Global existence for wave and beam equations with double damping and a new power nonlinearity

2024/05/27 by Said, Khaldi, Zahra, Arioui Fatima
#35A01 #35L05 #35L15 #35L71 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2405.17274

Abstract

We consider the Cauchy problem in ℝⁿ for wave and beam equations with frictional, viscoelastic damping, and a new power nonlinearity. In addition to the solution and its total energy, we define the following quantity: Q[u](t):=‖u_t(t,⋅)+(-Δ)^σu(t,⋅)‖_L²(ℝⁿ). Our aim is to show that the interaction between frictional and viscoelastic damping in a linear model leads to an exponential decay of Q[u](t) as t→ ∞. This decay motivates us to define a new power nonlinearity of the form N[u]:=|u_t+(-Δ)^σu|^p. Surprisingly, N[u] can be considered a small perturbation for any p>1, in the sense that, the decay estimates of the unique global solution, the total energy and Q[u](t) coincide with those for solutions to the corresponding linear Cauchy problem with vanishing right-hand side.

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