In this section, we prove the existence of global weak solutions to problem (1.1)–(1.3) with subcritical energy \(E(0)< d\).
Definition 4.1
Function \(u = u(x,t)\) is called a global weak solution to problem (1.1)–(1.3) on \(Q_{T}=\Omega \times [0,T)\). If \(u(x,t)\in L^{\infty } (0,T; H_{0}^{M}(\Omega ) )\) with \(u_{t}(x,t)\in L^{\infty } (0,T; H_{0}^{M}(\Omega ) ) \) satisfies:
-
(i)
for any \(v(x)\in H_{0}^{M}(\Omega )\), \(t\in [0,T)\),
$$\begin{aligned} & \bigl(u(x,t),v(x) \bigr)+ \bigl(\nabla ^{M}u(x,t),\nabla ^{M}v(x) \bigr) + \int _{0}^{t} \bigl(\nabla ^{M}u(x, \tau ),\nabla ^{M}v(x) \bigr)\,d\tau \\ &\quad =\bigl(u_{0}(x),v(x)\bigr)+\bigl(\nabla ^{M}u_{0}(x), \nabla ^{M}v(x)\bigr)+ \int _{0}^{t} \bigl(f\bigl(u(x,\tau ) \bigr),v(x) \bigr)\,d\tau ; \end{aligned}$$
(4.1)
-
(ii)
\(u(x,0)= u_{0}(x)\) in \(H_{0}^{M}(\Omega )\).
Theorem 4.1
Let \(f(u)=a|u|^{p-1}u\) and condition (H) hold. If \(E(0)< d\), then problem (1.1)–(1.3) admits a global weak solution as long as \(u_{0}(x)\in W\). Moreover, for all \(t\in [0,T)\), \(u(x,t)\in W\).
Proof
Let \(\omega _{j}(x)\) (\(j=1,2,3,\ldots \)) be the eigenfunctions of the problem
$$ (-1)^{M}\Delta ^{M}\omega _{j}=\lambda _{j}\omega _{j},\qquad D^{\gamma } \omega _{j}|_{\partial \Omega }=0\quad \bigl(0\leq \vert \gamma \vert \leq M-1 \bigr). $$
Then \(\{\omega _{j}(x) \}_{j=1}^{+\infty }\) is a complete orthonormal system in \(L^{2}(\Omega )\), \(\{\omega _{j}(x) \}_{j=1}^{+\infty }\) is also a complete orthogonal system in \(H_{0}^{M}(\Omega )\) (see [4] and [30]). We construct an approximate solution to problem (1.1)–(1.3) as follows:
$$ u_{m}(x,t)=\sum_{j=1}^{m}g_{jm}(t) \omega _{j}(x),\quad m=1,2,3, \ldots . $$
By the Galerkin method, \(u_{m}(x,t)\) (\(m=1,2,3,\ldots \)) satisfies the following initial value problem of nonlinear ordinary differential system:
$$\begin{aligned} & (u_{mt},\omega _{j} ) + \bigl((-1)^{M} \Delta ^{M}u_{m}, \omega _{j} \bigr) + \bigl((-1)^{M}\Delta ^{M}u_{mt},\omega _{j} \bigr) = \bigl(f(u_{m}),\omega _{j} \bigr), \end{aligned}$$
(4.2)
$$\begin{aligned} & g_{jm}(0)=a_{jm}\quad (j=1,2,\ldots ,m). \end{aligned}$$
(4.3)
Since \(u_{0}(x)\in H_{0}^{M}(\Omega )\), \(\{\omega _{j}(x) \}_{j=1}^{+\infty }\) construct a complete orthogonal system in \(H_{0}^{M}(\Omega )\), we can choose \(a_{jm}\) (\(j=1,2,\ldots ,m\)) such that, when \(m\rightarrow +\infty \),
$$ u_{m}(x,0)=\sum_{j=1}^{m}a_{jm} \omega _{j}(x) \text{ is strongly convergent to } u_{0}(x) \text{ in } H_{0}^{M}( \Omega ). $$
(4.4)
Multiplying both sides of (4.2) by \(g_{jm}^{\prime }(t)\) and summing up from 1 to m with respect to j, we obtain
