We are going to prove the existence and uniqueness for problems (1.1)–(1.3) by the Galerkin method and the compactness theorem in this section.
Let \({y_{i}(x)}\) be the orthonormal basis in \(L^{2}(\Omega)\) composed of the eigenfunctions of the eigenvalue problem
$$ \textstyle\begin{cases} \Delta y+\lambda y = 0, \\ y|_{\partial\Omega}=0 \end{cases} $$
(2.1)
corresponding to eigenvalue \(\lambda_{i}\) (\(i=1,2,\ldots\)).
Let
$$ u_{N}(x,t)=\sum_{i=1}^{N} \gamma_{Ni}(t)y_{i}(x) $$
(2.2)
be the Galerkin approximate solution for problem (1.1)–(1.3), where \(\gamma_{Ni}(t)\) are the undetermined functions and N is a natural number. Suppose that the initial value functions \(\varphi(x)\) may be expressed as
$$ \varphi(x)=\sum_{i=1}^{\infty} \mu_{i}y_{i}(x),\qquad \psi(x)=\sum _{i=1}^{\infty}\nu_{i}y_{i}(x), $$
(2.3)
where \(\mu_{i}\) and \(\nu_{i}\) (\(i=1,2,\ldots\)) are constants.
Substituting the approximate solution \(u_{N}(x,t)\) into Eq. (1.1), multiplying both sides by \(y_{s}(x)\), we obtain
$$ \bigl(u_{Ntt}+u_{Nt}-k \Delta u_{Nt}+ \Delta^{2} u_{N}, y_{s}\bigr)= \bigl(\Delta f(u_{N}), y_{s} \bigr),\quad s=1,2,\ldots, N, $$
(2.4)
where \((\cdot,\cdot)\) denotes the inner product of \(L^{2}(\Omega)\).
Substituting the approximate solution \(u_{N}(x,t)\) and the approximations
$$ \varphi_{N}(x)=\sum_{i=1}^{N} \mu_{i}y_{i}(x),\qquad \psi_{N}(x)=\sum _{i=1}^{N}\nu_{i}y_{i}(x) $$
of the initial value functions into the initial condition (1.3), we get
$$ \gamma_{Ns}(0)=\mu_{s},\qquad \dot{ \gamma}_{Ns}(0)=\nu_{s}, \quad s=1, 2, \ldots, N, $$
(2.5)
where \(\dot{\gamma}_{Ns}(t)=\frac{d}{dt}\gamma_{Ns}(t)\).
In order to prove the existence of the global generalized solution for problem (1.1)–(1.3), we make a series of estimations for the approximate solution \(u_{N}(x,t)\).
Lemma 2.1
Suppose that
\(\varphi\in H^{2}(\Omega)\)
and
\(\psi\in L^{2}(\Omega)\)
satisfy the boundary condition (1.2), \(f \in C^{1}(R)\), \(0\leq F(s)=\int_{0}^{s}f(\eta)\,d\eta\), and
\(|f'(s)|\leq C_{1}|s|^{2}+C_{2}\), where
\(C_{1}>0\), \(C_{2}>0\)
are constants. Then the following estimate holds:
$$ \big\| u_{N}(\cdot,t)\big\| ^{2}_{H^{2}}+ \big\| u_{Nt}(\cdot,t)\big\| ^{2} \leq C,\quad t\in[0,T], $$
(2.6)
where and in the sequel
\(C>0\)
is a constant which only depends on
T.
