarithmic type.…”

“…for any z ≥ R. Due to the special structure of (40) and (41), in this case is not necessary to applying the slicing procedure in order to restrict step by step the domain of integration. Thus, the first step will be to prove…”

“…In the massless case, if we take into account a weaker kind of damping term, namely, scattering producing damping terms (which means that the time-dependent coefficients for the damping terms are nonnegative and summable functions), then, the critical condition for the powers in the nonlinear terms is exactly the same as in the corresponding classical not-damped case (cf. [29,30,31,58,41,42,43]). Really recently, in [16] the blow-up dynamic for the semilinear wave equation with time-dependent and scattering producing damping and mass terms has been studied both in the subcritical and critical case.…”

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

“…has been considered in [34] for the case with power nonlinearities G 1 (v, ∂ t v) = |v| p , G 2 (u, ∂ t u) = |u| q and in [35] for the case with semilinear terms of derivative type…”

“…In this paper our approach is based on the following methods: in the subcritical case we employ two multipliers, that are introduced in [25], in order to apply a standard iteration argument based on lower bound estimates for the spatial integrals of the nonlinear terms and on a coupled system of ordinary integral inequalities; in the critical case, we modify the approach introduced by Wakasa-Yordanov in [47,48] and adapted to weakly coupled systems in [34] with the purpose to deal with the nonlinear term of derivative type. We underline that in the case with time-dependent coefficients for the damping terms in the scattering case, we may not apply the revised test function method which is introduced by Ikeda-Sobajima-Wakasa in [16] for semilinear wave models.…”