We’d like to understand how you use our websites in order to improve them. Register your interest.

# Global existence of solutions and energy decay for a Kirchhoff-type equation with nonlinear dissipation

## Abstract

This paper deals with the initial boundary value problem for a class of nonlinear Kirchhoff-type equation with dissipative term

$u t t −φ ( ∥ ∇ u ∥ 2 2 ) Δu+a | u t | α − 2 u t =b | u | β − 2 u,x∈Ω,t>0$

in a bounded domain, where $a,b>0$ and $α,β>2$ are constants. We obtain the global existence of solutions by constructing a stable set in $H 0 1 (Ω)$ and show the energy decay estimate by applying a lemma of Komornik.

MSC:35B40, 35L70.

## 1 Introduction

In this paper, we investigate the existence and asymptotic stability of global solutions for the initial boundary value problem of the following Kirchhoff-type equation with nonlinear dissipative term in a bounded domain

$u t t −φ ( ∥ ∇ u ∥ 2 2 ) Δu+a | u t | α − 2 u t =b | u | β − 2 u,x∈Ω,t>0,$
(1.1)
$u(x,0)= u 0 (x), u t (x,0)= u 1 (x),x∈Ω,$
(1.2)
$u(x,t)=0,x∈∂Ω,t≥0,$
(1.3)

where Ω is a bounded domain in $R n$ with a smooth boundary Ω, $a,b>0$ and $α,β>2$ are constants, $φ(s)$ is a $C 1$-class function on $[0,+∞)$ satisfying

$φ(s)≥ m 0 ,sφ(s)≥ ∫ 0 s φ(θ)dθ,∀s∈[0,+∞)$
(1.4)

with $m 0 ≥1$ is a constant.

If $Ω=[0,L]$ is an interval of the real line, equation (1.1) describes a small amplitude vibration of an elastic string with fixed endpoints. The original equation is

$ρh u t t +δ u t +f= ( γ 0 + E h 2 L ∫ 0 L | u x | 2 d s ) u x x ,$

where L is the rest length, E is the Young modulus, ρ is the mass density, h is the cross-section area, $γ 0$ is the initial axial tension, δ is the resistance modulus and f is a nonlinear perturbation effect.

When $a=b=0$, $φ(s)= s r$, $r≥1$ and $u 0 ≠0$ (the mildly degenerate case), the local existence of solutions in Sobolev space was investigated by many author . Concerning a global existence of solutions for mildly degenerate Kirchhoff equations, it is natural to add a dissipative term $u t$ or $Δ u t$.

For $a=1$, $b=0$, $α=2$, $φ(s)= s r$, $r≥1$, the problem (1.1)-(1.3) was treated by Nishihara and Yamada . They proved the existence and uniqueness of a global solution $u(t)$ for small data $( u 0 , u 1 )∈( H 0 1 (Ω)∩ H 2 (Ω))× H 0 1 (Ω)$ with $u 0 ≠0$ and the polynomial decay of the solution. Aassila and Benaissa  extended the global existence part of  to the case where $φ(s)≥0$ with $φ( ∥ ∇ u 0 ∥ 2 )≠0$ and the case of nonlinear dissipative term case ($a≠0$).

In the case $a=0$, for large β and $φ(s)≥r>0$, D’Ancona and Spagnolo  proved that if $u 0 , u 1 ∈ C 0 ∞ ( R n )$ are small, then problem (1.1)-(1.3) has a global solution. The nondegenerate case with $α=2$, $a>0$ and $b=0$ was considered by De Brito, Yamada and Nishihara , they proved that for small initial data $( u 0 , u 1 )∈( H 0 1 (Ω)∩ H 2 (Ω))× H 0 1 (Ω)$ there exists a unique global solution of (1.1)-(1.3) that decays exponentially as $t→+∞$.

When $φ(s)≥0$, Ghisi and Gobbino  proved the existence and uniqueness of a global solution $u(t)$ of the problem (1.1)-(1.3) for small initial data $( u 0 , u 1 )∈( H 0 1 (Ω)∩ H 2 (Ω))× H 0 1 (Ω)$ with $m( ∥ ∇ u 0 ∥ 2 )≠0$ and the asymptotic behavior $(u(t), u t (t), u t t (t))→( u ∞ ,0,0)$ in $( H 0 1 (Ω)∩ H 2 (Ω))× H 0 1 (Ω)× L 2 (Ω)$ as $t→+∞$, where either $u ∞ =0$ or $φ( ∥ ∇ u ∞ ∥ 2 )=0$.