$$\begin{aligned} & \Biggl(u_{mt},\sum_{j=1}^{m}g_{jm}^{\prime }(t) \omega _{j}(x) \Biggr) + \Biggl((-1)^{M}\Delta ^{M}u_{mt},\sum_{j=1}^{m}g_{jm}^{\prime }(t) \omega _{j}(x) \Biggr) \\ &\qquad {}+ \Biggl((-1)^{M}\Delta ^{M}u_{m},\sum_{j=1}^{m}g_{jm}^{\prime }(t) \omega _{j}(x) \Biggr) \\ &\quad = \Biggl(f(u_{m}),\sum_{j=1}^{m}g_{jm}^{\prime }(t) \omega _{j}(x) \Biggr). \end{aligned}$$
Since \(u_{mt}(x,t)=\sum_{j=1}^{m}g_{jm}^{\prime }(t)\omega _{j}(x)\), that is,
$$ (u_{mt},u_{mt} ) + \bigl((-1)^{M}\Delta ^{M}u_{m},u_{mt} \bigr) + \bigl((-1)^{M}\Delta ^{M}u_{mt},u_{mt} \bigr) = \bigl(f(u_{m}),u_{mt} \bigr). $$
(4.5)
Integrating by parts with respect to x in (4.5), we get
$$ \bigl\Vert u_{mt}(x,t) \bigr\Vert ^{2} + \bigl\Vert \nabla ^{M}u_{mt}(x,t) \bigr\Vert ^{2} + \frac{1}{2}\frac{d}{dt} \bigl\Vert \nabla ^{M}u_{m}(x,t) \bigr\Vert ^{2} = \frac{d}{dt} \int _{\Omega }F\bigl(u_{m}(x,t)\bigr)\,dx, $$
where \(F(u_{m})=\int _{0}^{u_{m}}f(s)\,ds\), that is,
$$ \bigl\Vert u_{mt}(x,t) \bigr\Vert ^{2} + \bigl\Vert \nabla ^{M}u_{mt}(x,t) \bigr\Vert ^{2} + \frac{d}{dt} \biggl(\frac{1}{2} \bigl\Vert \nabla ^{M}u_{m}(x,t) \bigr\Vert ^{2}- \int _{\Omega }F\bigl(u_{m}(x,t)\bigr)\,dx \biggr)=0. $$
(4.6)
By integrating (4.6) with respect to t over \([0,t)\) and combining
$$ E_{m}(t)=\frac{1}{2} \bigl\Vert \nabla ^{M}u_{m}(x,t) \bigr\Vert ^{2} - \int _{ \Omega }F \bigl(u_{m}(x,t) \bigr)\,dx $$
with
$$ E_{m}(0)=\frac{1}{2} \bigl\Vert \nabla ^{M}u_{m}(x,0) \bigr\Vert ^{2} - \int _{ \Omega }F \bigl(u_{m}(x,0) \bigr)\,dx, $$
we can get that
$$ \int _{0}^{t} \bigl\Vert u_{m\tau }(x, \tau ) \bigr\Vert ^{2}\,d\tau + \int _{0}^{t} \bigl\Vert \nabla ^{M}u_{m\tau }(x,\tau ) \bigr\Vert ^{2}\,d\tau +E_{m}(t)=E_{m}(0); $$
it concludes that
$$ E_{m}(t)\leq E_{m}(0). $$
By (4.4), condition (H), and the Sobolev imbedding theorem, it shows that when \(m\rightarrow +\infty \),
$$\begin{aligned} & \bigl\Vert \nabla ^{M}u_{m}(x,0) \bigr\Vert \rightarrow \bigl\Vert \nabla ^{M}u_{0}(x) \bigr\Vert , \\ & \bigl\Vert u_{m}(x,0) \bigr\Vert _{p+1}^{p+1} \rightarrow \bigl\Vert u_{0}(x) \bigr\Vert _{p+1}^{p+1}. \end{aligned}$$
From \(u_{0}(x)\in W\), we get that \(I(u_{0})>0\) and \(J(u_{0})< d\) or \(\|\nabla ^{M}u_{0}(x) \|=0\). If \(I(u_{0})>0\), \(J(u_{0})< d\), when \(m\rightarrow +\infty \), we have \(I (u_{m}(x,0) )\rightarrow I (u_{0}(x) )\), \(J (u_{m}(x,0) )\rightarrow J (u_{0}(x) ) \). Then, for sufficiently large m, the inequalities \(I (u_{m}(x,0) )>0\) and \(J (u_{m}(x,0) )< d\) hold, that is, \(u_{m}(x,0)\in W\).
If \(\|\nabla ^{M}u_{0}(x) \|=0\), when \(m\rightarrow +\infty \), we see that \(\|\nabla ^{M}u_{m}(x,0) \|\rightarrow \|\nabla ^{M}u_{0}(x) \|=0\). Then, for sufficiently large m, \(u_{m}(x,0)\in W\).
