In this paper, \(\Omega\subset \mathbb{R}^{N}\) is an open and bounded set with a smooth boundary ∂Ω and the usual function spaces \(C^{m} ( \overline{\Omega} ) \), \(W^{m,p}=W^{m,p} ( \Omega ) \), \(L^{p}=W^{0,p} ( \Omega ) \), \(H^{m}=W^{m,2} ( \Omega ) \), \(1\leq p\leq\infty\), \(m=0,1,\ldots \) are used. Let \(\langle\cdot,\cdot\rangle\) be either the scalar product in \(L^{2}\) or the dual pairing of a continuous linear functional and an element of a function space. The notation \(\Vert \cdot \Vert \) stands for the norm in \(L^{2}\) and we denote by \(\Vert \cdot \Vert _{X}\) the norm in the Banach space X. We call \(X^{\prime}\) the dual space of X. We denote by \(L^{p}(0,T;X)\), \(1\leq p\leq\infty \), the Banach space of the real functions \(u:(0,T)\rightarrow X\) measurable, such that
$$ \Vert u\Vert _{L^{p}(0,T;X)}= \biggl( \int _{0}^{T}\bigl\Vert u(t)\bigr\Vert _{X}^{p}\,dt \biggr) ^{1/p}< \infty\quad \text{for }1\leq p< \infty $$
and
$$ \Vert u\Vert _{L^{\infty}(0,T;X)}= \mathop{\operatorname{ess} \sup}_{0< t< T}\bigl\Vert u(t)\bigr\Vert _{X}\quad \text{for }p=\infty. $$
Let \(u(t)\), \(u^{\prime}(t)=u_{t}(t)\), \(u^{\prime\prime }(t)=u_{tt}(t)\), \(\nabla u(t)\), \(\Delta u(t)\) denote \(u(x,t)\), \(\frac{\partial u}{\partial t}(x,t)\), \(\frac{\partial^{2}u}{\partial t^{2}}(x,t)\), \((\frac{\partial u}{\partial x_{1}}(x,t), \ldots, \frac{\partial u}{\partial x_{N}}(x,t))\), \(\sum_{i=1}^{N}\frac{\partial^{2}u}{\partial x_{i}^{2}}(x,t)\), respectively.
On \(H^{1}\) we shall use the following norm: \(\Vert v\Vert _{H^{1}}= ( \Vert v\Vert ^{2}+\Vert \nabla v \Vert ^{2} ) ^{1/2}\).
In cases \(N=1\) or \(N=2\), by the continuity and compactness of the injections \(H^{1}(\Omega)\hookrightarrow C^{0}(\overline{\Omega})\) with \(N=1\) or \(H^{1}(\Omega)\hookrightarrow L^{q}(\Omega)\) with \(N=2\), it is not difficult to study problem (1.1)-(1.3). On the other hand, it is obvious that the problem considered with \(a=1\) is more difficult than the one with \(a=-1\),so in what follows we only consider problem (1.1)-(1.3) with \(N\geq3\), \(a=1\). A remark in the end of this paper will give a note in the case \(a=-1\).
First, we recall the following results, see [26].
Lemma 2.1
Let
\(\Omega\subset \mathbb{R}^{N}\)
be an open and bounded set of class
\(C^{1}\). Then the embedding
\(H^{1}\hookrightarrow L^{q}\), is continuous if
\(1\leq q\leq2^{\ast}\)
and compact if
\(1\leq q<2^{\ast}\), where
\(2^{\ast}=\frac{2N}{N-2}\), \(N\geq3\).
Lemma 2.2
Let
\(\Omega\subset \mathbb{R}^{N}\)
be an open and bounded set with a smooth boundary
∂Ω. Then
$$ \biggl( \int _{\partial\Omega}v^{2}(x)\,dS_{x} \biggr) ^{1/2}\leq \gamma_{\Omega} \Vert v\Vert _{H^{1}}\quad \textit{for all } v\in H^{1}, $$
(2.1)
where
\(\gamma_{\Omega}\)
is a positive constant depending only on the domain Ω.
The proofs below also require the following lemma.
Lemma 2.3
Let
\(\Omega\subset \mathbb{R}^{N}\)
be an open and bounded set with a smooth boundary
∂Ω. Let
\(2\leq p\leq\frac{2N-2}{N-2}\), \(N\geq3\). Then there exists a constant
\(D_{p}>0\)
depending on
p, N
and Ω such that
$$ \begin{aligned} (\mathrm{i}) &\quad \bigl\Vert \vert u\vert ^{p-2}u-\vert v\vert ^{p-2}v\bigr\Vert \\ &\qquad \leq D_{p} \bigl[ 1+ \bigl( \Vert u\Vert _{H^{1}}+\Vert v\Vert _{H^{1}} \bigr) ^{1/N}+ \bigl( \Vert u\Vert _{H^{1}}+ \Vert v\Vert _{H^{1}} \bigr) ^{p-2} \bigr] \Vert u-v\Vert _{H^{1}}, \\ (\mathrm{ii}) &\quad \bigl\Vert \vert u\vert ^{p-2}v\bigr\Vert \leq D_{p} \bigl[ 1+\Vert u\Vert _{H^{1}}^{1/N}+ \Vert u\Vert _{H^{1}}^{p-2} \bigr] \Vert v\Vert _{H^{1}} \end{aligned} $$
(2.2)
for all
\(u, v\in H^{1}\).
Proof
(i) We have
$$\begin{aligned} \bigl\vert \vert u\vert ^{p-2}u-\vert v\vert ^{p-2}v \bigr\vert =&\biggl\vert \int _{0}^{1}\frac{d}{d\theta } \bigl[ \bigl\vert v+\theta(u-v)\bigr\vert ^{p-2} \bigl( v+\theta (u-v) \bigr) \bigr] \,d \theta\biggr\vert \\ =&(p-1)\vert u-v\vert \int _{0}^{1}\bigl\vert v+\theta(u-v) \bigr\vert ^{p-2}\,d\theta\leq(p-1)\vert u-v\vert \vert W \vert ^{p-2}, \end{aligned}$$
(2.3)
with \(W=\vert u\vert +\vert v\vert \).
Hence, by Hölder’s inequality we have
$$\begin{aligned} \begin{aligned}[b] \bigl\Vert \vert u\vert ^{p-2}u-\vert v\vert ^{p-2}v \bigr\Vert \leq{}&(p-1) \biggl( \int _{\Omega} \vert u-v\vert ^{2}\vert W\vert ^{2p-4}\,dx \biggr) ^{1/2} \\ \leq&{}(p-1) \biggl( \int _{\Omega} \vert u-v\vert ^{2\alpha}\,dx \biggr) ^{1/2\alpha} \biggl( \int _{\Omega} \vert W\vert ^{(2p-4)\alpha^{\prime}}\,dx \biggr) ^{1/2\alpha^{\prime}} \\ &\mbox{for all }\alpha>1.\end{aligned} \end{aligned}$$
(2.4)
Note that \(H^{1}\hookrightarrow L^{q}\), \(1\leq q\leq2^{\ast}=\frac {2N}{N-2}\), \(N\geq3\), and \(\Vert v\Vert _{L^{q}}\leq C_{q}\Vert v\Vert _{H^{1}}\), \(\forall v\in H^{1}\), \(1\leq q\leq2^{\ast}\).
Choose \(\alpha=\frac{2^{\ast}}{2}=\frac{N}{N-2}\), we have \(\alpha ^{\prime }=\frac{\alpha}{\alpha-1}=\frac{\frac{N}{N-2}}{\frac{N}{N-2}-1}=\frac {N}{2}\), and
$$ \biggl( \int _{\Omega} \vert u-v\vert ^{2\alpha }\,dx \biggr) ^{1/2\alpha}=\Vert u-v\Vert _{L^{2^{\ast}}}\leq C_{2^{\ast}} \Vert u-v\Vert _{H^{1}}. $$
(2.5)
By the condition \(2\leq p\leq\frac{2N-2}{N-2}=2+\frac{2}{N-2}\), \(N\geq3\) is equivalent to
$$ 0\leq(2p-4)\alpha^{\prime}\leq2^{\ast}=\frac{2N}{N-2}, $$
(2.6)
so we consider two cases as follows.
Case 1. \(1\leq(2p-4)\alpha^{\prime}\leq2^{\ast}=\frac {2N}{N-2}\):
$$\begin{aligned} \biggl( \int _{\Omega} \vert W\vert ^{(2p-4)\alpha ^{\prime}}\,dx \biggr) ^{1/2\alpha^{\prime}} =&\Vert W\Vert _{L^{2(p-2)\alpha^{\prime}}}^{p-2}\leq \bigl( C_{2(p-2)\alpha^{\prime }}\Vert W\Vert _{H^{1}} \bigr) ^{p-2} \\ =&C_{2(p-2)\alpha ^{\prime}}^{p-2} \Vert W\Vert _{H^{1}}^{p-2}. \end{aligned}$$
(2.7)
Case 2. \(0\leq\beta\equiv(2p-4)\alpha^{\prime}<1\leq 2^{\ast}=\frac{2N}{N-2}\):
$$\begin{aligned}& \vert W\vert ^{(2p-4)\alpha^{\prime}}=\vert W \vert ^{\beta}\leq1+\vert W\vert , \end{aligned}$$
(2.8)
$$\begin{aligned}& \begin{aligned}[b] \biggl( \int _{\Omega} \vert W\vert ^{(2p-4)\alpha ^{\prime}}\,dx \biggr) ^{1/2\alpha^{\prime}}&\leq \biggl( \int _{\Omega} \bigl( 1+\vert W\vert \bigr) \,dx \biggr) ^{1/2\alpha^{\prime}} \\ &\leq \bigl( \vert \Omega \vert + \vert \Omega \vert ^{1/2}\Vert W\Vert \bigr) ^{1/2\alpha ^{\prime}} \\ &\leq \bigl( \vert \Omega \vert +\vert \Omega \vert ^{1/2}\Vert W\Vert _{H^{1}} \bigr) ^{1/2\alpha^{\prime }} \\ &= \bigl( \vert \Omega \vert + \vert \Omega \vert ^{1/2}\Vert W\Vert _{H^{1}} \bigr) ^{1/N} \\ &\leq \vert \Omega \vert ^{1/N}+\vert \Omega \vert ^{1/2N}\Vert W\Vert _{H^{1}}^{1/N}. \end{aligned} \end{aligned}$$
(2.9)
Consequently, in both cases we get
$$ \biggl( \int _{\Omega} \vert W\vert ^{(2p-4)\alpha ^{\prime}}\,dx \biggr) ^{1/2\alpha^{\prime}}\leq \vert \Omega \vert ^{1/N}+\vert \Omega \vert ^{1/2N}\Vert W\Vert _{H^{1}}^{1/N}+C_{(p-2)N}^{p-2} \Vert W\Vert _{H^{1}}^{p-2}. $$
(2.10)
Hence
$$\begin{aligned} \bigl\Vert \vert u\vert ^{p-2}u-\vert v\vert ^{p-2}v \bigr\Vert \leq&(p-1)C_{2^{\ast}} \Vert u-v\Vert _{H^{1}} \\ &{}\times\bigl[ \vert \Omega \vert ^{1/N}+\vert \Omega \vert ^{1/2N}\Vert W\Vert _{H^{1}}^{1/N}+C_{(p-2)N}^{p-2} \Vert W\Vert _{H^{1}}^{p-2} \bigr] \\ \leq& D_{p}\Vert u-v\Vert _{H^{1}} \bigl[ 1+\Vert W \Vert _{H^{1}}^{1/N}+\Vert W\Vert _{H^{1}}^{p-2} \bigr] \\ \leq& D_{p} \bigl[ 1+ \bigl( \Vert u\Vert _{H^{1}}+\Vert v\Vert _{H^{1}} \bigr) ^{\frac{1}{N}}+ \bigl( \Vert u\Vert _{H^{1}}+\Vert v\Vert _{H^{1}} \bigr) ^{p-2} \bigr] \\ &{}\times \Vert u-v\Vert _{H^{1}}. \end{aligned}$$
(2.11)
Similarly (ii) is proved.
