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

Abstract

In this paper we consider the blow-up of solutions to a weakly coupled system of semilinear damped wave equations in the scattering case with nonlinearities of mixed type, namely, in one equation a power nonlinearity and in the other a semilinear term of derivative type. The proof of the blow-up results is based on an iteration argument. As expected, due to the assumptions on the coefficients of the damping terms, we find as critical curve in the pq plane for the pair of exponents (p, q) in the nonlinear terms the same one found by Hidano-Yokoyama and, recently, by Ikeda-Sobajima-Wakasa for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. In the critical and not-damped case we provide a different approach from the test function method applied by Ikeda-Sobajima-Wakasa to prove the blow-up of the solution on the critical curve, improving in some cases the upper bound estimate for the lifespan. More precisely, we combine an iteration argument with the so-called slicing method to show the blow-up dynamic of a weighted version of the functionals used in the subcritical case.

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

“…where C Φ,R is a suitable positive constant. Combing the estimate (25), (26) and (22), we find (19). This concludes the proof.…”

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.…”

“…We prove now the inductive step. Noticing that R + y ≤ 2y for y ≥ R, if we plug (42) in (41), then, it follows…”

“…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.…”

“…Let us point out that, according to the results we quoted above, the conjecture that the critical line for (6) is given by (7) has be shown to be true only partially, as the global existence of small data solutions has been proved only in the 3-dimensional and radial symmetric case. In the massless case (ν 2 1 = ν 2 2 = 0) and scattering producing case, that is, if we consider time-dependent, nonnegative and summable coefficients b 1 (t), b 2 (t) instead of µ 1 (1 + t) −1 , µ 2 (1 + t) −1 , really recently in [42] a blow-up result has been proved in the same range for the exponents (p, q) as for the corresponding not damped case, namely, for (p, q) such that max{Λ(n, p, q), Λ(n, q, p)} ≥ 0 is satisfied.…”