The case $φ(s)≥r>0$ has been considered by Hosoya and Yamada  under the following condition:

$0≤β< 2 n − 4 ,n≥5;0≤β<+∞,n≤4.$

They proved that, if the initial datas are small enough, the problem (1.1)-(1.3) has a global solution which decays exponentially as $t→+∞$.

In this paper, we prove the global existence for the problem (1.1)-(1.3) by applying the potential well theory introduced by Sattinger  and Payne and Sattinger . Meanwhile, we obtain the asymptotic stability of global solutions by use of the lemma of Komornik .

We adopt the usual notation and convention. Let $H m (Ω)$ denote the Sobolev space with the norm $∥ u ∥ H m ( Ω ) = ( ∑ | α | ≤ m ∥ D α u ∥ L 2 ( Ω ) 2 ) 1 2$, $H 0 m (Ω)$ denotes the closure in $H m (Ω)$ of $C 0 ∞ (Ω)$. For simplicity of notation, hereafter we denote by $∥ ⋅ ∥ p$ the Lebesgue space $L p (Ω)$ norm, $∥⋅∥$ denotes $L 2 (Ω)$ norm and we write equivalent norm $∥∇⋅∥$ instead of $H 0 1 (Ω)$ norm $∥ ⋅ ∥ H 0 1 ( Ω )$. Moreover, M denotes various positive constants depending on the known constants and it may be difference in each appearance.

This paper is organized as follows: In the next section, we will give some preliminaries. Then in Section 3, we state the main results and give their proof.

## 2 Preliminaries

In order to state and prove our main results, we first define the following functionals:

$K(u)= m 0 ∥ ∇ u ∥ 2 −b ∥ u ∥ β β ,J(u)= m 0 2 ∥ ∇ u ∥ 2 − b β ∥ u ∥ β β ,$

for $u∈ H 0 1 (Ω)$. Then we define the stable set S by

$S= { u ∈ H 0 1 ( Ω ) , K ( u ) > 0 } ∪{0}.$

We denote the total energy functional associated with (1.1)-(1.3) by

$E(t)= 1 2 ∥ u t ∥ 2 + 1 2 ∫ 0 ∥ ∇ u ∥ 2 φ(s)ds− b β ∥ u ∥ β β$
(2.1)

for $u∈ H 0 1 (Ω)$, $t≥0$, and $E(0)= 1 2 ∥ u 1 ∥ 2 + 1 2 ∫ 0 ∥ ∇ u 0 ∥ 2 φ(s)ds− b β ∥ u 0 ∥ β β$ is the total energy of the initial data.

Lemma 2.1 Let q be a number with $2≤q<+∞$, $n≤2$ and $2≤q≤ 2 n n − 2$, $n>2$. Then there is a constant C depending on Ω and q such that

$∥ u ∥ q ≤C ∥ u ∥ H 0 1 ( Ω ) ,∀u∈ H 0 1 (Ω).$

Lemma 2.2 

Let $y(t): R + → R +$ be a nonincreasing function and assume that there are two constants $μ≥1$ and $A>0$ such that

$∫ s + ∞ y ( t ) μ + 1 2 dt≤Ay(s),0≤s<+∞,$

then $y(t)≤Cy(0) ( 1 + t ) − 2 μ − 1$, $∀t≥0$, if $μ>1$, where C is positive constants independent of $y(0)$.

Lemma 2.3 Let $u(t,x)$ be a solutions to the problem (1.1)-(1.3). Then $E(t)$ is a nonincreasing function for $t>0$ and

$d d t E(t)=−a ∥ u t ( t ) ∥ α α .$
(2.2)

Proof By multiplying equation (1.1) by $u t$ and integrating over Ω, we get

$d d t E ( u ( t ) ) =−a ∥ u t ( t ) ∥ α α ≤0.$

Therefore, $E(t)$ is a nonincreasing function on t. □

We state a local existence result, which is known as a standard one (see [6, 19]).