Proof
Let \(w_{n}\) be the unique solution of the problem
$$\begin{gathered} \Delta w_{n}=u_{n}, \\ w_{n} |_{\partial\Omega}=0.\end{gathered} $$
Substituting the approximate solution \(u_{N}(x,t)\) into Eq. (1.1), multiplying both sides by \(2w_{Nt}\), we obtain
$$ \bigl(u_{Ntt}+u_{Nt}-k \Delta u_{Nt}+ \Delta^{2}u_{N},2w_{Nt}\bigr)= \bigl(\Delta f(u_{N}),2w_{Nt} \bigr). $$
Integrating by parts with respect to x on Ω, we have
$$\begin{gathered} \frac{d}{dt} \biggl[\big\| \nabla w_{Nt}(\cdot,t)\big\| ^{2}+\big\| \nabla u_{N}(\cdot ,t)\big\| ^{2}+2 \int_{0}^{1}F(u_{n})\,dx \biggr] \\ \quad{}+2\big\| \nabla w_{Nt}(\cdot,t)\big\| ^{2}+2k \big\| u_{Nt}(\cdot,t)\big\| ^{2}\leq0.\end{gathered} $$
Hence, we know
$$\begin{aligned}& \|\nabla w_{Nt}\|\leq C, \end{aligned}$$
(2.7)
$$\begin{aligned}& \|\nabla u_{N}\|\leq C. \end{aligned}$$
(2.8)
By the Sobolev imbedding theorem, it follows from (2.7) and (2.8) that
$$\begin{aligned}& \|u_{N}\|_{L^{q}}\leq C, \quad\hbox{for any $q< \infty$ ($n=2$)}, \end{aligned}$$
(2.9)
$$\begin{aligned}& \|u_{N}\|_{L^{6}}\leq C\quad(n=3). \end{aligned}$$
(2.10)
Multiplying both sides of (2.4) by \(2\gamma_{Nst}(t)\), summing up for \(s=1,2,\ldots, N\), we have
$$ \bigl(u_{Ntt}+u_{Nt}-k \Delta u_{Nt}+ \Delta^{2}u_{N},2u_{Nt}\bigr)= \bigl(\Delta f(u_{N}),2u_{Nt} \bigr). $$
Integrating by parts with respect to x on Ω, we get
$$\begin{gathered} \frac{d}{dt} \bigl[\big\| u_{Nt}(\cdot,t)\big\| ^{2}+\big\| \Delta u_{N}(\cdot,t)\big\| ^{2} \bigr]+2\|u_{Nt} \|^{2}+2k \big\| \nabla u_{Nt}(\cdot,t)\big\| ^{2} \\ \quad\leq\frac{1}{4k}\big\| f'(u_{N})\nabla u_{N}(\cdot,t)\big\| ^{2}+k\big\| \nabla u_{Nt}(\cdot ,t) \big\| ^{2}.\end{gathered} $$
By \(|f'(s)|\leq C_{1}|u|^{2}+C_{2}\), hence
$$ \begin{aligned}[b]&\frac{d}{dt} \bigl[\big\| u_{Nt}(\cdot,t) \big\| ^{2}+\big\| \Delta u_{N}(\cdot,t)\big\| ^{2} \bigr]+2 \|u_{Nt}\|^{2}+k \big\| \nabla u_{Nt}(\cdot,t)\big\| ^{2} \\ &\quad\leq C \big\| u_{N}\big\| _{\infty}^{4}+C.\end{aligned} $$
(2.11)
On the other hand, by the Gagliardo–Nirenberg inequality, (2.9), and (2.10), we see
$$\begin{gathered} \|u\|_{\infty}\leq C\|\Delta u\|^{a}\|u\|_{q}^{1-a} \leq C\|\Delta u\|^{a}, \quad a=\frac{2}{q+2}\text{ ($n=2$)}, \\ \|u\|_{\infty}\leq C\|\Delta u\|^{1/2}\|u\|_{6}^{1/2} \leq C\|\Delta u\| ^{1/2}\quad(n=3).\end{gathered} $$
Therefore, by (2.11),
$$ \begin{aligned}[b]&\frac{d}{dt} \bigl[\big\| u_{Nt}(\cdot,t) \big\| ^{2}+\big\| \Delta u_{N}(\cdot,t)\big\| ^{2} \bigr]+2 \|u_{Nt}\|^{2}+k \big\| \nabla u_{Nt}(\cdot,t)\big\| ^{2} \\ &\quad\leq C\|\Delta u_{N}\|^{2}+C.\end{aligned} $$
(2.12)
Then, integrating (2.12) on \([0,t]\) and using the Gronwall inequality, we deduce
$$ \big\| u_{N}(\cdot,t)\big\| ^{2}_{H^{2}}+ \big\| u_{Nt}(\cdot,t)\big\| ^{2}\leq Ce^{T} \bigl(\| \varphi \|^{2}_{H^{2}}+\|\psi\|^{2}+1 \bigr),\quad t\in[0,T]. $$
(2.13)
Immediately, we get (2.6) from (2.13). The proof is completed. □
Lemma 2.2
Suppose that the conditions of Lemma
2.1
hold. If
\(f\in C^{3}(R)\), \(\varphi\in H^{4}(\Omega)\), \(\psi\in H^{2}(\Omega)\), and