The proof of Lemma 2.3 is complete. □
Next, we state two local existence theorems. We make the following assumptions:
- (\(\mathrm{A}_{0}\)):
-
\(2< p\leq\frac{2N-2}{N-2}\), \(N\geq3\),
- (\(\mathrm{B}_{0}\)):
-
\(K, \lambda\in \mathbb{R}\),
- (\(\mathrm{A}_{1}\)):
-
\(f, f^{\prime}\in L^{1}(0,T;L^{2})\),
- (\(\mathrm{A}_{2}\)):
-
\(h\in L^{2}(0,T;L^{2} ( \partial\Omega\times\Omega ) )\), \(h^{\prime}, h^{\prime\prime}\in L^{2}(0,T;L^{2} ( \partial\Omega\times\Omega ) )\),
- (\(\mathrm{A}_{3}\)):
-
\(g\in L^{2} ( \partial\Omega\times\Omega ) \), \(g^{\prime}, g^{\prime\prime}\in L^{2} ( \partial\Omega\times \Omega ) \),
- (\(\mathrm{A}_{1}^{\prime}\)):
-
\(f\in L^{2}(Q_{T})\),
- (\(\mathrm{A}_{2}^{\prime}\)):
-
\(h\in L^{2}(0,T;L^{2} ( \partial\Omega\times \Omega ) )\), \(h^{\prime}\in L^{2}(0,T;L^{2} ( \partial\Omega \times\Omega ) )\),
- (\(\mathrm{A}_{3}^{\prime}\)):
-
\(g\in L^{2}(0,T;L^{2}(\partial\Omega))\), \(g^{\prime }\in L^{2}(0,T;L^{2}(\partial\Omega))\).
Then we have the following theorem as regards the existence of a ‘strong solution’.
Theorem 2.4
Suppose that (\(\mathrm{A}_{0}\)), (\(\mathrm{B}_{0}\)), (\(\mathrm{A}_{1}\))-(\(\mathrm{A}_{3}\)) hold and the initial data
\(( u_{0},u_{1} ) \in H^{2}\times H^{1}\)
satisfies the compatibility condition
$$ -\frac{\partial u_{0}}{\partial\nu}(x)=g(x,0)+ \int _{\Omega }h(x,y,0)u_{0}(y)\,dy. $$
(2.12)
Then problem (1.1)-(1.3) has a unique local solution
$$ u\in L^{\infty} \bigl( 0,T_{\ast};H^{2} \bigr) , \qquad u_{t}\in L^{\infty } \bigl( 0,T_{\ast};H^{1} \bigr) , \qquad u_{tt}\in L^{\infty} \bigl( 0,T_{\ast };L^{2} \bigr) $$
(2.13)
for
\(T_{\ast}>0\)
small enough.
Remark 2.1
The regularity obtained by (2.13) shows that problem (1.1)-(1.3) has a unique strong solution
$$ \left \{ \textstyle\begin{array}{l} u\in L^{\infty} ( 0,T_{\ast};H^{2} ) \cap C^{0} ( 0,T_{\ast };H^{1} ) \cap C^{1} ( 0,T_{\ast};L^{2} ) , \\ u_{t}\in L^{\infty} ( 0,T_{\ast};H^{1} ) \cap C^{0} ( 0,T_{\ast};L^{2} ) , \\ u_{tt}\in L^{\infty} ( 0,T_{\ast};L^{2} ) .\end{array}\displaystyle \right . $$
(2.14)
With less regular initial data, we obtain the following theorem as regards the existence of a weak solution.
Theorem 2.5
Let (\(\mathrm{A}_{0}\)), (\(\mathrm{B}_{0}\)), (\(\mathrm{A}_{1}^{\prime }\))-(\(\mathrm{A}_{3}^{\prime}\)) hold and
\(( u_{0},u_{1} ) \in H^{1}\times L^{2}\).
Then problem (1.1)-(1.3) has a unique local solution
$$ u\in C \bigl( [0,T_{\ast}];H^{1} \bigr) \cap C^{1} \bigl( [0,T_{\ast }];L^{2} \bigr) $$
(2.15)
for
\(T_{\ast}>0\)
small enough.
Proof of Theorem 2.4
Let \(\{w_{j}\}\) be a denumerable base of \(H^{2}\). Under the assumptions of Theorem 2.4, using the Faedo-Galerkin approximation and Lemmas 2.1-2.3, we find the approximate solution of problem (1.1)-(1.3) in the form
$$ u_{m}(t)=\sum_{j=1}^{m}c_{mj}(t)w_{j}, $$
(2.16)
where the coefficient functions \(c_{mj}\) satisfy the system of ordinary differential equations
$$ \left \{ \textstyle\begin{array}{l} \langle u_{m}^{\prime\prime}(t),w_{j} \rangle+ \langle \nabla u_{m}(t),\nabla w_{j} \rangle+ \langle Ku_{m}(t)+\lambda u_{m}^{\prime}(t),w_{j} \rangle \\ \qquad {}+\int _{\partial\Omega} ( \langle h(x,t),u_{m}(t)\rangle +g(x,t) ) w_{j}(x)\,dS_{x} \\ \quad = \langle \vert u_{m}(t)\vert ^{p-2}u_{m}(t),w_{j} \rangle+ \langle f(t),w_{j} \rangle,\quad 1\leq j\leq m, \\ u_{m}(0)=u_{0},\qquad u_{m}^{\prime}(0)=u_{1}.\end{array}\displaystyle \right . $$
(2.17)
From the assumptions of Theorem 2.4, system (2.17) has a solution \(u_{m}\) on an interval \([0,T_{m}]\subset[0,T]\). The following estimates allow one to take \(T_{m}=T_{\ast}\) for all m, consisting of two key estimates.
In the first key estimate, we put \(S_{m}(t)=\Vert u_{m}^{\prime }(t)\Vert ^{2}+\Vert \nabla u_{m}(t)\Vert ^{2}\), it implies from (2.17) that
$$\begin{aligned} S_{m}(t) =&S_{m}(0)+2 \int _{\partial\Omega} \bigl( \bigl\langle h(x,0),u_{0}\bigr\rangle +g(x,0) \bigr) u_{0}(x)\,dS_{x} \\ &{}-2 \int _{0}^{t} \bigl\langle Ku_{m}(s)+ \lambda u_{m}^{\prime}(s),u_{m}^{\prime}(s) \bigr\rangle \, ds \\ &{}+2 \int _{0}^{t} \bigl\langle f(s),u_{m}^{\prime }(s) \bigr\rangle \, ds+2 \int _{0}^{t} \bigl\langle \bigl\vert u_{m}(s)\bigr\vert ^{p-2}u_{m}(s),u_{m}^{\prime}(s) \bigr\rangle \, ds \\ &{}-2 \int _{\partial\Omega }g(x,t)u_{m}(x,t)\,dS_{x}-2 \int _{\partial\Omega}\bigl\langle h(x,t),u_{m}(t)\bigr\rangle u_{m}(x,t)\,dS_{x} \\ &{}+2 \int _{0}^{t}ds \int _{\partial \Omega} \bigl[ \bigl\langle h^{\prime}(x,s),u_{m}(s) \bigr\rangle +\bigl\langle h(x,s),u_{m}^{\prime}(s)\bigr\rangle +g^{\prime}(x,s) \bigr] u_{m}(x,s)\,dS_{x} \\ \equiv& S_{m}(0)+\sum_{j=1}^{7}I_{j}. \end{aligned}$$
(2.18)
By Lemmas 2.1-2.3 and the following inequalities:
$$ \left \{ \textstyle\begin{array}{l} 2ab\leq\beta a^{2}+\frac{1}{\beta}b^{2}\quad \text{for all }a, b\in \mathbb{R}, \beta>0, \\ ( a+b+c ) ^{q}\leq3^{q-1} ( a^{q}+b^{q}+c^{q} ) \quad \text{for all } q\geq1 , a , b , c\geq0 \end{array}\displaystyle \right . $$
(2.19)
and
$$ \Vert v\Vert \leq \Vert v\Vert _{H^{1}},\qquad \Vert v\Vert _{L^{q}}\leq C_{q}\Vert v\Vert _{H^{1}},\quad \forall v\in H^{1}, 1\leq q\leq2^{\ast}=\frac{2N}{N-2}, N \geq3, $$
(2.20)
with computing explicitly, all terms in the right-hand side of (2.18) are estimated, in which the following estimates are worthy of note:
$$\begin{aligned}& S_{m}(0)+I_{1}=S_{m}(0)+2 \int _{\partial\Omega} \bigl( \bigl\langle h(x,0),u_{0}\bigr\rangle +g(x,0) \bigr) u_{0}(x)\,dS_{x} \\& \hphantom{S_{m}(0)+I_{1}}=\Vert u_{1}\Vert ^{2}+\Vert \nabla u_{0} \Vert ^{2}+2 \int _{\partial \Omega} \bigl( \bigl\langle h(x,0),u_{0}\bigr\rangle +g(x,0) \bigr) u_{0}(x)\,dS_{x}\equiv \frac{1}{2}\overline{C}_{0}; \end{aligned}$$
(2.21)
