DOI: 10.1515/math-2024-0124 ISSN: 2391-5455
Blow-up of solutions for Euler-Bernoulli equation with nonlinear time delay
Rongrui Lin, Yunlong Gao, Lianbing She Abstract
We study the Euler-Bernoulli equations with time delay:
u
t
t
+
Δ
2
u
−
g
1
∗
Δ
2
u
+
g
2
∗
Δ
u
+
μ
1
u
t
(
x
,
t
)
∣
u
t
(
x
,
t
)
∣
m
−
2
+
μ
2
u
t
(
x
,
t
−
τ
)
∣
u
t
(
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),
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.