In this paper we consider the existence and asymptotic behavior of global solutions for the initial boundary problem of the nonlinear higher-order wave equation with nonlinear damping and source term:
where
,
is a nature number,
and
are real numbers,
is a bounded domain of
with smooth boundary
,
is the Laplace operator, and
.
When
, the existence and uniqueness, as well as decay estimates, of global solutions and blow up of solutions for the initial boundary value problem and Cauchy problem of (1.1) have been investigated by many people through various approaches and assumptive conditions [1–8]. Rammaha [9] deals with wave equations that feature two competing forces and analyzes the influence of these forces on the long-time behavior of solutions. Barbu et al. [10] study the following initial-boundary value problem:
where
is a bounded domain in
with a smooth boundary
,
is a
convex, real value function defined on
, and
denotes the derivative of
. They prove that every generalized solution to the above problem and additional regularity blows up in finite time, whenever the exponent
is greater than the critical value
, and the initial energy is negative.
For the following model of semilinear wave equation with a nonlinear boundary dissipation and nonlinear boundary(interior) sources,
where the operators
are Nemytskii operators associated with scalar, continuous functions
defined for
. The function
is assumed monotone. The paper [11, 12] proves the existence and uniqueness of both local and global solutions of this system on the finite energy space and derive uniform decay rates of the energy when
.
When
, Guesmia [13] considered the equation
with initial boundary value conditions (1.2) and (1.3), where
is a continuous and increasing function with
, and
is a bounded function. He prove a global existence and a regularity result of the problem (1.6), (1.2), and (1.3). Under suitable growth conditions on
, he also established decay results for weak and strong solutions. Precisely, In [13], Guesmia showed that the solution decays exponentially if
behaves like a linear function, whereas the decay is of a polynomial order otherwise. Results similar to the above system, coupled with a semilinear wave equation, have been established by Guesmia [14]. As
in (1.6) is replaced by
. Aassila and Guesmia [15] have obtained a exponential decay theorem through the use of an important lemma of Komornik [16]. Moreover, Messaoudi [17] sets up an existence result of this problem and shows that the solution continues to exist globally if
; however, it blows up in finite time if
.
Nakao [18] has used Galerkin method to present the existence and uniqueness of the bounded solutions, and periodic and almost periodic solutions to the problem (1.1)–(1.3) as the dissipative term is a linear function
. Nakao and Kuwahara [19] studied decay estimates of global solutions to the problem (1.1)–(1.3) by using a difference inequality when the dissipative term is a degenerate case
. When there is no dissipative term in (1.1), Brenner and von Wahl [20] proved the existence and uniqueness of classical solutions to the initial boundary problem for (1.1) in Hilbert space. Pecher [21] investigated the existence and uniqueness of Cauchy problem for (1.1) by the use of the potential well method due to Payne and Sattinger [6] and Sattinger [22].
When
, for the semilinear higher-order wave equation (1.1), Wang [23] shows that the scattering operators map a band in
into
if the nonlinearities have critical or subcritical powers in
. Miao [24] obtains the scattering theory at low energy using time-space estimates and nonlinear estimates. Meanwhile, he also gives the global existence and uniqueness of solutions under the condition of low energy.
The proof of global existence for problem (1.1)–(1.3) is based on the use of the potential well theory [6, 22]. See also Todorova [7, 25] for more recent work. And we study the asymptotic behavior of global solutions by applying the lemma of Komornik [16].
We adopt the usual notation and convention. Let
denote the Sobolev space with the norm
, let
denote the closure in
of
. For simplicity of notation, hereafter we denote by
the Lebesgue space
norm and
denotes
norm, we write equivalent norm
instead of
norm
. Moreover,
denotes various positive constants depending on the known constants and may be different at each appearance.
This paper is organized as follows. In the next section, we will study the existence of global solutions of problem (1.1)–(1.3). Then in Section 3, we are devoted to the proof of decay estimate.
We conclude this introduction by stating a local existence result, which is known as a standard one (see [17]).
Theorem 1.1.
Suppose that
satisfies
and
, then there exists
such that the problem (1.1)–(1.3) has a unique local solution
in the class
Theorem 1.2.
Under the assumptions in Theorem 1.1, if
then
, where
is the maximum time interval on which the solution
of problem (1.1)–(1.3) exists.
Please notice that in [17], we can also construct the following space
in proving the existence of local solution by using contraction mapping principle:
which is equipment with norm
Let
, and
We define a distance
on
, and then
is a complete distance space. This show that, for small enough
, there exists an unique fixed point on
and
only depends on
. Therefore, with the standard extension method of solution, we obtain
for
Here we omit the detailed proof of extension.