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

Rongrui Lin; Yunlong Gao; Lianbing She

doi:10.1515/math-2024-0124

Article Open Access

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

Rongrui Lin , Yunlong Gao and Lianbing She

Published/Copyright: February 6, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

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 ) ,

where τ 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

MSC 2010: 35L05; 35B44; 93D15

1 Introduction

In this article, we are concerned with the following delayed nonlinear Euler-Bernoulli problem:

(1.1) 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 ) , ( x , t ) ∈ Ω × ( 0 , + ∞ ) , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ Ω , u t ( x , t − τ ) = f 0 ( x , t − τ ) , ( x , t ) ∈ Ω × ( 0 , τ ) , u ( x , t ) = 0 , Δ u ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , + ∞ ) ,

where Ω is a bounded domain in R n with a smooth boundary ∂ Ω , μ 1 ≥ ∣ μ 2 ∣ > 0 , m ≥ 2 , g 1 , g 2 , and f ( u ) are functions satisfying some conditions to be specified later, and

g i ∗ v ( t ) = ∫ 0 t g i ( t − s ) v ( s ) d s , i = 1 , 2 .

The Euler-Bernoulli type equation is formulated as follows:

(1.2) u t t + Δ 2 u = g ( x , t , u , u t ) .

In addition, it finds extensive a applications in numerous branches of physics, including nuclear physics, optics, geophysics, and ocean acoustics [1,2]. Therefore, an increasing number of references have investigated the global existence, energy decay, and blow-up phenomena of solutions for equation (1.2) [1–14]. When g 1 ( ⋅ ) = g 2 ( ⋅ ) = 0 , μ 1 = 0 , Messaoudi [4] considered a more simple equation

(1.3) u t t + Δ 2 u + a u t ∣ u t ∣ m − 2 = b u ∣ u ∣ p − 2 , x ∈ Ω , t > 0 .

The authors demonstrated the existence of a local weak solution that exhibits finite-time blowing-up behavior when p > m and the energy is negative. The authors additionally found that the solution remains global if m ≥ p . Subsequently, Chen and Zhou [5] demonstrated that the solution experiences finite-time blowing-up with a positive initial energy. When equation (1.3) contains a viscoelastic term ( g 1 ( ⋅ ) ≠ 0 ), Li and Gao [6] established that the solution with upper-bounded initial energy experiences finite-time blowing-up. Moreover, Liu et al. [7] demonstrated that the solution blows up in finite time under strong damping and with initial energy E ( 0 ) = R for any given R ≥ 0 . In [8], Ye examined the following initial boundary value problem concerning the higher-order nonlinear viscoelastic wave equation

u t t + ( − Δ ) m u − ∫ 0 t g ( t − s ) ( − Δ ) m u ( s ) d s = ∣ u ∣ p − 2 u , ( x , t ) ∈ Ω × R + ,

where m ≥ 1 denotes a natural number, and p > 2 represents a real number. By using the Galerkin method, Ye demonstrated the existence of global weak solutions. Simultaneously, it is established that the solution blows up in finite time under both positive and nonpositive initial energy, and lifetime estimates for the solutions are provided. In the absence of memory term for (1.1), Benaissa and Messaoudi [9] consider the wave equation in a bounded domain with a delay term in the nonlinear internal feedback

(1.4) u t t − Δ u + μ 1 σ ( t ) g 1 ( u t ) + μ 2 σ ( t ) g 2 ( u t ( x , t − τ ( t ) ) ) = 0 , ( x , t ) ∈ Ω × R + .

The authors demonstrated the global existence of solutions by employing the energy method in conjunction with the Faedo-Galerkin procedure, and investigated their asymptotic behavior utilizing a perturbed energy method. When the nonlinearity and delay of equation (1.4) are variable-exponent, Kafini and Messaoudi [10] established a global nonexistence result and exponential decay. Numerous researchers have examined both the delay term and the memory term for references [11–13].

In the absence of a time-delayed nonlinear term and with μ 2 = 0 , Mellah and Hakem [14] investigated the global existence and uniqueness of a solution to an initial boundary value problem for the Euler-Bernoulli viscoelastic equation:

u t t + Δ 2 u − g 1 ∗ Δ 2 u + g 2 ∗ Δ u + u t = 0 , ( x , t ) ∈ Ω × R + .

The authors also demonstrated an exponential decay. In [3], Feng et al. examined the extendable viscoelastic plate equation with a nonlinear time-varying delay feedback and a nonlinear source term

u t t + Δ 2 u − M ( ∥ ∇ u ∥ 2 ) Δ u − ∫ 0 t h ( t − s ) Δ 2 u ( s ) d s + μ 1 g 1 ( u t ( t ) ) + μ 2 g 2 ( u t ( t − τ ( t ) ) ) + f ( u ( t ) ) = 0 , ( x , t ) ∈ Ω × ( 0 , + ∞ ) .

Under suitable assumptions regarding the relaxation function, nonlinear internal delay feedback, and source term, the authors demonstrated the global existence of solutions and the general decay of energy by employing the multiplier method.

