In this paper, we consider the higher-order nonlinear Kirchhoff-type equation with degenerate damping and source:

where m > 1 is an integer constant, p, q, k > 0, Ω is a bounded domain of Rⁿ, with a smooth boundary $\partial \Omega$ and a unit outer normal $\nu .j(·):R\to R$ is a given function to be specified later and we use $\partial j$ to denote its sub-differential for example(see [1, 4, 9]). Moreover, in [14], Messaodui and Houari consider the higher-order nonlinear Kirchhoff-type hyperbolic equation

where m ≥ 1, p, q, r ≥ 0, Ω is a bounded domain of Rⁿ, with a smooth boundary ∂Ω and a unit outer normal ν. They established a blow-up result for certain solutions with positive initial energy. In [4], Han and Wang investigated the global existence and blow-up of solutions for nonlinear viscoelastic wave equation with degenerate damping and source term

where Ω is a bounded domain of Rⁿ, with a smooth boundary ∂Ω in Rⁿ, k, p ≥ 0, and $g(\cdot ):R_{+}\to R_{+},j(\cdot ):R\to R$ are given functions to be specified. Here, we use ∂j to denote its sub-differential(see [1, 4, 9]). Furthermore, in [9], Kim et al. consider a stochastic quasilinear viscoelastic wave equation with degenerate damping and sources

where $\overline{D}$ is a bounded domain in $R^n(n\ge 1)$ with a smooth boundary $\partial D,k,p\ge 0$ and $g(\cdot ):R_{+}\to R_{+},j(\cdot ):R\to R$ is given functions to be specified. Here, we use ∂j to denote its sub-differential, W (x, t) is an infinite dimensional Winner process, and σ(x, t) is L²(D) valued progressively measurable. Authors proved the blow-up of solution for stochastic/no stochastic quasilinear viscoelastic wave equation with positive probability or explosive in energy sense [2, 3, 5, 6, 7, 8, 10, 11, 13, 15, 16, 17].

Motivated by the previous works, we studied the blow-up of solutions for higher-order nonlinear Kirchhoff-type equation with degenerate damping and source. To the best of our knowledge. there are no results of a higher-order nonlinear Kirchhoff-type equation with degenerate damping and source. The main result was proved in section 2.

For $j(\cdot ):R\to R,$ similarly to [1, 4, 9], we have the following assumptions :

(H1) Let $j(\cdot ):R\to R$ be continuous, convex real valued function, there exist constants $c>0,c_0>0,c_1>0,c_2\ge 0$ such that for all $s,v\in R,j(\cdot )$ satisfies

• (1) coercivity: $j(s)\ge c{\left|s\right|}^{r+1},$

• (2) strict monotonicity: $(\partial j(s)-\partial j(v))(s-v)\ge c_1{\left|s-v\right|}^{r+1},$

• (3) continuity: $\partial j(x)$ is single-valued and $\left|\partial j(s)\right|\le c_0{\left|s\right|}^r+c_2.$

(H2)$p\le max\left\{\frac{p*}{2},\frac{p*r+k}{r+1}\right\};p*=\frac{2n}{n-2}.$

(H3)$k,r\ge 0,p>0.$ In addition, $k\le \frac{n}{n-2},p+1<\frac{2n}{n-2},ifn\ge 3.$

Here $‖\cdot ‖={‖\cdot ‖}_2.$ Let B be the best constant of the embedding inequality ${‖u‖}_{p+2}\le B‖D^{m}u‖.$ We set

and

Lemma 2.1. ([14]) Let u(t) be solution of Eq.(1.1). Assume that$E(0)<E_1$and$‖D^m{u}_0‖>{\alpha }_1$. Then there exists a constant${\alpha }_2>{\alpha }_1$such that

Under the assumptions (H1)-(H3), we get the two theorems.

Theorem 2.1.Suppose that$p\le k+r-1$and

Then for any initial data$(u_0,u_1)\in H^{2m}(\Omega )\cap H_0^m(\Omega )\times H_0^m(\Omega ),$the solution of Eq.(1.1) is global.

Proof. By using Theorem 1.1 of [12] and Theorem 2.3 of [4], we derive to the solution.

Our main results is the Theorem 2.2. Here c₂ = 0.

Theorem 2.2.Suppose that$p>max\left\{k+r-1,2q\right\}$and

Then for any initial data$(u_{0,}{u}_1)\in H^{2m}(\Omega )\cap H_0^m(\Omega )\times H_0^m(\Omega ),$any solution of Eq.(1.1) with initial data satisfying

blows up in finite time T.