Theorem 2.1 Suppose that $α,β>2$ satisfy

$2<β<+∞,n≤2;2<β≤ 2 ( n − 1 ) n − 2 ,n>2,$
(2.3)
$2<α<+∞,n≤2;2<α≤ 2 n n − 2 ,n>2,$
(2.4)

and let $( u 0 , u 1 )∈ H 0 1 (Ω)× L 2 (Ω)$. Then there exists $T>0$ such that the problem (1.1)-(1.3) has a unique local solution $u(t)$ in the class

$u∈C ( [ 0 , T ) ; H 0 1 ( Ω ) ) , u t ∈C ( [ 0 , T ) ; L 2 ( Ω ) ) ∩ L α ( Ω × [ 0 , T ) ) .$
(2.5)

In order to prove the existence of global solutions of the problem (1.1)-(1.3), we need the following lemma.

Lemma 2.4 Supposed that (2.3) holds, If $u 0 ∈S$, $u 1 ∈ L 2 (Ω)$ such that

$δ=b C β ( 2 β ( β − 2 ) m 0 E ( 0 ) ) β − 2 2 <1,$
(2.6)

then $u∈S$, for each $t∈[0,T)$.

Proof Assume that there exists a number $t ∗ ∈[0,T)$ such that $u(t)∈S$ on $[0, t ∗ )$ and $u( t ∗ )∉S$. Then we have

$K ( u ( t ∗ ) ) =0,u ( t ∗ ) ≠0.$
(2.7)

Since $u(t)∈S$ on $[0, t ∗ )$, it holds that

$J ( u ( t ) ) = m 0 2 ∥ ∇ u ( t ) ∥ 2 − b β ∥ u ( t ) ∥ β β ≥ m 0 2 ∥ ∇ u ( t ) ∥ 2 − m 0 β ∥ ∇ u ( t ) ∥ 2 = ( β − 2 ) m 0 2 β ∥ ∇ u ( t ) ∥ 2 ,$
(2.8)

we have from $K(u( t ∗ ))=0$ that

$J ( u ( t ∗ ) ) = m 0 2 ∥ ∇ u ( t ∗ ) ∥ 2 − b β ∥ u ( t ∗ ) ∥ β β = m 0 2 ∥ ∇ u ( t ∗ ) ∥ 2 − m 0 β ∥ ∇ u ( t ∗ ) ∥ 2 = ( β − 2 ) m 0 2 β ∥ ∇ u ( t ∗ ) ∥ 2 ,$
(2.9)

we conclude from (1.4) and (2.1) that

$E ( t ) ≥ 1 2 ∥ u t ( t ) ∥ 2 + m 0 2 ∥ ∇ u ( t ) ∥ 2 − b β ∥ u ( t ) ∥ β β = 1 2 ∥ u t ( t ) ∥ 2 + J ( u ( t ) ) .$
(2.10)

Therefore, we obtain from (2.8), (2.9) and (2.10) that

$∥ ∇ u ( t ) ∥ 2 ≤ 2 β ( β − 2 ) m 0 J ( u ( t ) ) ≤ 2 β ( β − 2 ) m 0 E(t)≤ 2 β ( β − 2 ) m 0 E(0),$
(2.11)

for $∀t∈[0, t ∗ ]$.

By exploiting Lemma 2.1, (2.6) and (2.11), we easily arrive at

$b ∥ u ( t ) ∥ β β ≤ b C β ∥ ∇ u ( t ) ∥ β = b C β ∥ ∇ u ( t ) ∥ β − 2 ∥ ∇ u ( t ) ∥ 2 ≤ b C β ( 2 β ( β − 2 ) m 0 E ( 0 ) ) β − 2 2 ∥ ∇ u ( t ) ∥ 2 < ∥ ∇ u ( t ) ∥ 2 ,$
(2.12)

for all $t∈[0, t ∗ ]$.