For [14], global existence and energy decay are investigated in the absence of time-delay and nonlinearity terms. In [3], the author demonstrated energy decay with g 2 ( ⋅ ) = 0 . In this article, we expand upon the results from [13] to [3]. When g 1 ( ⋅ ) and g 2 ( ⋅ ) satisfy the condition ∫ 0 + ∞ g 1 ( s ) d s + ∫ 0 + ∞ g 2 ( s ) d s ≤ ( p − p a − 1 ) 2 − 1 ( p − p a − 1 ) 2 for every 0 < a < p − 2 p , we obtain the blow-up of the solution. Our work is structured as follows. In Section 2, we present some lemmas and a local existence theorem. In Section 3, we prove several lemmas related to blow-up. Moreover, we demonstrate the existence of finite-time blow-up when E ( 0 ) < E 1 and E ( 0 ) < 0 .

2 Preliminaries

In this section, we will provide the necessary materials for the proof of our main results. We employ the standard Lebesgue space L p ( Ω ) and Sobolev space H 0 2 ( Ω ) along with their conventional products and norms ∥ ⋅ ∥ p ≔ ∥ ⋅ ∥ L p ( Ω ) . In particular, ∥ ⋅ ∥ 2 is shorthanded for ∥ ⋅ ∥ when p = 2 . The constant λ 1 is the embedding constant λ 1 2 ∥ u ∥ 2 ≤ λ 1 ‖ ▿ u ‖ 2 ≤ ∥ △ u ∥ 2 for every u ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) .

Now, we introduce a new variable, as in [10],

z ( x , ρ , t ) = u t ( x , t − τ ρ ) , ( x , ρ , t ) ∈ Ω × ( 0 , 1 ) × ( 0 , + ∞ ) .

Thus, we have

τ z t ( x , ρ , t ) + z ρ ( x , ρ , t ) = 0 , ( x , ρ , t ) ∈ Ω × ( 0 , 1 ) × ( 0 , + ∞ ) .

Then, problem (1.1) assumes the following form:

(2.1) 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 z ( x , 1 , t ) ∣ z ( x , 1 , t ) ∣ m − 2 = f ( u ) , ( x , t ) ∈ Ω × ( 0 , + ∞ ) , τ z t ( x , ρ , t ) + z ρ ( x , ρ , t ) = 0 , ( x , ρ , t ) ∈ Ω × ( 0 , 1 ) × ( 0 , + ∞ ) , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ Ω , z ( x , ρ , 0 ) = f 0 ( x , − ρ τ ) , ( x , ρ ) ∈ Ω × ( 0 , 1 ) , u ( x , t ) = 0 , Δ u ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , + ∞ ) .

First, to state and prove our result, we require several assumptions.

( H 0 ) g 1 : R + → R + is a bounded function, which satisfies

g 1 ( t ) ∈ C 2 ( R + ) ∩ L 1 ( R + ) ,

and there exist positive constants α 1 , α 2 , and α 3 such that

− α 1 g 1 ( t ) ≤ g ′ 1 ( t ) ≤ − α 2 g 1 ( t ) , ∀ t ≥ 0 , 0 ≤ g ′ ′ 1 ( t ) ≤ α 3 g 1 ( t ) , ∀ t ≥ 0 .

( H 1 ) g 2 : R + → R + is a bounded function, which satisfies

g 2 ( t ) ∈ C 1 ( R + ) ∩ L 1 ( R + ) ,

and there exist positive constants β 1 , β 2 , and β 3 such that

− β 1 g 2 ( t ) ≤ g ′ 2 ( t ) ≤ − β 2 g 2 ( t ) , ∀ t ≥ 0 , 1 − ∫ 0 + ∞ g 1 ( s ) d s − 1 λ 1 ∫ 0 + ∞ g 2 ( s ) d s = l > 0 .

( H 2 ) The source term f ( u ) is a nonlinear function that f ( 0 ) = 0 , and

k 0 ∣ u ∣ p ≤ p F ( u ) ≤ f ( u ) u ≤ k 1 ∣ u ∣ p ,

where k 0 , k 1 are positive constants, and

F ( z ) = ∫ 0 z f ( s ) d s , 2 < p < + ∞ if n ≤ 4 and 2 < p ≤ 2 ( n − 2 ) n − 4 if n ≥ 5 .

( H 3 ) Under the assumptions of g 1 , g 2 , and ( H 2 ) , concurrently, we also assume the relaxation functions satisfy:

∫ 0 + ∞ g 1 ( s ) d s + 1 λ 1 ∫ 0 + ∞ g 2 ( s ) d s ≤ ( p − p a − 1 ) 2 − 1 ( p − p a − 1 ) 2 ,

where 0 < a < p − 2 p is a fixed number.

Next, we present several lemmas and a local existence theorem throughout the text.

Lemma 2.1

[15] (Sobolev-Poincaré inequality): Let s be a number with

2 ≤ s < + ∞ ( n ≤ 4 ) or 2 ≤ s ≤ 2 n n − 4 .