$$\begin{aligned}& I_{2}=-2 \int _{0}^{t} \bigl\langle Ku_{m}(s)+ \lambda u_{m}^{\prime }(s),u_{m}^{\prime}(s) \bigr\rangle \,ds\leq \int _{0}^{t} \bigl\Vert u_{m}(s)\bigr\Vert ^{2}\,ds+ \bigl( K^{2}+2\vert \lambda \vert \bigr) \int _{0}^{t}\bigl\Vert u_{m}^{\prime}(s) \bigr\Vert ^{2}\,ds \\& \hphantom{I_{2}}\leq \int _{0}^{t} \biggl[ \Vert u_{0} \Vert + \int _{0}^{s}\bigl\Vert u_{m}^{\prime}(r) \bigr\Vert \,dr \biggr] ^{2}\,ds+ \bigl( K^{2}+2\vert \lambda \vert \bigr) \int _{0}^{t}S_{m}(s)\,ds \\& \hphantom{I_{2}}\leq2T\Vert u_{0}\Vert ^{2}+T^{2} \int _{0}^{t}\bigl\Vert u_{m}^{\prime}(r) \bigr\Vert ^{2}\,dr+ \bigl( K^{2}+2\vert \lambda \vert \bigr) \int _{0}^{t}S_{m}(s)\,ds \\& \hphantom{I_{2}}\leq2T\Vert u_{0}\Vert ^{2}+ \bigl( T^{2}+K^{2}+2\vert \lambda \vert \bigr) \int _{0}^{t}S_{m}(s)\,ds\leq C_{T} \biggl( 1+ \int _{0}^{t}S_{m}(s)\,ds \biggr) ; \end{aligned}$$
(2.22)
$$\begin{aligned}& I_{3}=2 \int _{0}^{t} \bigl\langle f(s),u_{m}^{\prime }(s) \bigr\rangle \,ds\leq \int _{0}^{T}\bigl\Vert f(s) \bigr\Vert ^{2}\,ds+ \int _{0}^{t}\bigl\Vert u_{m}^{\prime}(s) \bigr\Vert ^{2}\,ds\leq C_{T}+ \int _{0}^{t}S_{m}(s)\,ds; \end{aligned}$$
(2.23)
$$\begin{aligned}& I_{4}=2 \int _{0}^{t} \bigl\langle \bigl\vert u_{m}(s) \bigr\vert ^{p-2}u_{m}(s),u_{m}^{\prime}(s) \bigr\rangle \,ds\leq 2 \int _{0}^{t}\bigl\Vert \bigl\vert u_{m}(s)\bigr\vert ^{p-1}\bigr\Vert \bigl\Vert u_{m}^{\prime}(s)\bigr\Vert \,ds \\& \hphantom{I_{4}}\leq \int _{0}^{t}\bigl\Vert \bigl\vert u_{m}(s)\bigr\vert ^{p-1}\bigr\Vert ^{2}\,ds+ \int _{0}^{t}S_{m}(s)\,ds \\& \hphantom{\hphantom{I_{4}}}= \int _{0}^{t}\bigl\Vert u_{m}(s)\bigr\Vert _{L^{2p-2}}^{2p-2}\,ds+ \int _{0}^{t}S_{m}(s)\,ds\leq C_{2p-2}^{2p-2} \int _{0}^{t}\bigl\Vert u_{m}(s)\bigr\Vert _{H^{1}}^{2p-2}\,ds+ \int _{0}^{t}S_{m}(s)\,ds, \end{aligned}$$
(2.24)
since \(1\leq2\leq2p-2\leq2^{\ast}\), and \(H^{1}(\Omega )\hookrightarrow L^{2p-2}(\Omega)\), we have
$$\begin{aligned} \bigl\Vert u_{m}(t)\bigr\Vert _{H^{1}}^{2p-2} \leq& \biggl[ 2\Vert u_{0}\Vert ^{2}+S_{m}(t)+2t \int _{0}^{t}S_{m} ( s ) \,ds \biggr] ^{p-1} \\ \leq&3^{p-2}2^{p-1}\Vert u_{0}\Vert ^{2p-2}+3^{p-2} \bigl( S_{m}(t) \bigr) ^{p-1}+3^{p-2}2^{p-1}t^{2p-3} \int _{0}^{t} \bigl( S_{m} ( s ) \bigr) ^{p-1}\,ds, \end{aligned}$$
it leads to
$$\begin{aligned}& I_{4}=2 \int _{0}^{t} \bigl\langle \bigl\vert u_{m}(s) \bigr\vert ^{p-2}u_{m}(s),u_{m}^{\prime}(s) \bigr\rangle \,ds\leq C_{T}+C_{T} \int _{0}^{t} \bigl( S_{m} ( s ) \bigr) ^{p-1}\,ds+ \int _{0}^{t}S_{m}(s)\,ds; \end{aligned}$$
(2.25)
$$\begin{aligned}& I_{5}=-2 \int _{\partial\Omega}g(x,t)u_{m}(x,t)\,dS_{x}\leq 2 \gamma _{\Omega} \Vert g\Vert _{L^{\infty}(0,T;L^{2}(\partial \Omega ))}\bigl\Vert u_{m}(t)\bigr\Vert _{H^{1}} \\& \hphantom{I_{5}}\leq\frac{1}{\beta}\gamma_{\Omega}^{2}\Vert g\Vert _{L^{\infty}(0,T;L^{2}(\partial\Omega))}^{2}+\beta\bigl\Vert u_{m}(t)\bigr\Vert _{H^{1}}^{2} \\& \hphantom{\hphantom{I_{5}}}\leq\frac{1}{\beta}\gamma_{\Omega}^{2}\Vert g\Vert _{L^{\infty}(0,T;L^{2}(\partial\Omega))}^{2}+\beta \biggl[ 2\Vert u_{0}\Vert ^{2}+S_{m}(t)+2t \int _{0}^{t}S_{m} ( s ) \,ds \biggr] \\& \hphantom{\hphantom{\hphantom{I_{5}}}}\leq\frac{1}{\beta}C_{T}+\beta S_{m}(t)+C_{T} \int _{0}^{t}S_{m} ( s ) \,ds\quad \mbox{for all }0< \beta< 1; \end{aligned}$$
(2.26)
$$\begin{aligned}& I_{6}=-2 \int _{\partial\Omega}\bigl\langle h(x,t),u_{m}(t)\bigr\rangle u_{m}(x,t)\,dS_{x}\leq2\gamma_{\Omega} \Vert h\Vert _{L^{\infty }(0,T;L^{2} ( \partial\Omega\times\Omega ) )}\bigl\Vert u_{m}(t)\bigr\Vert \bigl\Vert u_{m}(t)\bigr\Vert _{H^{1}} \\& \hphantom{I_{6}}\leq \frac{1}{\beta}\gamma_{\Omega}^{2}\Vert h\Vert _{L^{\infty}(0,T;L^{2} ( \partial\Omega\times\Omega ) )}^{2}\bigl\Vert u_{m}(t)\bigr\Vert ^{2}+\beta\bigl\Vert u_{m}(t)\bigr\Vert _{H^{1}}^{2} \\& \hphantom{I_{6}}\leq\frac{1}{\beta}\gamma_{\Omega}^{2}\Vert h\Vert _{L^{\infty}(0,T;L^{2} ( \partial\Omega\times\Omega ) )}^{2} \biggl[ 2\Vert u_{0}\Vert ^{2}+2t \int _{0}^{t}S_{m} ( s ) \,ds \biggr] \\& \hphantom{I_{6}={}}{} +\beta \biggl[ 2\Vert u_{0}\Vert ^{2}+S_{m}(t)+2t \int _{0}^{t}S_{m} ( s ) \,ds \biggr] \\& \hphantom{I_{6}}\leq\frac{1}{\beta}C_{T}+\beta S_{m}(t)+ \frac{1}{\beta}C_{T} \int _{0}^{t}S_{m} ( s ) \,ds\quad \mbox{for all }\beta >0, \beta< 1; \end{aligned}$$
(2.27)
$$\begin{aligned}& I_{7}=2 \int _{0}^{t}\,ds \int _{\partial\Omega} \bigl[ \bigl\langle h^{\prime}(x,s),u_{m}(s) \bigr\rangle +\bigl\langle h(x,s),u_{m}^{\prime }(s)\bigr\rangle +g^{\prime}(x,s) \bigr] u_{m}(x,s)\,dS_{x} \\& \hphantom{I_{7}}\leq 2\gamma_{\Omega}\bigl\Vert h^{\prime}\bigr\Vert _{L^{\infty}(0,T;L^{2} ( \partial\Omega\times\Omega ) )} \int _{0}^{t}\bigl\Vert u_{m}(s)\bigr\Vert _{H^{1}}^{2}\,ds \\& \hphantom{I_{7}={}}{}+2\gamma_{\Omega} \Vert h\Vert _{L^{\infty }(0,T;L^{2} ( \partial\Omega\times\Omega ) )} \int _{0}^{t}\bigl\Vert u_{m}^{\prime}(s) \bigr\Vert \bigl\Vert u_{m}(s)\bigr\Vert _{H^{1}}\,ds \\& \hphantom{I_{7}={}}{}+2\gamma_{\Omega}\bigl\Vert g^{\prime}\bigr\Vert _{L^{\infty }(0,T;L^{2}(\partial\Omega))} \int _{0}^{t}\bigl\Vert u_{m}(s)\bigr\Vert _{H^{1}}\,ds \\& \hphantom{I_{7}}\leq C_{T}+C_{T} \int _{0}^{t}\bigl\Vert u_{m}(s)\bigr\Vert _{H^{1}}^{2}\,ds+ \int _{0}^{t}\bigl\Vert u_{m}^{\prime}(s) \bigr\Vert ^{2}\,ds\leq C_{T} \biggl( 1+ \int _{0}^{t}S_{m} ( s ) \,ds \biggr) . \end{aligned}$$
(2.28)
Combining estimations of all terms and choosing \(\beta=\frac{1}{4}\), we obtain after some rearrangements
$$ S_{m}(t)\leq C_{T} \biggl( 1+ \int _{0}^{t}S_{m} ( s ) \,ds+ \int _{0}^{t} \bigl( S_{m} ( s ) \bigr) ^{p-1}\,ds \biggr) ,\quad 0\leq t\leq T_{m}, $$
(2.29)
where \(C_{T}\) always indicates a constant depending on T.
Then, by solving a nonlinear Volterra integral inequality (2.29) (based on the methods in [27]), the following lemma is proved.
Lemma 2.6
There exists a constant
\(T_{\ast}>0\)
depending on
T (independent of
m) such that
$$ S_{m}(t)\leq C_{T}, \quad \forall m\in \mathbb{N},\forall t \in[0,T_{\ast}], $$
(2.30)
where
\(C_{T}\)
is a constant depending only on
T.