Therefore, we obtain

$K ( u ( t ∗ ) ) = m 0 ∥ ∇ u ( t ∗ ) ∥ 2 −b ∥ u ( t ∗ ) ∥ β β >0,$
(2.13)

which contradicts (2.7). Thus, we conclude that $u(t)∈S$ on $[0,T)$. □

## 3 The global existence and nonexistence

Theorem 3.1 Suppose that (2.3) and (2.4) hold, and $u(t)$ is a local solution of problem (1.1)-(1.3) on $[0,T)$. If $u 0 ∈S$ and $u 1 ∈ L 2 (Ω)$ satisfy (2.6), then $u(x,t)$ is a global solution of the problem (1.1)-(1.3).

Proof It suffices to show that $∥ ∇ u ( t ) ∥ 2 + ∥ u t ( t ) ∥ 2$ is bounded independently of t.

Under the hypotheses in Theorem 3.1, we get from Lemma 2.4 that $u(t)∈S$ on $[0,T)$. So the formula (2.8) holds on $[0,T)$.

Therefore, we have from (2.8) that

$1 2 ∥ u t ∥ 2 + ( β − 2 ) m 0 2 β ∥ ∇ u ( t ) ∥ 2 ≤ 1 2 ∥ u t ( t ) ∥ 2 +J ( u ( t ) ) =E(t)≤E(0).$
(3.1)

Hence, we get

$∥ u t ( t ) ∥ 2 + ∥ ∇ u ( t ) ∥ 2 ≤max ( 2 , 2 β ( β − 2 ) m 0 ) E(0)<+∞.$

The above inequality and the continuation principle lead to the global existence of the solution, that is, $T=+∞$. Thus, the solution $u(t)$ is a global solution of the problem (1.1)-(1.3). □

Now we employ the analysis method to discuss the solution of the problem (1.1)-(1.3) occurs blow-up in finite time. Our result reads as follows.

Theorem 3.2 Assume that (i) $2<β< 2 n n − 2$, if $n>2$; (ii) $0<β<+∞$, if $n≤2$. If $u 0 ∈S$ and $u 1 ∈ L 2 (Ω)$ such that

$E(0)< Q 0 , ∥ u 0 ∥ β > S 0 ,$

where

$Q 0 = ( β − 2 ) b 2 β ( m 0 b C 2 ) β β − 2 , S 0 = ( m 0 b C 2 ) 1 β − 2$

with $C>0$ is a positive Sobolev constant. Then the solution of the problem (1.1)-(1.3) does not exist globally in time.

Proof On the contrary, under the conditions in Theorem 3.2, suppose that $u(x,t)$ is a global solution of the problem (1.1)-(1.3); then by Lemma 2.1, it is well known that there exists a constant $C>0$ depending only n, β such that $∥ u ∥ β ≤C∥∇u∥$ for all $u∈ H 0 1 (Ω)$.

From the above inequality, we conclude that

$∥ ∇ u ∥ 2 ≥ C − 2 ∥ u ∥ β 2 .$
(3.2)

It follows from (1.4), (2.1) and (3.2) that

$E ( t ) = 1 2 ∥ u t ∥ 2 + 1 2 ∫ 0 ∥ ∇ u ∥ 2 φ ( s ) d s − b β ∥ u ∥ β β ≥ m 0 2 ∥ ∇ u ∥ 2 − b β ∥ u ∥ β β ≥ m 0 2 C 2 ∥ u ∥ β 2 − b β ∥ u ∥ β β .$
(3.3)

Setting

$s=s(t)= ∥ u ( t ) ∥ β = { ∫ Ω | u ( x , t ) | β d x } 1 β .$

We denote the right side of (3.3) by $Q(s)=Q( ∥ u ( t ) ∥ β )$, then

$Q(s)= m 0 2 C 2 s 2 − b β s β ,s≥0.$
(3.4)

By (3.4), we obtain

$Q ′ (s)= m 0 C 2 s−b s β − 1 .$

Let $Q ′ (s)=0$, then we obtain $S 0 = ( m 0 b C 2 ) 1 β − 2$.