Then there exists a positive constant B depending Ω and s such that

∥ u ∥ s ≤ B ∥ Δ u ∥ , ∀ u ∈ H 0 2 ( Ω ) .

Lemma 2.2

Assume that psatisfies ( H 2 ) . Then there exists a positive constant C such that

∥ u ( t ) ∥ p s ≤ C ( ∥ Δ u ( t ) ∥ 2 + ∥ u ( t ) ∥ p p )

for all t ∈ [ 0 , T ) , 2 ≤ s ≤ p .

Proof

By employing continuous embedding inequalities and Sobolev-Poincaré inequality, we can establish the conclusion parallel to Lemma 2.5 in [6].□

The localized existence results for problem (2.1) are as follows:

Theorem 2.3

(Local existence) Assume the assumptions ( H 0 ) – ( H 2 ) hold. Let

(2.2) ( u 0 , u 1 , f 0 ) ∈ ( H 2 ( Ω ) ∩ H 0 1 ( Ω ) ) × L 2 ( Ω ) × L 2 ( Ω × ( 0 , 1 ) )

satisfy the compatibility condition f 0 ( ⋅ , 0 ) = u t ( ⋅ , 0 ) = u 1 . Then problem (2.1) possesses a unique weak solution satisfying

u ∈ C ( [ 0 , T ] ; H 2 ( Ω ) ∩ H 0 1 ( Ω ) ) , u t ∈ C ( [ 0 , T ] ; L 2 ( Ω ) ) ∩ L m ( Ω × ( 0 , T ) ) , z ∈ C ( [ 0 , T ) ; L m ( Ω × ( 0 , 1 ) ) ) .

Remark 2.4

Although the sign of the nonlinear source term in [3] is opposite to that of our nonlinear source term, local well-posedness can be further established by combining the methods in [3, 14] with the contraction mapping theorem. A similar approach can be referred to in reference [16].

3 Blow up result

To state and prove our main results, we introduce the following notation and the associated energy functional related to (2.1).

(3.1) B 1 ≔ k 1 p B l , Φ ( α ( t ) ) ≔ 1 2 α 2 ( t ) − B 1 p p α p ( t ) ,

where

α ( t ) = l ∥ Δ u ∥ 2 + ( g 1 ∘ Δ u ) ( t ) + ( g 2 ∘ ∇ u ) ( t ) + 2 ξ m ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x 1 2 .

We can define energy function of (2.1):

(3.2) E ( t ) ≔ 1 2 ∥ u t ∥ 2 + 1 2 1 − ∫ 0 t g 1 ( s ) d s ∥ Δ u ∥ 2 − 1 2 ∫ 0 t g 2 ( s ) d s ∥ ∇ u ∥ 2 + 1 2 [ ( g 1 ∘ Δ u ) ( t ) + ( g 2 ∘ ∇ u ) ( t ) ] + ξ m ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x − ∫ Ω F ( u ) d x ,

where

(3.3) τ ∣ μ 2 ∣ ( m − 1 ) < ξ < τ ( μ 1 m − ∣ μ 2 ∣ ) ,

( g 1 ∘ Δ u ) ( t ) = ∫ 0 t g 1 ( t − s ) ∥ Δ u ( t ) − Δ u ( s ) ∥ 2 d s , ( g 2 ∘ ∇ u ) ( t ) = ∫ 0 t g 2 ( t − s ) ∥ ∇ u ( t ) − ∇ u ( s ) ∥ 2 d s .

Lemma 3.1

Let ( u , z ) be a weak solution of (2.1), and assume ( H 0 ) – ( H 1 ) , (3.3) hold. Then, for some C 0 > 0 ,

d dt E ( t ) ≤ − C 0 ( ∥ u t ( t ) ∥ m m + ∥ z ( ⋅ , 1 , t ) ∥ m m ) ≤ 0 ,

where

0 < C 0 ≤ min μ 1 − ∣ μ 2 ∣ m − ξ τ m , ξ τ m − ∣ μ 2 ∣ ( m − 1 ) m .

Proof

After multiplying the first equation within (2.1) by factor u t and performing integration across region Ω , and subsequently multiplying the second equation from (2.1) by factor ξ τ z ∣ z ∣ m − 2 with integration carried out over region Ω × ( 0 , 1 ) , we sum these integrated outcomes to derive the following result.

(3.4) d dt 1 2 ∥ u t ∥ 2 + 1 2 ∥ Δ u ∥ 2 + ξ m ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x − ∫ Ω F ( u ) d x − ∫ 0 t g 1 ( t − s ) ∫ Ω Δ u ( s ) Δ u t ( t ) d x d s − ∫ 0 t g 2 ( t − s ) ∫ Ω ∇ u ( s ) ∇ u t ( t ) d x d s = − μ 1 ∥ u t ( t ) ∥ m m − μ 2 ∫ Ω u t ( x , t ) z ( x , 1 , t ) ∣ z ( x , 1 , t ) ∣ m − 2 d x − ξ τ ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m − 2 z z ρ ( x , ρ , t ) d ρ d x .