By Lemma 2.6, we can take a constant \(T_{m}=T_{\ast}\) for all m.
In the second key estimate, we put \(X_{m}(t)=\Vert u_{m}^{\prime \prime }(t)\Vert ^{2}+\Vert \nabla u_{m}^{\prime}(t)\Vert ^{2}\),and it follows from (2.17) that
$$\begin{aligned} X_{m}(t) =&X_{m}(0)+2 \int _{\partial\Omega} \bigl( \bigl\langle h^{\prime}(x,0),u_{0} \bigr\rangle +\bigl\langle h(x,0),u_{1}\bigr\rangle +g^{\prime }(x,0) \bigr) u_{1}(x)\,dS_{x} \\ &{}-2 \int _{0}^{t} \bigl\langle Ku_{m}^{\prime }(s)+ \lambda u_{m}^{\prime\prime}(s),u_{m}^{\prime\prime }(s) \bigr\rangle \,ds+2 \int _{0}^{t} \bigl\langle f^{\prime }(s),u_{m}^{\prime\prime}(s) \bigr\rangle \,ds \\ &{}+2(p-1) \int _{0}^{t} \bigl\langle \bigl\vert u_{m}(s)\bigr\vert ^{p-2}u_{m}^{\prime}(s),u_{m}^{\prime\prime }(s) \bigr\rangle \,ds \\ &{}-2 \int _{\partial\Omega} \bigl( \bigl\langle h^{\prime}(x,t),u_{m}(t) \bigr\rangle +\bigl\langle h(x,t),u_{m}^{\prime }(t)\bigr\rangle +g^{\prime}(x,t) \bigr) u_{m}^{\prime}(x,t) \,dS_{x} \\ &{}+2 \int _{0}^{t}\,ds \int _{\partial \Omega}\bigl[ \bigl\langle h^{\prime\prime}(x,s),u_{m}(s) \bigr\rangle +2\bigl\langle h^{\prime}(x,s),u_{m}^{\prime}(s) \bigr\rangle \\ &{}+\bigl\langle h(x,s),u_{m}^{\prime\prime }(s)\bigr\rangle +g^{\prime\prime}(x,s)\bigr] u_{m}^{\prime }(x,s) \,dS_{x} \\ \equiv& X_{m}(0)+\sum_{i=1}^{6}J_{i} . \end{aligned}$$
(2.31)
Letting \(t\rightarrow0_{+}\) in equation (2.17)1, multiplying the result by \(c_{mj}^{\prime\prime}(0)\), and using the compatibility (2.12), we get
$$ \bigl\Vert u_{m}^{\prime\prime}(0)\bigr\Vert ^{2}= \bigl\langle \Delta u_{0},u_{m}^{\prime\prime}(0) \bigr\rangle - \bigl\langle Ku_{0}+\lambda u_{1},u_{m}^{\prime\prime}(0) \bigr\rangle + \bigl\langle \vert u_{0}\vert ^{p-2}u_{0},u_{m}^{\prime\prime}(0) \bigr\rangle + \bigl\langle f(0),u_{m}^{\prime\prime}(0) \bigr\rangle . $$
This implies that
$$ \bigl\Vert u_{m}^{\prime\prime}(0)\bigr\Vert \leq \Vert \Delta u_{0}\Vert +\vert K\vert \Vert u_{0}\Vert + \vert \lambda \vert \Vert u_{1}\Vert +\bigl\Vert \vert u_{0}\vert ^{p-1}\bigr\Vert +\bigl\Vert f(0) \bigr\Vert =\overline{X}_{0} \quad \mbox{for all }m, $$
(2.32)
where \(\overline{X}_{0}\) is a constant depending only on p, K, λ, \(u_{0}\), \(u_{1}\), f.
Also note the following estimations:
$$\begin{aligned}& \begin{aligned}[b] X_{m}(0)+J_{1}&=X_{m}(0)+2 \int _{\partial\Omega} \bigl( \bigl\langle h^{\prime}(x,0),u_{0} \bigr\rangle +\bigl\langle h(x,0),u_{1}\bigr\rangle +g^{\prime }(x,0) \bigr) u_{1}(x)\,dS_{x} \\ &\leq\overline{X}_{0}^{2}+\Vert \nabla u_{1} \Vert ^{2}+2 \int _{\partial\Omega} \bigl( \bigl\langle h^{\prime}(x,0),u_{0} \bigr\rangle +\bigl\langle h(x,0),u_{1}\bigr\rangle +g^{\prime }(x,0) \bigr) u_{1}(x)\,dS_{x} \\ &\equiv\frac{1}{2}X_{0}; \end{aligned} \end{aligned}$$
(2.33)
$$\begin{aligned}& \begin{aligned}[b] J_{2}&=-2 \int _{0}^{t} \bigl\langle Ku_{m}^{\prime}(s)+ \lambda u_{m}^{\prime\prime}(s),u_{m}^{\prime\prime}(s) \bigr\rangle \,ds \\ &\leq \int _{0}^{t}\bigl\Vert u_{m}^{\prime}(s) \bigr\Vert ^{2}\,ds+ \bigl( K^{2}+2\vert \lambda \vert \bigr) \int _{0}^{t}\bigl\Vert u_{m}^{\prime\prime}(s) \bigr\Vert ^{2}\,ds \\ &\leq C_{T}+ \bigl( K^{2}+2\vert \lambda \vert \bigr) \int _{0}^{t}X_{m}(s)\,ds; \end{aligned} \end{aligned}$$
(2.34)
$$\begin{aligned}& \begin{aligned}[b]J_{3}&=2 \int _{0}^{t} \bigl\langle f^{\prime}(s),u_{m}^{\prime \prime}(s) \bigr\rangle \,ds\leq \int _{0}^{t}\bigl\Vert f^{\prime }(s)\bigr\Vert \,ds+ \int _{0}^{t}\bigl\Vert f^{\prime }(s)\bigr\Vert \bigl\Vert u_{m}^{\prime\prime}(s)\bigr\Vert ^{2} \,ds \\ &\leq C_{T}+ \int _{0}^{t}\bigl\Vert f^{\prime}(s)\bigr\Vert X_{m}(s)\,ds. \end{aligned} \end{aligned}$$
(2.35)
From
$$ \bigl\Vert \bigl\vert u_{m}(s)\bigr\vert ^{p-2}u_{m}^{\prime }(s) \bigr\Vert \leq D_{p} \bigl[ 1+\bigl\Vert u_{m}(s)\bigr\Vert _{H^{1}}^{1/N}+\bigl\Vert u_{m}(s)\bigr\Vert _{H^{1}}^{p-2} \bigr] \bigl\Vert u_{m}^{\prime}(s) \bigr\Vert _{H^{1}}\leq D_{p}C_{T} \bigl\Vert u_{m}^{\prime}(s)\bigr\Vert _{H^{1}},$$
by Lemma 2.3(ii), it gives
$$\begin{aligned}& J_{4}=2(p-1) \int _{0}^{t} \bigl\langle \bigl\vert u_{m}(s)\bigr\vert ^{p-2}u_{m}^{\prime}(s),u_{m}^{\prime\prime }(s) \bigr\rangle \,ds \\& \hphantom{J_{4}}\leq2(p-1) \int _{0}^{t}\bigl\Vert \bigl\vert u_{m}(s)\bigr\vert ^{p-2}u_{m}^{\prime}(s) \bigr\Vert \bigl\Vert u_{m}^{\prime\prime}(s)\bigr\Vert \,ds \\& \hphantom{J_{4}}\leq2(p-1)D_{p}C_{T} \int _{0}^{t}\bigl\Vert u_{m}^{\prime }(s) \bigr\Vert _{H^{1}}\bigl\Vert u_{m}^{\prime\prime}(s)\bigr\Vert \,ds \\& \hphantom{J_{4}}\leq(p-1)^{2}D_{p}^{2}C_{T}^{2} \int _{0}^{t}\bigl\Vert u_{m}^{\prime}(s) \bigr\Vert _{H^{1}}^{2}\,ds+ \int _{0}^{t}\bigl\Vert u_{m}^{\prime\prime }(s) \bigr\Vert ^{2}\,ds \\& \hphantom{J_{4}}=(p-1)^{2}D_{p}^{2}C_{T}^{2} \biggl[ \int _{0}^{t}\bigl\Vert u_{m}^{\prime}(s) \bigr\Vert ^{2}\,ds+ \int _{0}^{t}\bigl\Vert \nabla u_{m}^{\prime}(s)\bigr\Vert ^{2}\,ds \biggr] + \int _{0}^{t} \bigl\Vert u_{m}^{\prime\prime}(s) \bigr\Vert ^{2}\,ds \\& \hphantom{J_{4}}\leq C_{T} \biggl( 1+ \int _{0}^{t}X_{m} ( s ) \,ds \biggr) ; \end{aligned}$$
(2.36)
$$\begin{aligned}& J_{5}=-2 \int _{\partial\Omega} \bigl( \bigl\langle h^{\prime }(x,t),u_{m}(t) \bigr\rangle +\bigl\langle h(x,t),u_{m}^{\prime}(t)\bigr\rangle +g^{\prime }(x,t) \bigr) u_{m}^{\prime}(x,t) \,dS_{x} \\& \hphantom{J_{5}}\leq2\gamma_{\Omega}\bigl[ \bigl\Vert u_{m}(t) \bigr\Vert \bigl\Vert h^{\prime}\bigr\Vert _{L^{\infty}(0,T;L^{2}(\partial \Omega \times\Omega))}+\bigl\Vert u_{m}^{\prime}(t)\bigr\Vert \Vert h\Vert _{L^{\infty}(0,T;L^{2}(\partial\Omega\times\Omega ))} \\& \hphantom{J_{5}={}}{}+\bigl\Vert g^{\prime}\bigr\Vert _{L^{\infty}(0,T;L^{2}(\partial\Omega))}\bigr] \bigl\Vert u_{m}^{\prime}(t)\bigr\Vert _{H^{1}} \\& \hphantom{J_{5}}\leq2C_{T}\bigl\Vert u_{m}^{\prime}(t) \bigr\Vert _{H^{1}}\leq \frac{1}{\beta}C_{T}+\beta\bigl\Vert u_{m}^{\prime}(t)\bigr\Vert _{H^{1}}^{2} \\& \hphantom{J_{5}}\leq\frac{1}{\beta}C_{T}+\beta \biggl[ 2\Vert u_{1}\Vert ^{2}+X_{m}(t)+2t \int _{0}^{t}X_{m} ( s ) \,ds \biggr] \\& \hphantom{J_{5}}\leq\frac{1}{\beta}C_{T}+\beta X_{m}(t)+C_{T} \biggl( 1+ \int _{0}^{t}X_{m} ( s ) \,ds \biggr) \quad \mbox{for all } \beta\in (0,1); \end{aligned}$$
