Arbitrary energy global existence for wave equation with combined power-type nonlinearities of different signs
Xu et al. Boundary Value Problems (2016) 2016:214
DOI 10.1186/s13661-016-0722-4
RESEARCH
Open Access
Arbitrary energy global existence for wave
equation with combined power-type
nonlinearities of different signs
Runzhang Xu1* , Xingchang Wang2 , Huichao Xu1 and Mingyou Zhang2
*
Correspondence:
1
College of Science, Harbin
Engineering University, Harbin,
150001, P.R. China
Full list of author information is
available at the end of the article
Abstract
This paper proves the global existence of solution for a class of nonlinear wave
equations with nonlinear combined power-type nonlinearities of different signs for
the initial data at sup-critical energy level.
Keywords: initial boundary value; wave equation; global existence; sup-critical
energy
1 Introduction
In the present paper, we mainly consider the initial boundary value problem for the nonlinear wave equation with combined power-type nonlinearities of different signs,
utt – u = f (u),
u(x, t) = ,
x ∈ , t ∈ [, ∞),
()
x ∈ ∂, t ∈ [, ∞),
u(x, ) = u (x),
ut (x, ) = u (x),
()
x ∈ ,
()
where ⊂ Rn (n ≥ ) is an open bounded domain with smooth boundary ∂, is the
Laplace operator on Rn , f (u) is for nonlinear combined power-type nonlinearities of different signs function of u, i.e.
f (u) ≡
l
k=
ak |u|pk – u –
s
bj |u|qj – u,
j=
ak ≥ , ≤ k ≤ l, bj > , ≤ j ≤ s. In addition, pk and qj satisfy the following conditions
(H):
(H)
⎧
⎨ < q < q · · · < q = q < p = p < p < · · · < p < ∞
s
s–
l
l–
⎩ < qs < qs– · · · < q = q < p = pl < pl– < · · · < p ≤ n+
n–
if n = , ,
if n ≥ .
Equation () is a class of important mathematical physical models, so there has been a
lot of important work, such as [–], focused on it. Recently Li and Zhang [] and Tao et
© The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and
indicate if changes were made.
Xu et al. Boundary Value Problems (2016) 2016:214
Page 2 of 6
al. [] considered combined power-type nonlinearities since this kind of nonlinearities is
more general than a single source term. The effects of this kind of nonlinearities on the
prosperities of solution were well treated. However, the nonlinear terms considered in our
paper are more general, we aim to provide some new results in this direction. This paper
is a continued study of [, ], and [], so we give a quick introduction here; for detailed
background of this problem, we refer the reader to [] and the references therein. The
authors in [] first considered problem ()-() and obtained the global existence and blow
up of solutions for the sub-critical case E() < d, where E() is the initial energy and d
is the depth of the potential well or the mountain pass level, which will be defined later.
In the same paper the critical case E() = d was also considered and the global existence
was derived. Later, Yu proved blow up of the solution for the critical case E() = d in [].
Furthermore, the high energy case E() > was treated in [], and the blow up result
was also given. Observing the above results for problem ()-(), helpful in the potential
well method which was introduced by Payne and Sattinger [], the global existence for the
sup-critical case, i.e. E() > , is still not solved. So the present paper solves this problem
by introducing a new stable invariant set and, using the method of [], we are focusing
on proving the global existence of the solution for problem ()-() in the sup-critical case
E() > .
Throughout the present paper, the following notations are used for a precise statement:
p
L denotes the space consisting of all Lp -functions on with norm up = uLp () , u =
uL () , and the inner product (u, v) = uv dx.
2 Global existence at sup-critical case E(0) > 0
For problem ()-() we introduce the energy functional and the Nehari functional as follows:
l
s
bj
ak
qj +
p +
upkk + +
uqj +
E(t) = ut + ∇u –
pk +
q +
j= j
()
k=
and
I(u) = ∇u –
l
k=
p +
ak upkk + +
s
qj +
bj uqj + .
()
j=
Furthermore, for problem ()-(), we define a new stable set
(p + )
W = u ∈ H ()I(u) >
ut ,
()
which will be used to investigate the existence of a global solution with sup-critical initial
energy.
Next, we give a definition of the weak solution for problem ()-().
Definition . (Weak solution []) We say that u = u(x, t) is a weak solution of problem
()-() on × [, T ) if u ∈ L∞ (, T; H ()), ut ∈ L∞ (, T; L ()) and
t
t
(i) (ut , v) + (∇u, ∇v) dτ = (f (u), v) dτ + (u , v) for all v ∈ H (), t ∈ (, T );
(ii) u(x, ) = u (x) in H (), ut (x, ) = u (x) in L ();
(iii) E(t) = E(), t ∈ [, T).
Xu et al. Boundary Value Problems (2016) 2016:214
Page 3 of 6
Theorem . (Local existence []) Suppose that u(x, ) ∈ H (), ut (x, ) ∈ L (). Then
problem ()-() admits a unique local solution u(x, t) defined on a maximal time interval
[, T ). Moreover, if
sup
t∈[,T )
u(x, t) H < ∞,
then T = ∞.
The invariance of the stable set W under the flow of ()-() plays an essential role while
proving the global existence of the weak solution for ()-(). In order to obtain the invariance, we need to prove the following lemma at first.
Lemma . Let u (x) ∈ H (), u (x) ∈ L (), and u(x, t) be solution of problem ()-()
with initial data (u , u ). Assume that E() > and the initial data satisfy
u + (u , u ) + E() < .
()
Then the map {t → u(t) + (u, ut )} is strictly decreasing as long as u(x, t) ∈ W .
Proof Let F(t) = u(t) , then
F (t) = (u, ut ),
F
()
(t) = (u, utt ) + ut .
()
Multiplying equation () by u, and integrating the obtained result with respect to x over
, we have
(utt , u) + ∇u =
l
p +
ak upkk + –
s
qj +
bj uqj +
for t ∈ [, ∞).
()
j=
k=
From () and (), we have
(utt , u) =
l
p +
ak upkk + –
k=
s
qj +
bj uqj + – ∇u = –I(u).
()
j=
Furthermore, from u(t) ∈ W we get
F (t) = ut – I(u) < for t ∈ [, ∞),
()
which shows that F (t) is strictly decreasing on the interval [, ∞). Obviously from E() >
and (), we can get
F () = (u , u ) < .
Then
F (t) < F () < .
()
Xu et al. Boundary Value Problems (2016) 2016:214
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We let H(t) = u(t) + (u, ut ), then
H (t) = (u, ut ) + (u, utt ) + ut .
()
From (), () becomes
H (t) = (u, ut ) – I(u) + ut .
By () and (), we have
H (t) < for t ∈ [, +∞),
which completes the proof.
In the following, we show the invariance of the new stable set W under the flow of problem ()-().
Lemma . (In (...truncated)