For the last two terms on the left side of (3.4), a direct calculation reveals that

(3.5) − ∫ 0 t g 1 ( t − s ) ∫ Ω Δ u ( s ) Δ u t ( t ) d x d s = 1 2 d dt ( g 1 ∘ Δ u ) ( t ) − ∫ 0 t g 1 ( s ) d s ∥ Δ u ∥ 2 + 1 2 g 1 ( t ) ∥ Δ u ∥ 2 − 1 2 ( g 1 ′ ∘ Δ u ) ( t ) ,

(3.6) − ∫ 0 t g 2 ( t − s ) ∫ Ω ∇ u ( s ) ∇ u t ( t ) d x d s = 1 2 d dt ( g 2 ∘ ∇ u ) ( t ) − ∫ 0 t g 2 ( s ) d s ∥ ∇ u ∥ 2 + 1 2 g 2 ( t ) ∥ ∇ u ∥ 2 − 1 2 ( g 2 ′ ∘ ∇ u ) ( t ) .

By Young’s inequality for the second term on the right side of (3.4), we can obtain

(3.7) − μ 2 ∫ Ω u t ( x , t ) z ( x , 1 , t ) ∣ z ( x , 1 , t ) ∣ m − 2 d x ≤ ∣ μ 2 ∣ ∫ Ω u t ( x , t ) ∣ z ( x , 1 , t ) ∣ m − 1 d x ≤ ∣ μ 2 ∣ m ∥ u t ( t ) ∥ m m + ∣ μ 2 ∣ ( m − 1 ) m ∥ z ( ⋅ , 1 , t ) ∥ m m .

For the last term on the right side of (3.4), a direct computation shows that

(3.8) − ξ τ ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m − 2 z z ρ ( x , ρ , t ) d ρ d x = − ξ τ m ∫ Ω ∫ 0 1 ∂ ∂ ρ ∣ z ( x , ρ , t ) ∣ m d ρ d x = − ξ τ m ∫ Ω ( ∣ z ( x , 1 , t ) ∣ m − ∣ z ( x , 0 , t ) ∣ m ) d x = ξ τ m ∥ u t ( t ) ∥ m m − ξ τ m ∥ z ( ⋅ , 1 , t ) ∥ m m .

Inserting (3.5)–(3.8) into (3.4), we obtain

d dt E ( t ) ≤ − 1 2 g 1 ( t ) ∥ Δ u ∥ 2 − 1 2 g 2 ( t ) ∥ ∇ u ∥ + 1 2 ( g 1 ′ ∘ Δ u ) ( t ) + 1 2 ( g 2 ′ ∘ ∇ u ) ( t ) − μ 1 − ∣ μ 2 ∣ m − ξ τ m ∥ u t ( t ) ∥ m m − ξ τ m − ∣ μ 2 ∣ ( m − 1 ) m ∥ z ( ⋅ , 1 , t ) ∥ m m .

According to ( H 0 ) – ( H 1 ) and (3.3), then

d dt E ( t ) ≤ − μ 1 − ∣ μ 2 ∣ m − ξ τ m ∥ u t ( t ) ∥ m m − ξ τ m − ∣ μ 2 ∣ ( m − 1 ) m ∥ z ( ⋅ , 1 , t ) ∥ m m < 0 .

Therefore, we can obtain the desired result.□

Lemma 3.2

If the assumptions ( H 0 ) – ( H 2 ) hold, then

E ( t ) ≥ 1 2 α 2 ( t ) − B 1 p p α p ( t )

and Φ ( α ) has a maximum value Φ ( α 1 ) = 1 2 − 1 p α 1 2 ≔ E 1 when α ( t ) = B 1 − p p − 2 ≔ α 1 , where α ( t ) is given in (3.1).

Proof

By ( H 0 ) – ( H 2 ) , (3.1) and applying Sobolev-Poincaré inequality, we have

E ( t ) ≥ 1 2 1 − ∫ 0 t g 1 ( s ) d s ∥ Δ u ∥ 2 − 1 2 ∫ 0 t g 2 ( s ) d s ∥ ∇ u ∥ 2 + 1 2 [ ( g 1 ∘ Δ u ) ( t ) + ( g 2 ∘ ∇ u ) ( t ) ] + ξ m ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x − ∫ Ω F ( u ) d x ≥ 1 2 1 − ∫ 0 t g 1 ( s ) d s − 1 λ 1 ∫ 0 t g 2 ( s ) d s ∥ Δ u ∥ 2 + 1 2 [ ( g 1 ∘ Δ u ) ( t ) + ( g 2 ∘ ∇ u ) ( t ) ] + ξ m ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x − k 1 p ∥ u ∥ p p ≥ 1 2 l ∥ Δ u ∥ 2 + 1 2 [ ( g 1 ∘ Δ u ) ( t ) + ( g 2 ∘ ∇ u ) ( t ) ] + ξ m ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x − k 1 B p p ∥ Δ u ∥ p ≥ 1 2 α 2 ( t ) − B 1 p p α p ( t ) . = Φ ( α ( t ) ) .