As $s= S 0$, we have

$Q ″ (s) | s = S 0 = ( m 0 C 2 − b ( β − 1 ) s β − 2 ) | s = S 0 =− m 0 ( β − 2 ) C 2 <0.$

Consequently, the function $Q(s)$ has a single maximum value $Q 0$ at $S 0$, where

$Q 0 =Q( S 0 )= ( β − 2 ) b 2 β ( m 0 b C 2 ) β β − 2 .$

Since the initial data is such that $E(0)$, $s(0)$ satisfies $E(0)< Q 0$, $∥ u 0 ∥ β > S 0$.

Therefore, we have from Lemma 2.3 that

$E(t)≤E(0)< Q 0 ,∀t>0.$

At the same time, by (3.3) and (3.4) it is evident that there can be no time $t>0$ for which

$E(t)< Q 0 ,s(t)= S 0 .$

Hence, we have also $s(t)> S 0$ for all $t>0$ from the continuity of $E(t)$ and $s(t)$.

According to the above contradiction we know that the global solution of the problem (1.1)-(1.3) does not exist, i.e., the solution blows up in some finite time.

This completes the proof of Theorem 3.2. □

## 4 Energy decay estimate

The following theorem shows the asymptotic behavior of global solutions of the problem (1.1)-(1.3).

Theorem 4.1 If the hypotheses in Theorem  3.2 are valid, then the global solutions of the problem (1.1)-(1.3) has the following asymptotic property:

$E(t)≤M ( 1 + t ) − 2 α − 2 ,$

where $M>0$ is a constant depending on initial energy $E(0)$.

Proof Multiplying by $E ( t ) α − 2 2 u$ on both sides of the equation (1.1) and integrating over $Ω×[S,T]$, we obtain that

$0= ∫ S T ∫ Ω E ( t ) α − 2 2 u [ u t t − φ ( ∥ ∇ u ∥ 2 ) Δ u + a | u t | α − 2 u t − b u | u | β − 2 ] dxdt,$
(4.1)

where $0≤S.

Since

$∫ S T ∫ Ω E ( t ) α − 2 2 u u t t d x d t = ∫ Ω E ( t ) α − 2 2 u u t d x | S T − ∫ S T ∫ Ω E ( t ) α − 2 2 | u t | 2 d x d t − α − 2 2 ∫ S T ∫ Ω E ( t ) α − 4 2 E ′ ( t ) u u t d x d t .$
(4.2)

So, substituting the formula (4.2) into the right-hand side of (4.1), we get that

$0 = ∫ S T E ( t ) α − 2 2 ( ∥ u t ∥ 2 + φ ( ∥ ∇ u ∥ 2 ) ∥ ∇ u ∥ 2 − 2 b β ∥ u ∥ β β ) d t − ∫ S T ∫ Ω E ( t ) α − 2 2 [ 2 | u t | 2 − a | u t | α − 2 u t u ] d x d t − α − 2 2 ∫ S T ∫ Ω E ( t ) α − 4 2 E ′ ( t ) u u t d x d t + ∫ Ω E ( t ) α − 2 2 u u t d x | S T + ( 2 β − 1 ) b ∫ S T E ( t ) α − 2 2 ∥ u ∥ β β d t .$
(4.3)

We obtain from (2.12) and (2.11) that

$b ( 1 − 2 β ) ∥ u ∥ β β ≤δ β − 2 β ∥ ∇ u ∥ 2 ≤δ β − 2 β ⋅ 2 β ( β − 2 ) m 0 E(t)= 2 δ m 0 E(t).$
(4.4)

We derive from (1.4) that

$∫ 0 ∥ ∇ u ∥ 2 φ(s)ds≤φ ( ∥ ∇ u ∥ 2 ) ∥ ∇ u ∥ 2 .$
(4.5)

It follows from (4.3), (4.4) and (4.5) that

$2 ( 1 − δ m 0 ) ∫ S T E ( t ) α 2 d t ≤ ∫ S T ∫ Ω E ( t ) α − 2 2 [ 2 | u t | 2 − a | u t | α − 2 u t u ] d x d t + α − 2 2 ∫ S T ∫ Ω E ( t ) α − 4 2 E ′ ( t ) u u t d x d t − ∫ Ω E ( t ) α − 2 2 u u t d x | S T .$
(4.6)