(2.37)
$$\begin{aligned}& J_{6}=2 \int _{0}^{t}ds \int _{\partial\Omega}\bigl[ \bigl\langle h^{\prime\prime}(x,s),u_{m}(s) \bigr\rangle +2\bigl\langle h^{\prime }(x,s),u_{m}^{\prime}(s) \bigr\rangle \\& \hphantom{J_{6}={}}{}+ \bigl\langle h(x,s),u_{m}^{\prime\prime}(s)\bigr\rangle +g^{\prime\prime}(x,s)\bigr] u_{m}^{\prime }(x,s) \,dS_{x} \\& \hphantom{J_{6}}\leq2\gamma_{\Omega}C_{T} \int _{0}^{t}\bigl\Vert h^{\prime\prime}(s)\bigr\Vert _{L^{2}(\partial\Omega\times\Omega )}\bigl\Vert u_{m}^{\prime}(s)\bigr\Vert _{H^{1}}\,ds+4\gamma_{\Omega }C_{T} \int _{0}^{t}\bigl\Vert u_{m}^{\prime}(s) \bigr\Vert _{H^{1}}\,ds \\& \hphantom{J_{6}={}}{}+2\gamma_{\Omega}C_{T} \int _{0}^{t}\bigl\Vert u_{m}^{\prime\prime}(s) \bigr\Vert \bigl\Vert u_{m}^{\prime }(s)\bigr\Vert _{H^{1}}\,ds+2\gamma_{\Omega } \int _{0}^{t}\bigl\Vert g^{\prime\prime}(s)\bigr\Vert _{L^{2}(\partial\Omega)}\bigl\Vert u_{m}^{\prime}(s)\bigr\Vert _{H^{1}}\,ds \\& \hphantom{J_{6}}\leq\gamma_{\Omega}^{2}C_{T}^{2} \bigl\Vert h^{\prime\prime }\bigr\Vert _{L^{1}(0,T;L^{2}(\partial\Omega\times\Omega ))}+ \int _{0}^{t}\bigl\Vert h^{\prime\prime}(s)\bigr\Vert _{L^{2}(\partial\Omega\times\Omega)}\bigl\Vert u_{m}^{\prime }(s)\bigr\Vert _{H^{1}}^{2}\,ds \\& \hphantom{J_{6}={}}{}+4\gamma_{\Omega}^{2}C_{T}^{2}T+ \int _{0}^{t}\bigl\Vert u_{m}^{\prime}(s) \bigr\Vert _{H^{1}}^{2}\,ds+\gamma_{\Omega }^{2}C_{T}^{2} \int _{0}^{t}\bigl\Vert u_{m}^{\prime\prime }(s) \bigr\Vert ^{2}\,ds+ \int _{0}^{t}\bigl\Vert u_{m}^{\prime }(s) \bigr\Vert _{H^{1}}^{2}\,ds \\& \hphantom{J_{6}={}}{}+\gamma_{\Omega}^{2}\bigl\Vert g^{\prime\prime}\bigr\Vert _{L^{1}(0,T;L^{2}(\partial\Omega))}+ \int _{0}^{t}\bigl\Vert g^{\prime\prime}(s)\bigr\Vert _{L^{2}(\partial\Omega)}\bigl\Vert u_{m}^{\prime}(s)\bigr\Vert _{H^{1}}^{2}\,ds \\& \hphantom{J_{6}}\leq C_{T}+C_{T} \int _{0}^{t}X_{m}(s)\,ds+ \int _{0}^{t}\Phi (s)\bigl\Vert u_{m}^{\prime}(s)\bigr\Vert _{H^{1}}^{2}\,ds \\& \hphantom{J_{6}}\leq C_{T}+C_{T} \int _{0}^{t}X_{m}(s)\,ds+ \int _{0}^{t}\Phi (s) \biggl[ 2\Vert u_{1}\Vert ^{2}+X_{m}(s)+2s \int _{0}^{s}X_{m} ( r ) \,dr \biggr] \,ds \\& \hphantom{J_{6}}\leq C_{T}+C_{T} \int _{0}^{t}X_{m}(s)\,ds+ \int _{0}^{t}\Phi (s)X_{m}(s)\,ds, \end{aligned}$$
(2.38)
where
$$ \Phi(s)=2+\bigl\Vert h^{\prime\prime}(s)\bigr\Vert _{L^{2}(\partial \Omega\times\Omega)}+\bigl\Vert g^{\prime\prime}(s)\bigr\Vert _{L^{2}(\partial\Omega)}, \quad \Phi\in L^{1}(0,T). $$
(2.39)
Combining estimations and choosing \(\beta=\frac{1}{2}\), we obtain after some rearrangements
$$ X_{m}(t)\leq C_{T}+ \int _{0}^{t}\Psi(s)X_{m}(s) \,ds, $$
(2.40)
where \(C_{T}\) always indicates a constant depending on T, and
$$ \Psi(s)=C_{T} \bigl[ 1+\bigl\Vert f^{\prime}(s)\bigr\Vert + \bigl\Vert h^{\prime\prime}(s)\bigr\Vert _{L^{2}(\partial\Omega\times\Omega )}+\bigl\Vert g^{\prime\prime}(s)\bigr\Vert _{L^{2}(\partial\Omega )} \bigr] , \quad \Psi \in L^{1}(0,T). $$
(2.41)
By Gronwall’s lemma, we deduce from (2.40) that
$$ X_{m}(t)\leq C_{T}\exp \biggl[ \int _{0}^{T}\Psi(s)\,ds \biggr] \leq C_{T} \quad \mbox{for all }t\in[0,T_{\ast}]. $$
(2.42)
It verifies the existence of a subsequence of \(\{u_{m}\}\), denoted by the same symbol, such that
$$ \left \{ \textstyle\begin{array}{l} u_{m}\rightarrow u \quad \text{in } L^{\infty}(0,T_{\ast};H^{1}) \text{ weakly}^{*}, \\ u_{m}^{\prime}\rightarrow u^{\prime} \quad \text{in } L^{\infty }(0,T_{\ast };H^{1}) \text{ weakly}^{*}, \\ u_{m}^{\prime\prime}\rightarrow u^{\prime\prime} \quad \text{in } L^{\infty }(0,T_{\ast};L^{2}) \text{ weakly}^{*}. \end{array}\displaystyle \right . $$
(2.43)
By the compactness lemma of Lions ([28], p.57), we can deduce from (2.43) the existence of a subsequence still denoted by \(\{u_{m}\}\), such that
$$ \left \{ \textstyle\begin{array}{l} u_{m}\rightarrow u \quad \text{strongly in } L^{2}(Q_{T_{\ast }}) \text{ and a.e. in }Q_{T_{\ast}}, \\ u_{m}^{\prime}\rightarrow u^{\prime}\quad \text{strongly in } L^{2}(Q_{T_{\ast}}) \text{ and a.e. in }Q_{T_{\ast}} .\end{array}\displaystyle \right . $$
(2.44)
By means of the continuity of the function \(t\longmapsto|t|^{p-2}t\), we have
$$ |u_{m}|^{p-2}u_{m}\rightarrow|u|^{p-2}u \quad \text{and a.e. in } Q_{T_{\ast }}. $$
(2.45)
On the other hand
$$\begin{aligned} \bigl\Vert |u_{m}|^{p-2}u_{m}\bigr\Vert _{L^{2}(Q_{T_{\ast }})}^{2} =& \int _{0}^{T_{\ast}}\,ds \int _{\Omega }\bigl|u_{m}(x,t)\bigr|^{2p-2}\,dx \\ =& \int _{0}^{T_{\ast}}\bigl\Vert u_{m}(t)\bigr\Vert _{L^{2p-2}}^{2p-2}\,dt \\ \leq& \int _{0}^{T_{\ast}} \bigl( C_{2p-2}\bigl\Vert u_{m}(t) \bigr\Vert _{H^{1}} \bigr) ^{2p-2}\,dt \\ \leq& C_{2p-2}^{2p-2}T_{\ast} \Vert u_{m}\Vert _{L^{\infty}(0,T_{\ast};H^{1})}^{2p-2}\leq C_{T} . \end{aligned}$$
(2.46)
Using the Lions lemma ([28], Lemma 1.3, p.12), it follows from (2.45) and (2.46) that
$$ |u_{m}|^{p-2}u_{m}\rightarrow|u|^{p-2}u \quad \text{in }L^{2}(Q_{T_{\ast }}) \text{ weakly}. $$
(2.47)
Passing to the limit in (2.17) by (2.43), (2.44), and (2.47), we have u satisfying the problem
$$ \left \{ \textstyle\begin{array}{l} \langle u^{\prime\prime}(t),v \rangle+ \langle\nabla u(t),\nabla v \rangle+ \langle Ku(t)+\lambda u^{\prime }(t),v \rangle \\ \qquad {}+\int _{\partial\Omega} ( \langle h(x,t),u(t)\rangle+g(x,t) ) v(x)\,dS_{x} \\ \quad = \langle \vert u(t)\vert ^{p-2}u(t),v \rangle+ \langle f(t),v \rangle \quad \mbox{for all } v\in H^{1}, \\ u(0)=u_{0} , \qquad u^{\prime}(0)=u_{1} .\end{array}\displaystyle \right . $$
(2.48)
On the other hand, we have from (2.43), (2.48)1
$$ \Delta u=u^{\prime\prime}+Ku+\lambda u^{\prime}-|u|^{p-2}u-f\in L^{\infty}\bigl(0,T_{\ast};L^{2}\bigr). $$
(2.49)
Thus \(u\in L^{\infty}(0,T_{\ast};H^{2})\) and the proof of existence is complete. The uniqueness of a weak solution is proved as follows.