Furthermore, according to the derivative of Φ ( α ( t ) ) , we have easy access to Φ ( α ) has a maximum value E 1 at α ( t ) = B 1 − p p − 2 .□

Lemma 3.3

Assume that ( H 0 ) – ( H 2 ) and (3.3) hold. Let ( u , z ) be a solution of (2.2) with initial data satisfying

E ( 0 ) < E 1 , α ( 0 ) > α 1 .

Then there exists a constant α 2 > α 1 such that

(3.9) α ( t ) = l ∥ Δ u ∥ 2 + ( g 1 ∘ Δ u ) ( t ) + ( g 2 ∘ ∇ u ) ( t ) + 2 ξ m ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x 1 2 ≥ α 2 ,

and

(3.10) ∥ u ∥ p ≥ B 1 k 1 p α 2 .

Proof

From the definition of Φ ( α ) and ( H 2 ) , we can easily obtain

Φ ′ ( α ) = α ( 1 − B 1 p α p − 2 ) > 0 , if α ∈ ( 0 , α 1 ) , < 0 , if α ∈ ( α 1 , + ∞ ) ,

which implies that

Φ ( α ) is strictly increasing in ( 0 , α 1 ) , Φ ( α ) is strictly decreasing in ( α 1 , + ∞ ) , Φ ( α ) → − ∞ as α → + ∞ .

Similar to the proof of Lemma 2.4 in [6], we easily obtain (3.9) holds.

Next, we will prove (3.10) holds. By (3.2) and ( H 2 ) , we have

k 1 p ∥ u ∥ p p ≥ ∫ Ω F ( u ) d x ≥ 1 2 1 − ∫ 0 t g 1 ( s ) d s ∥ Δ u ∥ 2 − 1 2 ∫ 0 t g 2 ( s ) d s ∥ ∇ u ∥ 2 + 1 2 [ ( g 1 ∘ Δ u ) ( t ) + ( g 2 ∘ ∇ u ) ( t ) ] + ξ m ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x − E ( t ) ≥ 1 2 l ∥ Δ u ∥ 2 + ( g 1 ∘ Δ u ) ( t ) + ( g 2 ∘ ∇ u ) ( t ) + 2 ξ m ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x − E ( 0 ) ≥ 1 2 α 2 2 − Φ ( α 2 ) = B 1 p α 2 p p .

Therefore, (3.10) holds.□

Theorem 3.4

Assume that hypothesis conditions ( H 0 ) – ( H 3 ) , (2.2), and (3.3) hold. Then the local solution of problem (2.1) with p > m and with initial conditions satisfying

E ( 0 ) < E 1 , α ( 0 ) > α 1 ;
E ( 0 ) < 0 .

Then any solution of (2.1) blows up in finite time T * .

Proof

We will prove the theorem in two parts based on conditions (i) and (ii).

Part (i): Setting H ( t ) = E 1 − E ( t ) , by Lemma 3.1 and (3.2), we obtain

H ′ ( t ) = − E ′ ( t ) ≥ C 0 ( ∥ u t ( t ) ∥ m m + ∥ z ( ⋅ , 1 , t ) ∥ m m ) .

By using ( H 2 ) , (3.2), and Lemma 3.3, then

H ( t ) = E 1 − 1 2 ∥ u t ∥ 2 − 1 2 1 − ∫ 0 t g 1 ( s ) d s ∥ Δ u ∥ 2 + 1 2 ∫ 0 t g 2 ( s ) d s ∥ ∇ u ∥ 2 − 1 2 [ ( g 1 ∘ Δ u ) ( t ) + ( g 2 ∘ ∇ u ) ( t ) ] − ξ m ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x + ∫ Ω F ( u ) d x ≤ E 1 − 1 2 1 − ∫ 0 t g 1 ( s ) d s − 1 λ 1 ∫ 0 t g 2 ( s ) d s ∥ Δ u ∥ 2 − 1 2 [ ( g 1 ∘ Δ u ) ( t ) + ( g 2 ∘ ∇ u ) ( t ) ] − ξ m ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x + k 1 p ∥ u ∥ p p ≤ E 1 − 1 2 α 2 ( t ) + k 1 p ∥ u ∥ p p ≤ E 1 − 1 2 α 1 2 + k 1 p ∥ u ∥ p p = − 1 p α 1 2 + k 1 p ∥ u ∥ p p .

So, for t ∈ [ 0 , T ) we can obtain

(3.11) 0 < E 1 − E ( 0 ) = H ( 0 ) ≤ H ( t ) ≤ − 1 p α 1 2 + k 1 p ∥ u ∥ p p ≤ k 1 p ∥ u ∥ p p .

Therefore, we define

L ( t ) = H 1 − θ ( t ) + ε M ( t ) + K E 1 t , ∀ t ∈ [ 0 , T ) ,

where ε , K > 0 are positive constants to be specified later, and

M ( t ) = ∫ Ω u u t d x , 0 < θ ≤ min p − 2 2 p , p − m p ( m − 1 ) .

