Decay characterization of solutions to semi-linear structurally damped σ-evolution equations with time-dependent damping

2024/04/10 by Anh, Cung The, An, Phan Duc, Duong, Pham Trieu
#35A01 #35G25 #35L71 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2404.06855

Abstract

In this paper, we study the Cauchy problem to the linear damped σ-evolution equation with time-dependent damping in the effective cases u_{t t}+(-Δ)^σu+b(t)(-Δ)^δu_t=0, and investigate the decay rates of the solution and its derivatives that are expressed in terms of the decay character of the initial data u₀(x)=u(0, x) and u₁(x)=u_t(0, x). We are interested also in the existence and decay rate of the global in time solution with small data for the corresponding semi-linear problem with the nonlinear term of power type ||D|^γu|^p. The blow-up results for solutions to the semi-linear problem in the case γ=0 are presented to show the sharpness of the exponent p.

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