\(|f''(s)|\leq C_{3}|s|+C_{4}\), \(|f'''(s)|\leq C\), then the approximate solution for problem (1.1)–(1.3) satisfies the following estimate:
$$ \big\| u_{N}(\cdot,t)\big\| ^{2}_{H^{4}}+ \big\| u_{Nt}(\cdot,t)\big\| ^{2}_{H^{2}}+\big\| u_{Ntt}( \cdot,t)\big\| ^{2}\leq C(T),\quad 0\leq t\leq T. $$
(2.14)
Proof
Multiplying both sides of (2.4) by \(2\lambda_{s}^{2}\gamma _{Nst}(t)\), summing up for \(s=1,2,\ldots, N\), we have
$$ \bigl(u_{Ntt}+u_{Nt}-k \Delta u_{Nt}+ \Delta^{2}u_{N},2\Delta^{2}u_{Nt}\bigr)= \bigl(\Delta f(u_{N}),2\Delta^{2}u_{Nt}\bigr). $$
Integrating by parts with respect to x, we get
$$\begin{gathered} \frac{d}{dt} \bigl(\big\| \Delta u_{Nt}(\cdot,t)\big\| ^{2}+\big\| \Delta^{2}u_{N}(\cdot ,t)\big\| ^{2} \bigr)+2\big\| \Delta u_{Nt}(\cdot,t)\big\| ^{2} +2k \big\| \nabla\Delta u_{Nt}( \cdot,t)\big\| ^{2} \\ \quad=2 \int_{\Omega}\Delta f(u_{N})\cdot\Delta^{2}u_{Nt} \,dx.\end{gathered} $$
On the other hand, we know
$$2 \int_{\Omega}\Delta f(u_{N})\cdot\Delta^{2} u_{Nt} \,dx=-2 \int_{\Omega}\nabla\Delta f(u_{N})\cdot\nabla\Delta u_{Nt} \,dx $$
and
$$\nabla\Delta f(u_{N})=f'''(u_{N})| \nabla u_{N}|^{2}\nabla u_{N}+3f''(u_{N}) \nabla u_{N}\Delta u_{N}+f'(u_{N}) \nabla\Delta u_{N}. $$
By (2.13), we know that \(\|u\|_{\infty}\leq C\), hence
$$\begin{gathered} 2 \int_{\Omega}\Delta f(u_{N})\cdot\Delta^{2}u_{Nt} \,dx \\ \quad\leq C \bigl(\big\| |\nabla u_{N}|^{3}\big\| ^{2}+\|\nabla u_{N}\Delta u_{N}\|^{2}+\|\nabla \Delta u_{N}\|^{2} \bigr) +k \big\| \nabla\Delta u_{Nt}( \cdot,t)\big\| ^{2}.\end{gathered} $$
Thus,
$$ \begin{aligned}[b] &\frac{d}{dt} \bigl(\big\| \Delta u_{Nt}(\cdot,t) \big\| ^{2}+\big\| \Delta^{2} u_{N}(\cdot,t)\big\| ^{2} \bigr)+2\big\| \Delta u_{Nt}(\cdot,t)\big\| ^{2} +k \big\| \nabla\Delta u_{Nt}(\cdot,t)\big\| ^{2} \\ &\quad\leq C \bigl(\big\| \nabla u_{N}(\cdot,t)\big\| _{L^{6}}^{2}+ \big\| \nabla u_{N}(\cdot,t)\big\| ^{2}_{L^{4}}\big\| \Delta u_{N}(\cdot,t)\big\| ^{2}_{L^{4}}+\big\| \nabla\Delta u_{N}(\cdot,t)\big\| ^{2} \bigr).\end{aligned} $$
(2.15)
By (2.13) and the Sobolev imbedding theorem, we see that
$$\begin{gathered} \|\nabla u_{N}\|_{L^{q}}\leq C, \quad\hbox{for any $q< \infty$ ($n=2$)}, \\ \|\nabla u_{N}\|_{L^{6}}\leq C\quad(n=3).\end{gathered} $$
Using the Gagliardo–Nirenberg inequality, we conclude
$$\begin{gathered}\begin{aligned} \big\| \Delta u_{N}(\cdot,t)\big\| _{L^{4}}^{2}\leq{}& C\big\| \Delta^{2} u_{N}(\cdot,t)\big\| ^{a}\big\| \nabla u_{N}(\cdot,t)\big\| _{L^{q}}^{1-a} \\ \leq{}& C\big\| \Delta^{2}u_{N}(\cdot,t)\big\| +C,\quad\hbox{where } a= \frac {q+4}{4+4q}\text{ ($n=2$)},\end{aligned} \\ \big\| \Delta u_{N}(\cdot,t)\big\| _{L^{4}}^{2}\leq C\big\| \Delta^{2} u_{N}(\cdot,t)\big\| ^{3/8}\big\| \nabla u_{N}(\cdot,t)\big\| _{L^{6}}^{5/8}\leq C\big\| \Delta^{2} u_{N}(\cdot,t)\big\| +C\quad (n=3).\end{gathered} $$
On the other hand, by boundary conditions (1.2), we obtain
$$\|\nabla\Delta u_{N}\|^{2}\leq C\|\Delta^{2}u_{N} \|^{2}. $$
Substituting the above inequalities into (2.15), we get