Then,

(3.12) L ′ ( t ) = ( 1 − θ ) H − θ ( t ) H ′ ( t ) + ε M ′ ( t ) + K E 1 , ∀ t ∈ [ 0 , T ) .

Next, we estimate the function M ′ ( t ) . By (2.1) and taking the derivative of M ( t ) with respect to time t , we obtain

(3.13) M ′ ( t ) = ∥ u t ∥ 2 + ( u , u t t ) = ∥ u t ∥ 2 − ∥ Δ u ∥ 2 + ∫ 0 t g 1 ( t − s ) ∫ Ω Δ u ( t ) Δ u ( s ) d x d s + ∫ 0 t g 2 ( t − s ) ∫ Ω ∇ u ( t ) ∇ u ( s ) d x d s − μ 1 ∫ Ω u u t ∣ u t ∣ m − 2 d x − μ 2 ∫ Ω u z ( x , 1 , t ) ∣ z ( x , 1 , t ) ∣ m − 2 d x + ∫ Ω u f ( u ) d x .

Applying Young’s inequality, for every η , δ > 0 , we have

(3.14) ∫ 0 t g 1 ( t − s ) ∫ Ω Δ u ( t ) Δ u ( s ) d x d s = ∫ 0 t g 1 ( t − s ) ∫ Ω Δ u ( t ) ( Δ u ( s ) − Δ u ( t ) ) d x d s + ∫ 0 t g 1 ( t − s ) d s ∥ Δ u ∥ 2 ≥ − η ( g 1 ∘ Δ u ) ( t ) + 1 − 1 4 η ∫ 0 t g 1 ( t − s ) d s ∥ Δ u ∥ 2 ,

(3.15) ∫ 0 t g 2 ( t − s ) ∫ Ω ∇ u ( t ) ∇ u ( s ) d x d s = ∫ 0 t g 2 ( t − s ) ∫ Ω ∇ u ( t ) ( ∇ u ( s ) − ∇ u ( t ) ) d x d s + ∫ 0 t g 2 ( t − s ) d s ∥ ∇ u ∥ 2 ≥ − η ( g 2 ∘ ∇ u ) ( t ) + 1 − 1 4 η ∫ 0 t g 2 ( t − s ) d s ∥ ∇ u ∥ 2 ,

(3.16) − μ 1 ∫ Ω u u t ∣ u t ∣ m − 2 d x ≥ − μ 1 δ m m ∥ u ∥ m m − μ 1 ( m − 1 ) δ − m m − 1 m ∥ u t ( t ) ∥ m m ,

(3.17) − μ 2 ∫ Ω u z ( x , 1 , t ) ∣ z ( x , 1 , t ) ∣ m − 2 d x ≥ − ∣ μ 2 ∣ δ m m ∥ u ∥ m m − ∣ μ 2 ∣ ( m − 1 ) δ − m m − 1 m ∥ z ( ⋅ , 1 , t ) ∥ m m .

According to the definition of H ( t ) , adding ( 1 − a ) p ( H ( t ) + E ( t ) − E 1 ) ( for every 0 < a < p − 2 p ) to the right-hand side of inequality (3.13) and inserting (3.14)–(3.17) into (3.13), we arrive at

(3.18) M ′ ( t ) ≥ 1 + ( 1 − a ) p 2 ∥ u t ∥ 2 + 1 − 1 4 η − ( 1 − a ) p 2 ∫ 0 t g 1 ( s ) d s + ( 1 − a ) p 2 − 1 ∥ Δ u ∥ 2 + 1 − 1 4 η − ( 1 − a ) p 2 ∫ 0 t g 2 ( s ) d s ∥ ∇ u ∥ 2 + ( 1 − a ) p 2 − η [ ( g 1 ∘ Δ u ) ( t ) + ( g 2 ∘ ∇ u ) ( t ) ] + p ξ ( 1 − a ) m ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x + ( 1 − a ) p H ( t ) + ∫ Ω u f ( u ) d x − ( 1 − a ) p ∫ Ω F ( u ) d x − μ 1 δ m m + ∣ μ 2 ∣ δ m m ∥ u ∥ m m − μ 1 ( m − 1 ) δ − m m − 1 m ∥ u t ( t ) ∥ m m − ∣ μ 2 ∣ ( m − 1 ) δ − m m − 1 m ∥ z ( ⋅ , 1 , t ) ∥ m m − ( 1 − a ) p E 1 .

