Lower and upper bounds for lifespan of solutions to viscoelastic hyperbolic equations with variable sources and damping terms - Lili Dai; Zhuo Zhang (Journal of Inequalities and Applications, 2019)
Abstract The aim of this paper is to study bounds for lifespan of solutions to the following equation: utt−Δu+∫0tg(t−τ)Δu(τ)dτ+|ut|m(x,t)−2ut=|u|p(x,t)−2u $$ u_{tt}-Δ u+ \int _{0}^{t}g(t-τ )Δ u(τ )\,dτ + \vert u_{t} \vert ^{m(x,t)-2}u _{t}= \vert u \vert ^{p(x,t)-2}u $$ under homogeneous Dirichlet boundary conditions. It is worth pointing out that it is not a trivial generalization for constant-exponent problems because we have to face some essential difficulties in studying such problems. The first difficulty is that the monotonicity of the energy functional fails. Another one is that there exists a gap between the norm and the modular to the generalized function space, which leads to the failure of the Poincaré inequality for modular form. To overcome such difficulties, the authors construct control function and apply new energy estimates to establish the quantitative relationship between the source ∫Ω|u|p(x,t)dx $\int _{\varOmega }|u|^{p(x,t)}\,dx$ and the initial energy, and then obtain the finite-time blow-up of solutions for a positive initial energy, especially, the authors only assume that pt(x,t) $p_{t}(x,t)$ is integrable rather than uniformly bounded. Such weak conditions are seldom seen for the variable exponent case. At last, an estimate of lower bound for lifespan is established by applying differential inequality argument and energy inequalities.