$$\frac{d}{dt} \bigl(\big\| \Delta u_{Nt}(\cdot,t)\big\| ^{2}+\big\| \Delta^{2}u_{N}(\cdot,t)\big\| ^{2} \bigr)\leq C(T)+\big\| \Delta^{2} u_{N}(\cdot,t)\big\| ^{2}. $$
Integrating the above inequality, and using the Gronwall inequality, we have
$$ \big\| \Delta u_{Nt}(\cdot,t)\big\| ^{2}+\big\| \Delta^{2}u_{N}(\cdot,t)\big\| ^{2}\leq C(T) \bigl(\| \varphi\|_{H^{4}}^{2}+\|\psi\|_{H^{2}}^{2}+1 \bigr),\quad t\in[0,T]. $$
(2.16)
Similarly, multiplying both sides of (2.4) by \(\gamma_{Nstt}(t)\), summing up for \(s=1,2,\ldots, N\), we deduce
$$\bigl(u_{Ntt}+u_{Nt}-k \Delta u_{Nt}+ \Delta^{2}u_{N}, u_{Ntt}\bigr)=\bigl(\Delta f(u_{N}), u_{Ntt}\bigr). $$
Integrating by parts with respect to x and using the Cauchy inequality, we have
$$\begin{gathered} \big\| u_{Ntt}(\cdot,t)\big\| _{L^{2}}^{2} \\ \quad=- \int_{\Omega}u_{Nt}u_{Ntt}\,dx+k \int_{\Omega}\Delta u_{Nt}u_{Ntt}\,dx - \int_{\Omega}\Delta^{2} u_{N}u_{Ntt} \,dx+ \int_{\Omega}\Delta f(u_{N})u_{Ntt}\,dx \\ \quad\leq 2\big\| u_{Nt}(\cdot,t)\big\| ^{2}+2k^{2}\big\| \Delta u_{Nt}(\cdot,t)\big\| ^{2}+2\big\| \Delta^{2}u_{N}( \cdot,t)\big\| ^{2} \\ \qquad{}+2\big\| \Delta f(u_{N})\big\| ^{2}+\frac{1}{2} \big\| u_{Ntt}(\cdot,t)\big\| ^{2}.\end{gathered} $$
Therefore, we conclude
$$ \big\| u_{Ntt}(\cdot,t)\big\| _{L^{2}}^{2}\leq C(T),\quad t\in[0,T]. $$
(2.17)
Immediately, we get (2.14) from (2.16) and (2.17). This completes the proof. □
Theorem 2.1
Suppose that
\(\varphi\in H^{4}(\Omega)\)
and
\(\psi\in H^{2}(\Omega)\)
satisfy the boundary conditions (1.2), \(f\in C^{3}(R)\), \(0\leq F(s)=\int_{0}^{s}f(\eta)\,d\eta\), \(|f'(s)|\leq C_{1}|s|^{2}+C_{2}\), \(|f''(s)|\leq C_{3}|s|+C_{4}\), and
\(|f'''(s)|\leq C\). Then problem (1.1)–(1.3) has a unique global generalized solution
$$ u\in C \bigl([0,T];H^{4}(\Omega) \bigr)\cap C^{1} \bigl([0,T];H^{2}(\Omega) \bigr)\cap C^{2} \bigl([0,T];L^{2}(\Omega) \bigr). $$
(2.18)
Proof
From (2.14) we know that \(u_{N}\in C ([0,T];H^{4}(\Omega) )\), \(u_{Nt}\in C ([0,T];H^{2}(\Omega) )\), \(u_{Ntt}\in C ([0,T];L^{2}(\Omega) )\). Using the Sobolev imbedding theorem, we have \(D^{k}u_{N}\in C ([0,T]\times\Omega )\), \(0\leq k\leq2\). It follows from the above two relations and the Ascoli–Arzelá theorem that there exist a function \(u(x,t)\) and a subsequence of \({u_{N}(x,t)}\), still denoted by \({u_{N}(x,t)}\), such that as \(N\rightarrow\infty\), \({u_{N}(x,t)}\) uniformly converges to \(u(x,t)\) in \([0,T]\times\Omega\). The corresponding subsequence of \(\Delta u_{N}(x,t)\) also uniformly converges to \(\Delta u(x,t)\) in \([0,T]\times\Omega\). According to the compactness theorem, the subsequence \(D^{k}u_{N}(x,t)\) (\(0\leq k\leq4\)), \(D^{k}u_{Nt}(x,t)\) (\(0\leq k\leq2\)), and \(u_{Ntt}(x,t)\) weakly converge to \(D^{k}u(x,t)\) (\(0\leq k\leq4\)), \(D^{k}u_{t}(x,t)\) (\(0\leq k\leq2\)), and \(u_{tt}(x,t)\) in \(L^{2} ([0,T]\times\Omega )\), respectively. Hence, we know that \(u(x,t)\) satisfies (2.18). Therefore \(u(x,t)\) is the generalized solution for problem (1.1)–(1.3). It is easy to prove the uniqueness of the solutions for problem (1.1)–(1.3). This completes the proof of the theorem. □