Let \(u_{1}\), \(u_{2}\) be two weak solutions of problem (1.1)-(1.3), such that
$$ u_{i}\in L^{\infty} \bigl( 0,T_{\ast};H^{2} \bigr) , \qquad u_{i}^{\prime }\in L^{\infty} \bigl( 0,T_{\ast};H^{1} \bigr) , \qquad u_{i}^{\prime \prime} \in L^{\infty} \bigl( 0,T_{\ast};L^{2} \bigr) ,\quad i=1,2. $$
(2.50)
Then \(u=u_{1}-u_{2}\) satisfy the variational problem
$$ \left \{ \textstyle\begin{array}{l} \langle u^{\prime\prime}(t),v \rangle+ \langle\nabla u(t),\nabla v \rangle+ \langle Ku(t)+\lambda u^{\prime }(t),v \rangle+\int _{\partial\Omega}\langle h(x,t),u(t)\rangle v(x)\,dS_{x} \\ \quad = \langle|u_{1}|^{p-2}u_{1}-|u_{2}|^{p-2}u_{2},v \rangle\quad \mbox{for all }v\in H^{1}, \\ u(0)=u^{\prime}(0)=0.\end{array}\displaystyle \right . $$
(2.51)
We take \(v=u^{\prime}=u_{1}^{\prime}-u_{2}^{\prime} \) in (2.51) and integrating with respect to t, we obtain
$$\begin{aligned} \sigma(t) =&-2 \int _{0}^{t} \bigl\langle Ku(s)+\lambda u^{\prime }(s),u^{\prime}(s) \bigr\rangle \,ds-2 \int _{\partial\Omega }\bigl\langle h(x,t),u(t)\bigr\rangle u(x,t) \,dS_{x} \\ &{}+2 \int _{0}^{t}\,ds \int _{\partial\Omega } \bigl[ \bigl\langle h^{\prime}(x,s),u(s)\bigr\rangle +\bigl\langle h(x,s),u^{\prime }(s)\bigr\rangle \bigr] u(x,s) \,dS_{x} \\ &{}+2 \int _{0}^{t} \bigl\langle |u_{1}|^{p-2}u_{1}-|u_{2}|^{p-2}u_{2},u^{\prime}(s) \bigr\rangle \,ds=\sum_{j=1}^{4} \sigma_{j}, \end{aligned}$$
(2.52)
where
$$ \sigma(t)=\bigl\Vert u^{\prime}(t)\bigr\Vert ^{2}+\bigl\Vert \nabla u(t)\bigr\Vert ^{2}. $$
(2.53)
By (2.53) and the following inequalities:
$$\begin{aligned}& 2ab\leq\beta a^{2}+\frac{1}{\beta}b^{2}\quad \text{for all }a, b\in \mathbb{R} , \beta>0, \end{aligned}$$
(2.54)
$$\begin{aligned}& \bigl\Vert u(t)\bigr\Vert ^{2}= \biggl( \int _{0}^{t}\bigl\Vert u^{\prime}(s)\bigr\Vert \,ds \biggr) ^{2}\leq t \int _{0}^{t}\bigl\Vert u^{\prime}(s)\bigr\Vert ^{2}\,ds\leq t \int _{0}^{t}\sigma(s)\,ds, \\& \bigl\Vert u(t)\bigr\Vert _{H^{1}}^{2}=\bigl\Vert \nabla u(t) \bigr\Vert ^{2}+\bigl\Vert u(t)\bigr\Vert ^{2}\leq \sigma (t)+t \int _{0}^{t}\sigma(s)\,ds, \\& \int _{0}^{t}\bigl\Vert u(s)\bigr\Vert _{H^{1}}^{2}\,ds\leq \int _{0}^{t} \biggl[ \sigma(s)+s \int _{0}^{s}\sigma (r)\,dr \biggr] \,ds \leq \bigl( 1+t^{2} \bigr) \int _{0}^{t}\sigma (s)\,ds, \end{aligned}$$
(2.55)
we estimate the following integrals in the right-hand side of (2.52):
$$\begin{aligned}& \sigma_{1}=-2 \int _{0}^{t} \bigl\langle Ku(s)+\lambda u^{\prime }(s),u^{\prime}(s) \bigr\rangle \,ds \\& \hphantom{\sigma_{1}}\leq \int _{0}^{t}\bigl\Vert u(s)\bigr\Vert ^{2}\,ds+ \bigl( K^{2}+2\vert \lambda \vert \bigr) \int _{0}^{t}\bigl\Vert u^{\prime}(s)\bigr\Vert ^{2}\,ds \\& \hphantom{\sigma_{1}}\leq \int _{0}^{t} \biggl( s \int _{0}^{s}\sigma (r)\,dr \biggr) \,ds+ \bigl( K^{2}+2\vert \lambda \vert \bigr) \int _{0}^{t}\sigma(s)\,ds \\& \hphantom{\sigma_{1}}\leq T^{2} \int _{0}^{t}\sigma(r)\,dr+ \bigl( K^{2}+2 \vert \lambda \vert \bigr) \int _{0}^{t}\sigma (s)\,ds\leq C_{T} \int _{0}^{t}\sigma(s)\,ds; \end{aligned}$$
(2.56)
$$\begin{aligned}& \sigma_{2}=-2 \int _{\partial\Omega}\bigl\langle h(x,t),u(t)\bigr\rangle u(x,t) \,dS_{x} \\& \hphantom{\sigma_{2}}\leq2\gamma_{\Omega} \Vert h\Vert _{L^{\infty }(0,T;L^{2}(\partial\Omega\times\Omega))} \bigl\Vert u(t)\bigr\Vert \bigl\Vert u(t)\bigr\Vert _{H^{1}} \\& \hphantom{\sigma_{2}} \leq\frac{1}{\beta}\gamma_{\Omega}^{2}\Vert h\Vert _{L^{\infty}(0,T;L^{2}(\partial\Omega\times\Omega))}^{2}\bigl\Vert u(t)\bigr\Vert ^{2}+\beta\bigl\Vert u(t)\bigr\Vert _{H^{1}}^{2} \\& \hphantom{\sigma_{2}}\leq\frac{1}{\beta}\gamma_{\Omega}^{2}\Vert h\Vert _{L^{\infty}(0,T;L^{2}(\partial\Omega\times\Omega ))}^{2}t \int _{0}^{t}\sigma(s)\,ds+\beta \biggl[ \sigma (t)+t \int _{0}^{t}\sigma(s)\,ds \biggr] \\& \hphantom{\sigma_{2}}\leq\beta\sigma(t)+\frac{1}{\beta}C_{T} \int _{0}^{t}\sigma(s) \,ds; \end{aligned}$$
(2.57)
$$\begin{aligned}& \sigma_{3}=2 \int _{0}^{t}\,ds \int _{\partial\Omega } \bigl[ \bigl\langle h^{\prime}(x,s),u(s)\bigr\rangle +\bigl\langle h(x,s),u^{\prime }(s)\bigr\rangle \bigr] u(x,s)\,dS_{x} \\& \hphantom{\sigma_{3}}\leq2\gamma_{\Omega}\bigl\Vert h^{\prime}\bigr\Vert _{L^{\infty }(0,T;L^{2}(\partial\Omega\times\Omega ))} \int _{0}^{t}\bigl\Vert u(s)\bigr\Vert _{H^{1}}^{2}\,ds \\& \hphantom{\sigma_{3}\leq{}}{}+2\gamma_{\Omega} \Vert h\Vert _{L^{\infty }(0,T;L^{2}(\partial\Omega\times\Omega ))} \int _{0}^{t}\bigl\Vert u^{\prime}(s)\bigr\Vert \bigl\Vert u(s)\bigr\Vert _{H^{1}}\,ds \\& \hphantom{\sigma_{3}}\leq C_{T} \int _{0}^{t}\bigl\Vert u(s)\bigr\Vert _{H^{1}}^{2}\,ds+C_{T} \int _{0}^{t}\bigl\Vert u^{\prime }(s)\bigr\Vert ^{2}\,ds\leq C_{T} \int _{0}^{t}\sigma(s)\,ds. \end{aligned}$$
(2.58)
By Lemma 2.3(i), we have
$$\begin{aligned}& \bigl\Vert |u_{1}|^{p-2}u_{1}-|u_{2}|^{p-2}u_{2} \bigr\Vert \\& \quad \leq D_{p} \bigl[ 1+ \bigl( \Vert u_{1} \Vert _{H^{1}}+\Vert u_{2} \Vert _{H^{1}} \bigr) ^{1/N}+ \bigl( \Vert u_{1}\Vert _{H^{1}}+\Vert u_{2}\Vert _{H^{1}} \bigr) ^{p-2} \bigr] \bigl\Vert u(s)\bigr\Vert _{H^{1}} \\& \quad \leq D_{p} \bigl[ 1+M_{1}^{1/N}+M_{1}^{p-2} \bigr] \bigl\Vert u(s)\bigr\Vert _{H^{1}}\leq C_{T}\bigl\Vert u(s)\bigr\Vert _{H^{1}}, \end{aligned}$$
(2.59)
where \(M_{1}=\Vert u_{1}\Vert _{L^{\infty} ( 0,T_{\ast };H^{1} ) }+\Vert u_{2}\Vert _{L^{\infty} ( 0,T_{\ast };H^{1} ) }\). Hence
$$\begin{aligned} \sigma_{4} =&2 \int _{0}^{t} \bigl\langle |u_{1}|^{p-2}u_{1}-|u_{2}|^{p-2}u_{2},u^{\prime}(s) \bigr\rangle \,ds \\ \leq& 2C_{T} \int _{0}^{t}\bigl\Vert u(s)\bigr\Vert _{H^{1}} \bigl\Vert u^{\prime}(s)\bigr\Vert \,ds \\ \leq &C_{T} \int _{0}^{t}\bigl\Vert u(s)\bigr\Vert _{H^{1}}^{2}\,ds+C_{T} \int _{0}^{t}\bigl\Vert u^{\prime }(s)\bigr\Vert ^{2}\,ds \\ \leq& C_{T} \int _{0}^{t}\sigma(s)\,ds. \end{aligned}$$
(2.60)
Combining (2.52), (2.56)-(2.58), (2.60) and choosing \(\beta=\frac{1}{2}\), we obtain
$$ \sigma(t)\leq C_{T} \int _{0}^{t}\sigma(s)\,ds. $$
(2.61)
By Gronwall’s lemma, it follows from (2.61) that \(\sigma\equiv0\), i.e., \(u_{1}\equiv u_{2}\). Theorem 2.4 is proved completely. □
Proof of Theorem 2.5
In order to prove this theorem, we use standard arguments of density.