Since ( H 2 ) , then

∫ Ω u f ( u ) d x − ( 1 − a ) p ∫ Ω F ( u ) d x ≥ a k 0 ∥ u ∥ p p ,

and choosing δ − m m − 1 = N H − θ ( t ) , where N > 0 is a positive constant to be specified later, such that (3.18) becomes

(3.19) M ′ ( t ) ≥ 1 + ( 1 − a ) p 2 ∥ u t ∥ 2 + 1 − 1 4 η − ( 1 − a ) p 2 ∫ 0 t g 1 ( s ) d s + ( 1 − a ) p 2 − 1 ∥ Δ u ∥ 2 + 1 − 1 4 η − ( 1 − a ) p 2 ∫ 0 t g 2 ( s ) d s ∥ ∇ u ∥ 2 + ( 1 − a ) p 2 − η [ ( g 1 ∘ Δ u ) ( t ) + ( g 2 ∘ ∇ u ) ( t ) ] + p ξ ( 1 − a ) m ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x + ( 1 − a ) p H ( t ) + a k 0 ∥ u ∥ p p − ( μ 1 + ∣ μ 2 ∣ ) m N m − 1 H θ ( m − 1 ) ( t ) ∥ u ∥ m m − ( m − 1 ) N m H − θ ( t ) ( μ 1 ∥ u t ( t ) ∥ m m + ∣ μ 2 ∣ ∥ z ( ⋅ , 1 , t ) ∥ m m ) − ( 1 − a ) p E 1 .

By using (3.11), θ ≤ p − m p ( m − 1 ) and Lemma 2.2, we obtain

(3.20) − ( μ 1 + ∣ μ 2 ∣ ) m N m − 1 H θ ( m − 1 ) ( t ) ∥ u ∥ m m ≥ − ( μ 1 + ∣ μ 2 ∣ ) k 1 θ ( m − 1 ) m N m − 1 p θ ( m − 1 ) ∥ u ∥ p θ p ( m − 1 ) ∥ u ∥ m m ≥ − ( μ 1 + ∣ μ 2 ∣ ) ∣ Ω ∣ p − m p k 1 θ ( m − 1 ) m N m − 1 p θ ( m − 1 ) ∥ u ∥ p θ p ( m − 1 ) + m ≥ − C 1 m N m − 1 ( ∥ Δ u ∥ 2 + ∥ u ∥ p p ) ,

where C 1 ≔ ( μ 1 + ∣ μ 2 ∣ ) ∣ Ω ∣ p − m p k 1 θ ( m − 1 ) C p θ ( m − 1 ) .

Let η = ( 1 − a ) p 2 , we now use Sobolev-Poincaré inequality, and by inserting (3.20) into (3.19), we obtain

(3.21) M ′ ( t ) ≥ 1 + ( 1 − a ) p 2 ∥ u t ∥ 2 + ( 1 − a ) p 2 − 1 + 1 − 1 2 ( 1 − a ) p − ( 1 − a ) p 2 ∫ 0 + ∞ g 1 ( s ) d s + 1 λ 1 1 − 1 2 ( 1 − a ) p − ( 1 − a ) p 2 ∫ 0 + ∞ g 2 ( s ) d s ∥ Δ u ∥ 2 − C 1 m N m − 1 ∥ Δ u ∥ 2 + p ξ ( 1 − a ) m ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x + ( 1 − a ) p H ( t ) + a k 0 − C 1 m N m − 1 ∥ u ∥ p p − ( m − 1 ) N m H − θ ( t ) ( μ 1 ∥ u t ( t ) ∥ m m + ∣ μ 2 ∣ ∥ z ( ⋅ , 1 , t ) ∥ m m ) − ( 1 − a ) p E 1 .

Therefore, by substituting (3.21) into (3.12) and using Lemma 3.1, we have

(3.22) L ′ ( t ) ≥ ( 1 − θ ) C 0 − ( m − 1 ) μ 1 N ε m H − θ ( t ) ∥ u t ( t ) ∥ m m + ( 1 − θ ) C 0 − ( m − 1 ) ∣ μ 2 ∣ N ε m ∥ z ( ⋅ , 1 , t ) ∥ m m H − θ ( t ) + ε 1 + ( 1 − a ) p 2 ∥ u t ∥ 2 + ε ( 1 − a ) p 2 − 1 − C 1 m N m − 1 + 1 − 1 2 ( 1 − a ) p − ( 1 − a ) p 2 ∫ 0 + ∞ g 1 ( s ) d s + 1 λ 1 1 − 1 2 ( 1 − a ) p − ( 1 − a ) p 2 ∫ 0 + ∞ g 2 ( s ) d s ∥ Δ u ∥ 2 + ε p ξ ( 1 − a ) m ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x + ε ( 1 − a ) p H ( t ) + ε a k 0 − C 1 m N m − 1 ∥ u ∥ p p + ( K − ε ( 1 − a ) p ) E 1 .

At this point, we choose a small enough and using ( H 3 ) , such that

( 1 − a ) p 2 − 1 + 1 − 1 2 ( 1 − a ) p − ( 1 − a ) p 2 ∫ 0 + ∞ g 1 ( s ) d s + 1 λ 1 1 − 1 2 ( 1 − a ) p − ( 1 − a ) p 2 ∫ 0 + ∞ g 2 ( s ) d s > 0 .

Simplified, we obtain

∫ 0 + ∞ g 1 ( s ) d s + 1 λ 1 ∫ 0 + ∞ g 2 ( s ) d s < 1 − 1 ( p − p a − 1 ) 2 ,

and N so large that

a k 0 − C 1 m N m − 1 > 0 ,

( 1 − a ) p 2 − 1 − C 1 m N m − 1 + 1 − 1 2 ( 1 − a ) p − ( 1 − a ) p 2 ∫ 0 + ∞ g 1 ( s ) d s + 1 λ 1 1 − 1 2 ( 1 − a ) p − ( 1 − a ) p 2 ∫ 0 + ∞ g 2 ( s ) d s > 0 .