We have from Lemma 2.1 and (3.1) that

$| α − 2 2 ∫ S T ∫ Ω E ( t ) α − 4 2 E ′ ( t ) u u t d x d t | ≤ α − 2 2 ∫ S T E ( t ) α − 4 2 ( − E ′ ( t ) ) ( 1 2 ∥ u ∥ 2 + 1 2 ∥ u t ∥ 2 ) d t ≤ − α − 2 2 ∫ S T E ( t ) α − 4 2 E ′ ( t ) ( β C 2 ( β − 2 ) m 0 ⋅ ( β − 2 ) m 0 2 β ∥ ∇ u ∥ 2 + 1 2 ∥ u t ∥ 2 ) d t ≤ − α − 2 2 max ( β C 2 ( β − 2 ) m 0 , 1 ) ∫ S T E ( t ) α − 2 2 E ′ ( t ) d t = − α − 2 α max ( β C 2 ( β − 2 ) m 0 , 1 ) E ( t ) α 2 | S T ≤ M E ( S ) α 2 ,$
(4.7)

similarly, we have

$| − ∫ Ω E ( t ) α − 2 2 u u t d x | S T | ≤ max ( β C 2 ( β − 2 ) m 0 , 1 ) E ( t ) α 2 | S T ≤ M E ( S ) α 2 .$
(4.8)

Substituting the estimates (4.7) and (4.8) into (4.6), we conclude that

$2 ( 1 − δ m 0 ) ∫ S T E ( t ) α 2 d t ≤ ∫ S T ∫ Ω E ( t ) α − 2 2 [ 2 | u t | 2 − a | u t | α − 2 u t u ] d x d t + M E ( S ) α 2 .$
(4.9)

We get from Young inequality and Lemma 2.3 that

$2 ∫ S T ∫ Ω E ( t ) α − 2 2 | u t | 2 d x d t ≤ ∫ S T ∫ Ω ( ε 1 E ( t ) α 2 + M ( ε 1 ) | u t | α ) d x d t ≤ M ε 1 ∫ S T E ( t ) α 2 d t + M ( ε 1 ) ∫ S T ∥ u t ∥ α α d t = M ε 1 ∫ S T E ( t ) α 2 d t − M ( ε 1 ) a ( E ( T ) − E ( S ) ) ≤ M ε 1 ∫ S T E ( t ) α 2 d t + M E ( S ) .$
(4.10)

From Young inequality, Lemma 2.1, Lemma 2.3 and (2.11), We receive that

$− a ∫ S T ∫ Ω E ( t ) α − 2 2 u u t | u t | α − 2 d x d t ≤ a ∫ S T E ( t ) α − 2 2 ( ε 2 ∥ u ∥ α α + M ( ε 2 ) ∥ u t ∥ α α ) d t ≤ a C α ε 2 E ( 0 ) α − 2 2 ∫ S T ∥ ∇ u ∥ α d t + a M ( ε 2 ) E ( S ) α − 2 2 ∫ S T ∥ u t ∥ α α d t = a C α ε 2 E ( 0 ) α − 2 2 ∫ S T ( 2 β ( β − 2 ) m 0 E ( t ) ) α 2 d t + M ( ε 2 ) E ( S ) α − 2 2 ( E ( S ) − E ( T ) ) ≤ C α ε 2 E ( 0 ) α − 2 2 ( 2 β ( β − 2 ) m 0 ) α 2 ∫ S T E ( t ) α 2 d t + M E ( S ) α 2 .$
(4.11)

Choosing small enough $ε 1$ and $ε 2$, such that

$1 2 [ M ε 1 + E ( 0 ) α − 2 2 ( 2 β C 2 ( β − 2 ) m 0 ) α 2 ε 2 ] + δ m 0 <1,$

then, substituting (4.10) and (4.11) into (4.9), we get

$∫ S T E ( t ) α 2 dt≤ME(S)+ME ( S ) α 2 ≤M ( 1 + E ( 0 ) ) α − 2 2 E(S).$

Therefore, we have from Lemma 2.2 that

$E(t)≤M ( 1 + t ) − α − 2 2 ,t∈[0,+∞).$

The proof of Theorem 4.1 is thus finished. □

## References

1. 1.

Arosio A, Garavaldi S: On the mildly degenerate Kirchhoff string. Math. Methods Appl. Sci. 1991, 14: 177–195.

