A scale of critical exponents for semilinear waves with time-dependent damping and mass terms. (February 2019)

• Record Type: Journal Article
• Authors: D'Abbicco, M.; Girardi, G.; Reissig, M.
• Is Part Of: Nonlinear analysis. Volume 179(2019)
• Page Start: 15
• Page End: 40
• Publication Date: 2019-02
• Subjects: 35L15 -- 35L71 -- 35B33; Critical exponent -- Wave equation -- Effective damping -- Damped Klein–Gordon equation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.08.006
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the following Cauchy problem for a wave equation with time-dependent damping term b ( t ) u t and mass term m ( t ) 2 u, and a power nonlinearity | u | p : u t t − Δ u + b ( t ) u t + m 2 ( t ) u = | u | p, t ≥ 0, x ∈ R n, u ( 0, x ) = f ( x ), u t ( 0, x ) = g ( x ) . We discuss how the interplay between an effective time-dependent damping term and a time-dependent mass term influences the decay rate of the solution to the corresponding linear Cauchy problem, in the case in which the mass is dominated by the damping term, i.e. m ( t ) = o ( b ( t ) ) as t → ∞ . Then we use the obtained estimates of solutions to linear Cauchy problems to prove the existence of global in-time energy solutions to the Cauchy problem with power nonlinearity | u | p at the right-hand side of the equation, in a supercritical range p > p ̄, assuming small data in the energy space ( f, g ) ∈ H 1 × L 2 . In particular, we find a threshold case for the behavior of m ( t ) with respect to b ( t ) . Below the threshold, the mass has no influence on the critical exponent, so that p ̄ = 1 + 4 ∕ n, as in the case with m = 0 . Above the threshold, p ̄ = 1, global (in time) small data energy solutions exist for any p > 1, as it happens for the classical damped Klein–Gordon equation ( b = m = 1 ). Along the threshold, varying a parameter β ∈ [ 0, ∞ ] which depends on the behavior of m ( t ) with respect to b ( t ), we find a scale of critical exponents, namely, p ̄ = 1 + 4 ∕ ( n + 4 β ) . This scale of critical exponents is consistent with the diffusion phenomenon, that is, it is the same scale of critical exponents of the Cauchy problem for the corresponding diffusive equation − Δ v + b ( t ) v t + m 2 ( t ) v = | v | p, t ≥ 0, x ∈ R n, v ( 0, x ) = φ ( x ) . Highlights: We study a transition case between damped wave and damped Klein–Gordon equation. We study the interplay between time dependent damping and mass in a wave equation. We find a scale of critical exponents for global small data solutions.