Once a and N are fixed, we pick ε small enough that

( 1 − θ ) C 0 − ( m − 1 ) μ 1 N ε m > 0 , ( 1 − θ ) C 0 − ( m − 1 ) ∣ μ 2 ∣ N ε m > 0 , K − ε ( 1 − a ) p > 0

and

L ( 0 ) = H 1 − θ ( 0 ) + ε ∫ Ω u 0 ( x ) u 1 ( x ) d x > 0 .

Thus, (3.22) takes the form

(3.23) L ′ ( t ) ≥ ε C 2 H ( t ) + ∥ u t ∥ 2 + ∥ Δ u ∥ 2 + ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x + ∥ u ∥ p p ,

for a constant C 2 > 0 . Consequently,

L ( t ) ≥ L ( 0 ) > 0 , ∀ t ∈ [ 0 , T ) .

Next, according to inequality ( a + b ) p ≤ 2 p − 1 ( a p + b p ) ( a > 0 , b > 0 , p ≥ 1 ) and Lemma 3.3, we obtain that

(3.24) L 1 1 − θ ( t ) ≤ C 2 H ( t ) + ∫ Ω u t u d x 1 1 − θ + ( K t E 1 ) 1 1 − θ ≤ C 2 H ( t ) + ∫ Ω u t u d x 1 1 − θ + ( K T E 1 ) 1 1 − θ ≤ C 2 H ( t ) + ∫ Ω u t u d x 1 1 − θ + k 1 ( K T E 1 ) 1 1 − θ ( B 1 α 2 ) p ∥ u ∥ p p .

Similar to literature [10] and using Lemma 2.1, we know

∫ Ω u t u d x 1 1 − θ ≤ C 3 ( ∥ u t ∥ 2 + ∥ Δ u ∥ 2 + ∥ u ∥ p p ) ,

which implies for a constant C 4 > 0 , then

(3.25) L 1 1 − θ ( t ) ≤ C 4 H ( t ) + ∥ u t ∥ 2 + ∥ Δ u ∥ 2 + ∫ Ω ∫ 0 1 ∣ z ( x , ρ , t ) ∣ m d ρ d x + ∥ u ∥ p p .

Combining (3.23) and (3.25), we deduce that for some C 5 > 0 ,

L ′ ( t ) ≥ C 5 L 1 1 − θ ( t ) .

Integration over ( 0 , t ) yields

(3.26) L ( t ) ≥ L − θ 1 − θ ( 0 ) − C 5 θ 1 − θ t − 1 − θ θ .

Therefore, by L ( 0 ) > 0 and (3.26), we have lim t → T * L ( t ) = + ∞ , where T * ≤ 1 − θ θ C 5 L − θ 1 − θ ( 0 ) .

Part (ii): Setting H 1 ( t ) = − E ( t ) , by Lemma 3.1 and (3.2), we obtain

H ′ ( t ) = − E ′ ( t ) ≥ C 0 ( ∥ u t ( t ) ∥ m m + ∥ z ( ⋅ , 1 , t ) ∥ m m ) .

Thus, using ( H 2 ) , such that

0 < − E ( 0 ) = H 1 ( 0 ) ≤ H 1 ( t ) ≤ k 1 p ∥ u ∥ p p .

We then define

L 1 ( t ) = H 1 1 − θ 1 ( t ) + ε M 1 ( t ) , ∀ t ∈ [ 0 , T ) ,

where ε are positive constants to be specified later, and

M 1 ( t ) = ∫ Ω u u t d x , 0 < θ 1 ≤ min p − 2 2 p , p − m p ( m − 1 ) .

Then,

L ′ ( t ) = ( 1 − θ 1 ) H 1 − θ 1 ( t ) H ′ ( t ) + ε M ′ ( t ) , ∀ t ∈ [ 0 , T ) .

Similar to (3.13)–(3.26), we also prove any solution of (2.1) blows up in finite time T * for E ( 0 ) < 0 . This completes the proof.□

Funding information: This work was supported by the Natural Fund Project of Guizhou Education Department Youth Science and Technology Talents Growth Project (KY [2019]143; KY [2019]139; KY [2022]330), the School level Foundation of Liupanshui Normal University (LPSSYKYJJ201801; LPSSKJTD201907), and the Guizhou Provincial Science and Technology Projects (QKH[2024]Y018).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. RL: conceptualization, methodology, mathematical modeling, theoretical analysis of partial differential equations, writing-review and editing, visualization. YG: conceptualization, methodology, numerical Analysis, writing-review and editing, visualization. LS: conceptualization, methodology, supervision, visualization.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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Received: 2023-11-30

Revised: 2024-12-02

Accepted: 2024-12-22

Published Online: 2025-02-06

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2024-0124

Keywords for this article

Euler-Bernoulli equation; blow-up; nonlinear time delay

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