First, we note that \(W^{1}(0,T;L^{2}(\partial\Omega))=\{g\in L^{2}(0,T;L^{2}(\partial\Omega)):g^{\prime}\in L^{2}(0,T;L^{2}(\partial \Omega))\}\) is a Hilbert space with respect to the scalar product (see [27]):
$$ \langle f,g\rangle_{W^{1}(0,T;L^{2}(\partial\Omega ))}= \int _{0}^{T} \bigl[ \bigl\langle f(t),g(t)\bigr\rangle _{L^{2}(\partial \Omega)}+\bigl\langle f^{\prime}(t),g^{\prime}(t)\bigr\rangle _{L^{2}(\partial \Omega)} \bigr] \,dt. $$
(2.62)
Furthermore, we also have the embedding \(W^{1}(0,T;L^{2}(\partial\Omega ))\hookrightarrow C^{0}([0,T];L^{2}(\partial\Omega))\) is continuous and
$$\begin{aligned} \Vert g\Vert _{C^{0}([0,T];L^{2}(\partial\Omega))}&\leq \gamma _{T}\sqrt{ \bigl( \Vert g\Vert _{L^{2}(0,T;L^{2}(\partial \Omega ))}^{2}+\bigl\Vert g^{\prime}\bigr\Vert _{L^{2}(0,T;L^{2}(\partial \Omega ))}^{2} \bigr) } \\ &\equiv \gamma_{T}\Vert g\Vert _{W^{1}(0,T;L^{2}(\partial\Omega))} \end{aligned}$$
(2.63)
for all \(g\in W^{1}(0,T;L^{2}(\partial\Omega))\), where \(\gamma _{T}=\sqrt{\frac{1}{2T}+\sqrt{4+\frac{1}{4T^{2}}}}\) (see the Appendix).
Similarly, \(W^{1}(0,T;L^{2} ( \partial\Omega\times\Omega ) )=\{h\in L^{2}(0,T;L^{2} ( \partial\Omega\times\Omega ) ):h^{\prime}\in L^{2}(0,T;L^{2} ( \partial\Omega\times\Omega ) )\}\) is a Hilbert space with respect to the scalar product
$$ \langle h,k\rangle_{W^{1}(0,T;L^{2} ( \partial\Omega\times\Omega ) )}= \int _{0}^{T} \bigl[ \bigl\langle h(t),k(t)\bigr\rangle _{L^{2} ( \partial\Omega\times\Omega ) }+\bigl\langle h^{\prime }(t),k^{\prime}(t)\bigr\rangle _{L^{2} ( \partial\Omega\times\Omega ) } \bigr] \,dt, $$
(2.64)
and the embedding \(W^{1}(0,T;L^{2} ( \partial\Omega\times\Omega ) )\hookrightarrow C^{0}([0,T];L^{2} ( \partial\Omega\times \Omega ) )\) is continuous and
$$\begin{aligned} \Vert h\Vert _{C^{0}([0,T];L^{2} ( \partial\Omega \times \Omega ) )}&\leq\gamma_{T}\sqrt{ \bigl( \Vert h\Vert _{L^{2}(0,T;L^{2} ( \partial\Omega\times\Omega ) )}^{2}+\bigl\Vert h^{\prime}\bigr\Vert _{L^{2}(0,T;L^{2} ( \partial \Omega\times\Omega ) )}^{2} \bigr) } \\ &\equiv\gamma_{T}\Vert h\Vert _{W^{1}(0,T;L^{2} ( \partial\Omega\times\Omega ) )} \end{aligned}$$
(2.65)
for all \(h\in W^{1}(0,T;L^{2} ( \partial\Omega\times\Omega ) )\), where \(\gamma_{T}=\sqrt{\frac{1}{2T}+\sqrt{4+\frac{1}{4T^{2}}}}\) (see the Appendix).
Consider \(( u_{0},u_{1},f,g,h ) \in H^{1}\times L^{2}\times L^{2}(Q_{T})\times W^{1}(0,T;L^{2} ( \partial\Omega ) )\times W^{1}(0,T;L^{2} ( \partial\Omega\times\Omega ) )\).
Let the sequence \(\{ ( u_{0m},u_{1m},f_{m},g_{m},h_{m} ) \} \subset H^{2}\times H^{1}\times C_{0}^{\infty} ( \overline{Q}_{T} ) \times C_{0}^{\infty} ( \partial\Omega\times\overline{\Omega} ) \times C_{0}^{\infty} ( \partial\Omega\times\overline{\Omega }\times [0,T] ) \), such that
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} u_{0m}\rightarrow u_{0} \quad \text{strongly in } H^{1}, \\ u_{1m}\rightarrow u_{1} \quad \text{strongly in } L^{2}, \\ f_{m}\rightarrow f \quad \text{strongly in } L^{2}(Q_{T}) ,\end{array}\displaystyle \right . \end{aligned}$$
(2.66)
$$\begin{aligned}& \Vert g_{m}-g\Vert _{W^{1}(0,T;L^{2} ( \partial\Omega \times\Omega ) )}^{2}\equiv \Vert g_{m}-g\Vert _{L^{2}(0,T;L^{2}(\partial\Omega))}^{2}+\bigl\Vert g_{m}^{\prime }-g^{\prime}\bigr\Vert _{L^{2}(0,T;L^{2}(\partial\Omega ))}^{2} \rightarrow0 , \end{aligned}$$
(2.67)
$$\begin{aligned}& \Vert h_{m}-h\Vert _{W^{1}(0,T;L^{2} ( \partial\Omega \times\Omega ) )}^{2}\equiv \Vert h_{m}-h\Vert _{L^{2}(0,T;L^{2} ( \partial\Omega\times\Omega ) )}^{2}+\bigl\Vert h_{m}^{\prime}-h^{\prime}\bigr\Vert _{L^{2}(0,T;L^{2} ( \partial\Omega\times\Omega ) )}^{2} \\& \hphantom{\Vert h_{m}-h\Vert _{W^{1}(0,T;L^{2} ( \partial\Omega \times\Omega ) )}^{2}}\rightarrow0 . \end{aligned}$$
(2.68)
So \(\{ ( u_{0m},u_{1m} ) \}\) satisfy, for all \(m\in \mathbb{N}\), the compatibility condition
$$ -\frac{\partial u_{0m}}{\partial\nu}(x)=g_{m}(x,0)+ \int _{\Omega }h_{m}(x,y,0)u_{0m}(y)\,dy . $$
(2.69)
Then, for each \(m\in \mathbb{N}\), there exists a unique function \(u_{m}\) under the conditions of Theorem 2.4. Thus, we can verify
$$ \left \{ \textstyle\begin{array}{l} \langle u_{m}^{\prime\prime}(t),v \rangle+ \langle \nabla u_{m}(t),\nabla v \rangle+ \langle Ku_{m}(t)+\lambda u_{m}^{\prime }(t),v \rangle \\ \qquad {}+\int _{\partial\Omega} ( \langle h_{m}(x,t),u_{m}(t)\rangle +g_{m}(x,t) ) v(x)\,dS_{x} \\ \quad = \langle|u_{m}|^{p-2}u_{m},v \rangle+ \langle f_{m}(t),v \rangle \quad \mbox{for all } v\in H^{1}, \\ u_{m}(0)=u_{0m} ,\qquad u_{m}^{\prime}(0)=u_{1m} \end{array}\displaystyle \right . $$
(2.70)
and
$$ \left \{ \textstyle\begin{array}{l} u_{m}\in L^{\infty} ( 0,T_{\ast};H^{2} ) \cap C^{0} ( 0,T_{\ast};H^{1} ) \cap C^{1} ( 0,T_{\ast};L^{2} ) , \\ u_{m}^{\prime}\in L^{\infty} ( 0,T_{\ast};H^{1} ) \cap C^{0} ( 0,T_{\ast};L^{2} ) , \\ u_{m}^{\prime\prime}\in L^{\infty} ( 0,T_{\ast};L^{2} ) . \end{array}\displaystyle \right . $$
(2.71)
By the same arguments used to obtain the above estimates, we get
$$ \bigl\Vert u_{m}^{\prime}(t)\bigr\Vert ^{2}+\bigl\Vert u_{m}(t)\bigr\Vert _{H^{1}}^{2}\leq C_{T} , $$
(2.72)
\(\forall t\in[0,T_{\ast}]\), where \(C_{T}\) always indicates a constant depending on T as above.
