Blow-up of solutions for Euler-Bernoulli equation with nonlinear time delay

Abstract

We study the Euler-Bernoulli equations with time delay: utt+Δ2u−g1∗Δ2u+g2∗Δu+μ1ut(x,t)∣ut(x,t)∣m−2+μ2ut(x,t−τ)∣ut(x,t−τ)∣m−2=f(u),{u}_{tt}+{\Delta }^{2}u-{g}_{1}\ast {\Delta }^{2}u+{g}_{2}\ast \Delta u+{\mu }_{1}{u}_{t}\left(x,t){| {u}_{t}\left(x,t)| }^{m-2}+{\mu }_{2}{u}_{t}\left(x,t-\tau ){| {u}_{t}\left(x,t-\tau )| }^{m-2}=f\left(u), where τ\tau represents the time delay. We exhibit the blow-up behavior of solutions with both positive and nonpositive initial energy for the Euler-Bernoulli equations involving time delay.

Keywords

• euler-bernoulli equation
• blow-up
• nonlinear time delay
• 35l05
• 35b44
• 93d15