2. 2.

Crippa HR: On local solutions of some mildly degenerate hyperbolic equations. Nonlinear Anal. 1993, 21: 565–574. 10.1016/0362-546X(93)90001-9

3. 3.

Ebihara Y, Medeiros LA, Miranda MM: Local solutions for nonlinear degenerate hyperbolic equation. Nonlinear Anal. 1986, 10: 27–40. 10.1016/0362-546X(86)90009-X

4. 4.

Meideiros LA, Miranda M: Solutions for the equations of nonlinear vibrations in Sobolev spaces of fractionary order. Comput. Appl. Math. 1987, 6: 257–276.

5. 5.

Yamada Y: Some nonlinear degenerate wave equations. Nonlinear Anal. 1987, 11: 1155–1168. 10.1016/0362-546X(87)90004-6

6. 6.

Yamazaki T: On local solution of some quasilinear degenerate hyperbolic equations. Funkc. Ekvacioj 1988, 31: 439–457.

7. 7.

Nishihara K, Yamada Y: On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms. Funkc. Ekvacioj 1990, 33: 151–159.

8. 8.

Aassila M, Benaissa A: Existence globale et comportement asymptotique des solutions des équations de Kirchhoff moyennement dégénérées avec un terme nonlinear dissipatif. Funkc. Ekvacioj 2001, 44: 309–333.

9. 9.

D’Ancona P, Spagnolo S: Nolinear perturbations of the Kirchhoff equation. Commun. Pure Appl. Math. 1994, 47: 1005–1029. 10.1002/cpa.3160470705

10. 10.

De Brito EH: The damped elastic stretched string equation generalized: existence uniqueness, regularity and stability. Appl. Anal. 1982, 13: 219–233. 10.1080/00036818208839392

11. 11.

De Brito EH: Decay estimates for the generalized damped extensible string and beam equation. Nonlinear Anal. 1984, 8: 1489–1496. 10.1016/0362-546X(84)90059-2

12. 12.

Nishihara K: Global existence and asymptotic behavior of the solution of some quasilinear hyperbolic equation with linear damping. Funkc. Ekvacioj 1989, 32: 343–355.

13. 13.

Yamada Y: On some quasilinear wave equations with dissipative terms. Nagoya Math. J. 1982, 87: 17–39.

14. 14.

Ghisi M, Gobbino M: Global existence and asymptotic behaviour for a mildly degenerate dissipative hyperbolic equation of Kirchhoff type. Asymptot. Anal. 2004, 40: 25–36.

15. 15.

Hosoya M, Yamada Y: On some nonlinear wave equations II: global existence and energy decay of solutions. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 1991, 38: 239–250.

16. 16.

Sattinger DH: On global solutions for nonlinear hyperbolic equations. Arch. Ration. Mech. Anal. 1968, 30: 148–172.

17. 17.

Payne LE, Sattinger DH: Saddle points and instability of nonlinear hyperbolic equations. Isr. J. Math. 1975, 22: 273–303. 10.1007/BF02761595

18. 18.

Komornik V: Exact Controllability and Stabilization, the Multiplier Method. Masson, Paris; 1994.

19. 19.

Ono K: Global existence and decay properties of solutions for some mildly degenerate nonlinear dissipative Kirchhoff strings. Funkc. Ekvacioj 1997, 40: 255–270.

## Acknowledgements

This work was supported by the Natural Science Foundation of China (No. 61273016), the Natural Science Foundation of Zhejiang Province (No. Y6100016), the Middle-aged Academic Leader of Zhejiang University of Science and Technology (2008-2012), Interdisciplinary Pre-research Project of Zhejiang University of Science and Technology (2010-2012) and Zhejiang province universities scientific research key project (Z201017584).

## Author information

Authors

### Corresponding author

Correspondence to Yaojun Ye.

### Competing interests

The author declares that he has no competing interests.

## Rights and permissions

Reprints and Permissions

Ye, Y. Global existence of solutions and energy decay for a Kirchhoff-type equation with nonlinear dissipation. J Inequal Appl 2013, 195 (2013). https://doi.org/10.1186/1029-242X-2013-195 