Proof. A multiplication of Eq.(1.1) by u_t and integration over Ω give

Let

In view of (2.6) and (H1), we easily see that

By using (2.2), (2.6) and (2.7), we have

and exploiting (2.1) and (2.3) we obtain

Thus using (2.8) and (2.9), we get

As in [11], we construct a Lyapunov's function

where

and ε being a positive constant to be determined later. By taking a derivative of L(t) and using equation of (1.1), we have

On exploiting (2.2) and (2.4), estimate (2.13) takes the form

where

since ${\alpha }_2>B^{-(p+2)/(p-2q)}$. Thanks to $p+1>k+r$, then the continuity of $j(\cdot )$ in (H1) and using Hölder's inequality, we infer that

From (2.7),(2.15) and Young's inequality it yields

where $K=c_0{c}_1{}^{-\frac{r}{r+1}}{\left|\Omega \right|}^{\frac{p-r-k+1}{(p+1)(r+1)}}$ and δ is a positive constant to be determined later. From (2.14) and (2.16), we obtain

Putting

in (2.17), then

Thus, we get

In view of (2.10), we deduce

Since ${‖u‖}_{p+2}\ge {((p+2)G(0))}^{\frac{1}{p+2}}>0$ and α was chosen such that $r+k-p-1+\alpha r(p+2)\le 0$, it follows from (2.19) that

where

Now, we choose $0<\epsilon <(1-\alpha ){(F{K}^{1+1/r})}^{-1}$ small enough such that

In the sequel, we may adjust ε again. From (2.21), it follows

where $\theta =\frac{r(p+2)}{p-r-k+1-\alpha r(p+2)},c:={\left[{(c_{*}/2)}^{1/r}{(p+2)}^{(r+k-p-1)/r(p+2)}{K}^{1+1/r}{(1-\alpha )}^{-1}\right]}^{\theta }.$ Here and in what follows we use c to denote a generic positive constant which is independent of ε and initial data. Therefore, we conclude that from (2.18),(2.20) and (2.21) that

for $t\in \left[0,T\right)$. Obviously, (2.23) indicates that L(t) is increasing on $\left[0,T\right)$ and

If ${\displaystyle {\int }_{\Omega }{u}_0{u}_{1}dx\ge 0,}$ then no further restriction on ε is needed. However if ${\displaystyle {\int }_{\Omega }{u}_0{u}_{1}dx<0,}$ we further require ε satisfies

In both cases, we have $L(t)\ge L(0)>0,t\in \left[0,T\right).$ Now we prove that L(t) satisfies the following differential inequality

where c is a positive constant and $\sigma =\theta \left(1-\frac{2}{1-2\alpha (p+2)}\right)\ge 0.$ To this end, we consider two cases.

Case 1.${\displaystyle {\int }_{\Omega }u{u}_{t}dx}\le 0$ for some $t\in \left[0,T\right).$ Then we have for such t

Thus (2.24) is valid for all $t\in \left[0,T\right)$ for which ${\displaystyle {\int }_{\Omega }u{u}_{t}dx}\le 0.$

Case 2.${\displaystyle {\int }_{\Omega }u{u}_{t}dx}\le 0$ for some $t\in \left[0,T\right)$. By (2.12), it easily follows $0<\alpha <\frac{1}{2}.$ Noting $0<\epsilon <1$, then by convexity we have

Using Höolder's inequality, we obtain the following estimate

which implies

Noting $1<\frac{1}{1-\alpha }<2$, then by Young's inequality we have

From (2.12), it easy to see that $\frac{2}{1-2\alpha }\le p+2$. Then it follows from (2.10) that

Thanks to $\frac{2}{(1-2\alpha )(p+2)}-1\le 0$ and (2.22), it yields

where

Then, from (2.26) and (2.27), we infer that

Combing (2.25) and (2.28), we get

Therefore (2.24) follows from (2.23)and (2.29).

From (2.24), we obtain

From (2.30), we deduce that ${\lim }_{t\to T^{-}}L(t)=+\infty ,$ where

We obtain from(2.8) and (2.29)$L^{1/(1-\alpha )}(t)\le C{‖u‖}_{p+2}^{p+2}$ for some constant $C>0$ and also ${‖u‖}_{p+2}\le B‖D^{m}u‖.$ Hence we have ${\lim }_{t\to T^{-}}{‖u‖}_{p+2}=+\infty$ and ${\lim }_{t\to T^{-}}‖D^{m}u‖=+\infty$ i.e., the solution blow up at finite time T in $L^{p+2}(\Omega )$ and $H^m(\Omega )$ norm.

Thus the proof of Theorem 2.2. is complete.