On the other hand, we put \(w_{m, l}=u_{m}-u_{l}\), \(f_{m, l}=f_{m}-f_{l}\), \(h_{m, l}=h_{m}-h_{l}\), \(g_{m,l}=g_{m}-g_{l}\), \(h_{m, l}(x,y,0)=\bar{h}_{m, l}^{(0)}(x,y)\), \(g_{m, l}(x,0)=\bar{g}_{m, l}^{(0)}(x)\), from (2.70), it follows that
$$ \left \{ \textstyle\begin{array}{l} \langle w_{m,l}^{\prime\prime}(t),v \rangle+ \langle \nabla w_{m,l}(t),\nabla v \rangle+ \langle Kw_{m,l}(t)+\lambda w_{m,l}^{\prime}(t),v \rangle \\ \qquad {}+\int _{\partial\Omega} ( \langle h_{m}(x,t),w_{m,l}(t)\rangle+\langle h_{m, l}(x,t),u_{l}(t)\rangle+g_{m,l}(x,t) ) v(x)\,dS_{x} \\ \quad = \langle |u_{m}|^{p-2}u_{m}-|u_{l}|^{p-2}u_{l},v \rangle+ \langle f_{m, l}(t),v \rangle \quad \mbox{for all } v\in H^{1}, \\ w_{m,l}(0)=u_{0m}-u_{0l}\equiv\bar{w}_{m,l}^{(0)} ,\qquad w_{m,l}^{\prime }(0)=u_{1m}-u_{1l}\equiv\bar{w}_{m,l}^{(1)} .\end{array}\displaystyle \right . $$
(2.73)
We take \(v=w_{m,l}=u_{m}-u_{l}\), in (2.73) and integrating with respect to t, we get
$$\begin{aligned} S_{m,l}(t) =&S_{m,l}(0)+2 \int _{\partial\Omega} \bigl( \bigl\langle h_{m}(x,0), \bar{w}_{m,l}^{(0)}\bigr\rangle +\bigl\langle h_{m, l}(x,0),u_{0l}\bigr\rangle +g_{m,l}(x,0) \bigr) \bar{w}_{m,l}^{(0)}(x)\,dS_{x} \\ &{}+2 \int _{0}^{t} \bigl\langle f_{m, l}(s),w_{m,l}^{\prime}(s) \bigr\rangle \,ds-2 \int _{0}^{t} \bigl\langle Kw_{m,l}(s)+ \lambda w_{m,l}^{\prime }(s),w_{m,l}^{\prime}(s) \bigr\rangle \,ds \\ &{}-2 \int _{\partial\Omega} \bigl( \bigl\langle h_{m}(x,t),w_{m,l}(t) \bigr\rangle +\bigl\langle h_{m, l}(x,t),u_{l}(t)\bigr\rangle +g_{m,l}(x,t) \bigr) w_{m,l}(x,t)\,dS_{x} \\ &{}+2 \int _{0}^{t}\,ds \int _{\partial\Omega} \bigl( \bigl\langle h_{m}^{\prime}(x,s),w_{m,l}(s) \bigr\rangle +\bigl\langle h_{m}(x,s),w_{m,l}^{\prime }(s) \bigr\rangle \bigr) w_{m,l}(x,s)\,dS_{x} \\ &{}+2 \int _{0}^{t}\,ds \int _{\partial\Omega} \bigl( \bigl\langle h_{m, l}^{\prime}(x,s),u_{l}(s) \bigr\rangle +\bigl\langle h_{m, l}(x,s),u_{l}^{\prime}(s) \bigr\rangle \bigr) w_{m,l}(x,s)\,dS_{x} \\ &{}+2 \int _{0}^{t}\,ds \int _{\partial\Omega }g_{m,l}^{\prime }(x,s)w_{m,l}(x,s) \,dS_{x} \\ &{}+2 \int _{0}^{t} \bigl\langle |u_{m}|^{p-2}u_{m}-|u_{l}|^{p-2}u_{l},w_{m,l}^{\prime}(s) \bigr\rangle \,ds\equiv S_{m,l}(0)+\sum_{j=1}^{8}Z_{j} , \end{aligned}$$
(2.74)
where
$$\begin{aligned}& S_{m,l}(t)=\bigl\Vert w_{m,l}^{\prime}(t)\bigr\Vert ^{2}+\bigl\Vert \nabla w_{m,l}(t)\bigr\Vert ^{2} , \end{aligned}$$
(2.75)
$$\begin{aligned}& S_{m,l}(0)=\Vert u_{1m}-u_{1l}\Vert ^{2}+\Vert \nabla u_{0m}-\nabla u_{0l}\Vert ^{2} . \end{aligned}$$
(2.76)
After all terms of \(S_{m,l}(t)\) are estimated, in which we note the two main estimations \(Z_{1}\), \(Z_{8} \) as follows:
$$\begin{aligned} Z_{1} =&2 \int _{\partial\Omega} \bigl( \bigl\langle h_{m}(x,0), \bar{w}_{m,l}^{(0)}\bigr\rangle +\bigl\langle h_{m, l}(x,0),u_{0l}\bigr\rangle +g_{m,l}(x,0) \bigr) \bar{w}_{m,l}^{(0)}(x)\,dS_{x} \\ \leq&2\gamma_{\Omega} \bigl[ \bigl\Vert \bar{w}_{m,l}^{(0)} \bigr\Vert \bigl\Vert h_{m}(0)\bigr\Vert _{L^{2}(\partial \Omega\times\Omega)}+\Vert u_{0l}\Vert \bigl\Vert \bar{h}_{m, l}^{(0)} \bigr\Vert _{L^{2}(\partial\Omega\times\Omega )}+\bigl\Vert \bar{g}_{m, l}^{(0)} \bigr\Vert _{L^{2}(\partial \Omega)} \bigr] \bigl\Vert \bar{w}_{m,l}^{(0)} \bigr\Vert _{H^{1}} \\ \leq&2\gamma_{\Omega}\gamma_{T}\cdot \mathrm{const.} \bigl[ \bigl\Vert \bar{w}_{m,l}^{(0)}\bigr\Vert _{H^{1}}+ \Vert h_{m, l}\Vert _{W^{1}(0,T;L^{2} ( \partial\Omega\times\Omega ) )}+\Vert g_{m, l}\Vert _{W^{1}(0,T;L^{2} ( \partial \Omega ) )} \bigr] \bigl\Vert \bar{w}_{m,l}^{(0)}\bigr\Vert _{H^{1}} \\ \rightarrow&0 , \quad \mbox{as } m , l\rightarrow+\infty ; \end{aligned}$$
(2.77)
this result combined with (2.66)-(2.68) shows that
$$\begin{aligned} S_{m,l}(0)+Z_{1} =&\Vert u_{1m}-u_{1l} \Vert ^{2}+\Vert \nabla u_{0m}-\nabla u_{0l} \Vert ^{2}+Z_{1} \\ \equiv& R(m,l)\rightarrow0 ,\quad \mbox{as } m , l\rightarrow+\infty . \end{aligned}$$
(2.78)
On the other hand
$$\begin{aligned}& \bigl\Vert |u_{m}|^{p-2}u_{m}-|u_{l}|^{p-2}u_{l} \bigr\Vert \\& \quad \leq D_{p} \bigl[ 1+ \bigl( \Vert u_{m}\Vert _{H^{1}}+\Vert u_{l}\Vert _{H^{1}} \bigr) ^{1/N}+ \bigl( \Vert u_{m}\Vert _{H^{1}}+\Vert u_{l}\Vert _{H^{1}} \bigr) ^{p-2} \bigr] \bigl\Vert w_{m,l}(s)\bigr\Vert _{H^{1}} \\& \quad \leq C_{T}\bigl\Vert w_{m,l}(s)\bigr\Vert _{H^{1}} , \end{aligned}$$
(2.79)
by Lemma 2.3(i), we get
$$\begin{aligned} Z_{8} =&2 \int _{0}^{t} \bigl\langle |u_{m}|^{p-2}u_{m}-|u_{l}|^{p-2}u_{l},w_{m,l}^{\prime}(s) \bigr\rangle \,ds\leq2C_{T} \int _{0}^{t}\bigl\Vert w_{m,l}(s)\bigr\Vert _{H^{1}}\bigl\Vert w_{m,l}^{\prime}(s)\bigr\Vert \,ds \\ \leq& C_{T} \biggl[ 2t\bigl\Vert \bar{w}_{m,l}^{(0)} \bigr\Vert ^{2}+\bigl(1+t^{2}\bigr) \int _{0}^{t}S_{m,l}(s)\,ds \biggr] +C_{T}\int _{0}^{t}S_{m,l}(s) \,ds \\ \leq& C_{T} \biggl[ \bigl\Vert \bar{w}_{m,l}^{(0)} \bigr\Vert ^{2}+ \int _{0}^{t}S_{m,l}(s)\,ds \biggr] . \end{aligned}$$
(2.80)
We obtain
$$ S_{m,l}(t)\leq R_{T}^{(1)}(m,l)+C_{T} \int _{0}^{t}S_{m,l}(s)\,ds , $$
(2.81)
with
$$\begin{aligned} R_{T}^{(1)}(m,l) =&2R(m,l)+2\Vert f_{m, l}\Vert _{L^{2}(Q_{T})}^{2} \\ &{}+C_{T} \bigl( \bigl\Vert \bar{w}_{m,l}^{(0)} \bigr\Vert ^{2}+\Vert h_{m, l}\Vert _{W^{1}(0,T;L^{2} ( \partial\Omega\times\Omega ) )}^{2}+ \Vert g_{m,l}\Vert _{W^{1}(0,T;L^{2} ( \partial \Omega ) )}^{2} \bigr) \rightarrow0 , \end{aligned}$$
(2.82)
as \(m, l\rightarrow+\infty\). By Gronwall’s lemma, it follows from (2.81) that
$$ S_{m,l}(t)\leq R_{T}^{(1)}(m,l)\exp ( TC_{T} ) \leq C_{T}R_{T}^{(1)}(m,l), \quad \forall t\in[0,T_{\ast}] . $$
(2.83)
Thus, convergence of the sequence \(\{ ( u_{0m},u_{1m},f_{m},g_{m},h_{m} ) \}\) implies the convergence to zero as \(m, l\rightarrow+\infty\) of the term on the right-hand side of (2.83). Therefore, we get
$$ u_{m}\rightarrow u \quad \text{strongly in } C^{0} \bigl([0,T_{\ast }];H^{1}\bigr)\cap C^{1} \bigl([0,T_{\ast}];L^{2}\bigr) . $$
(2.84)
On the other hand, from (2.72), we get the existence of a subsequence of \(\{u_{m}\}\), still also so denoted, such that
$$ \left \{ \textstyle\begin{array}{l} u_{m}\rightarrow u \quad \text{in } L^{\infty}(0,T_{\ast};H^{1}) \text{ weakly}^{*}, \\ u_{m}^{\prime}\rightarrow u^{\prime} \quad \text{in } L^{\infty }(0,T_{\ast };L^{2}) \text{ weakly}^{*}.\end{array}\displaystyle \right . $$
(2.85)
By the compactness lemma of Lions ([28], p.57), we can deduce from (2.85) the existence of a subsequence, still denoted by \(\{u_{m}\}\), such that
$$ \textstyle\begin{array}{l} u_{m}\rightarrow u \quad \text{strongly in } L^{2}(Q_{T\ast })\text{ and a.e. in }Q_{T_{\ast}} .\end{array} $$
(2.86)
Similarly, by (2.72), it follows from (2.86) that
$$ |u_{m}|^{p-2}u_{m}\rightarrow|u|^{p-2}u \quad \text{in }L^{2}(Q_{T_{\ast }}) \text{ weakly}. $$
(2.87)
Passing to the limit in (2.70) by (2.84)-(2.87), we have u satisfying the problem
$$ \left \{ \textstyle\begin{array}{l} \frac{d}{dt} \langle u^{\prime}(t),v \rangle+ \langle \nabla u(t),\nabla v \rangle+ \langle Ku(t)+\lambda u^{\prime }(t),v \rangle \\ \qquad {}+\int _{\partial\Omega} ( \langle h(x,t),u(t)\rangle+g(x,t) ) v(x)\,dS_{x} \\ \quad = \langle|u|^{p-2}u,v \rangle+ \langle f(t),v \rangle \quad \mbox{for all } v\in H^{1}, \\ u(0)=u_{0} , \qquad u^{\prime}(0)=u_{1} .\end{array}\displaystyle \right . $$
(2.88)
Next, the uniqueness of a weak solution is obtained by using the well-known regularization procedure due to Lions. Theorem 2.5 is proved completely. □
Remark 2.2
In the case \(1< p\leq2\), \(f\in L^{2}(Q_{T})\), \(g\in W^{1}(0,T;L^{2} ( \partial\Omega ) )\), \(h\in W^{1}(0,T;L^{2} ( \partial\Omega\times\Omega ) )\), and \(( u_{0},u_{1} ) \in H^{1}\times L^{2}\), the integral inequality (2.29) leads to the following global estimation:
$$ S_{m}(t)\leq C_{T} ,\quad \forall m\in \mathbb{N} , \forall t\in[0,T] , \forall T>0 . $$
(2.89)
Then, by applying a similar argument to the proof of Theorem 2.4, we can obtain a global weak solution u of problem (1.1)-(1.3) satisfying
$$ u\in L^{\infty} \bigl( 0,T;H^{1} \bigr) , \qquad u_{t} \in L^{\infty} \bigl( 0,T;L^{2} \bigr) . $$
(2.90)
However, in the case \(1< p<2\), we do not imply that a weak solution obtained here belongs to \(C ( [0,T];H^{1} ) \cap C^{1} ( [0,T];L^{2} ) \). Furthermore, the uniqueness of a weak solution is also not asserted.