In what follows we will prove that, for sufficiently large m and for \(t\in [0,T)\), \(u_{m}(x,t)\in W\). If the above conclusion is wrong, there is sufficiently large m and \(t_{0}=t_{0}(m)\in (0,T)\) such that \(u_{m}(x,t_{0})\in \partial W\). From \(u_{m}(x,t_{0})\in \partial W\), it concludes that
$$ I \bigl(u_{m}(x,t_{0}) \bigr)=0,\qquad \bigl\Vert \nabla ^{M}u_{m}(x,t_{0}) \bigr\Vert \neq 0\quad \text{or}\quad J \bigl(u_{m}(x,t_{0}) \bigr)=d. $$
By \(f(u)=a|u|^{p-1}u\) and \(f(u)u\geq 0\), we can get
$$ 0\leq F(u)\leq \frac{a}{p+1} \vert u \vert ^{p+1} \quad \biggl(F(u)= \int _{0}^{u}f(s)\,ds \biggr), $$
therefore,
$$ \int _{\Omega }F(u)\,dx\leq \frac{a}{p+1} \Vert u \Vert _{p+1}^{p+1}, $$
that is,
$$ - \int _{\Omega }F(u)\,dx\geq -\frac{a}{p+1} \Vert u \Vert _{p+1}^{p+1}. $$
(4.7)
By a direct calculation, we know that
$$\begin{aligned} & \biggl\vert \int _{\Omega }F \bigl(u_{m}(x,0) \bigr)\,dx- \int _{\Omega }F \bigl(u_{0}(x) \bigr)\,dx \biggr\vert \\ &\quad = \biggl\vert \int _{\Omega }f(\varphi _{m}) \bigl(u_{m}(x,0)-u_{0}(x) \bigr)\,dx \biggr\vert \\ &\quad \leq \bigl\Vert f(\varphi _{m}) \bigr\Vert _{\frac{p+1}{p}} \bigl\Vert u_{m}(x,0)-u_{0}(x) \bigr\Vert _{p+1}, \end{aligned}$$
where \(\varphi _{m}=u_{0}(x)+\theta (u_{m}(x,0)-u_{0}(x))\), \(0<\theta <1\).
From \(f(u)=a|u|^{p-1}u\), we obtain
$$ \bigl\Vert f(\varphi _{m}) \bigr\Vert _{\frac{p+1}{p}} \bigl\Vert u_{m}(x,0)-u_{0}(x) \bigr\Vert _{p+1} = \bigl\Vert a \vert \varphi _{m} \vert ^{p} \bigr\Vert _{\frac{p+1}{p}} \bigl\Vert u_{m}(x,0)-u_{0}(x) \bigr\Vert _{p+1}. $$
From (4.4), we can get \(\|u_{m}(x,0)-u_{0}(x) \|_{H_{0}^{M}(\Omega )}\rightarrow 0\) as \(m\rightarrow +\infty \). By the Sobolev imbedding theorem and condition (H), it concludes that \(\|u_{m}(x,0)-u_{0}(x) \|_{p+1}\rightarrow 0\) as \(m\rightarrow +\infty \) and \(\|u_{m}(x,0) \|_{p+1}\rightarrow \|u_{0}(x) \|_{p+1}\) as \(m\rightarrow +\infty \).
By the Sobolev imbedding theorem and condition (H), \(\|a|\varphi _{m}|^{p} \|_{\frac{p+1}{p}}\) is bounded with respect to m. Therefore, when \(m\rightarrow +\infty \), we derive
$$ \biggl\vert \int _{\Omega }F \bigl(u_{m}(x,0) \bigr)\,dx- \int _{\Omega }F \bigl(u_{0}(x) \bigr)\,dx \biggr\vert \rightarrow 0, $$
that is,
$$ \int _{\Omega }F \bigl(u_{m}(x,0) \bigr)\,dx\rightarrow \int _{\Omega }F \bigl(u_{0}(x) \bigr)\,dx\quad \text{as $m\rightarrow +\infty $}. $$
(4.8)
From (4.8), it shows that
$$\begin{aligned} E_{m}(0)&=\frac{1}{2} \bigl\Vert \nabla ^{M}u_{m}(x,0) \bigr\Vert ^{2} - \int _{ \Omega }F \bigl(u_{m}(x,0) \bigr)\,dx \\ &\rightarrow \frac{1}{2} \bigl\Vert \nabla ^{M}u_{0}(x) \bigr\Vert ^{2} - \int _{ \Omega }F \bigl(u_{0}(x) \bigr)\,dx \quad \text{as $m\rightarrow +\infty $} \\ &=E(0). \end{aligned}$$
By \(E(0)< d\), we get that \(E_{m}(0)< d\) for sufficiently large m. Hence, for \(t\in [0,T)\) and for sufficiently large m, we can get
$$\begin{aligned} & \int _{0}^{t} \bigl\Vert u_{m\tau }(x, \tau ) \bigr\Vert ^{2}\,d\tau +J \bigl(u_{m}(x,t) \bigr) + \int _{0}^{t} \bigl\Vert \nabla ^{M}u_{m\tau }(x,\tau ) \bigr\Vert ^{2}\,d\tau \\ &\quad = \int _{0}^{t} \bigl\Vert u_{m\tau }(x, \tau ) \bigr\Vert ^{2}\,d\tau +\frac{1}{2} \bigl\Vert \nabla ^{M}u_{m}(x,t) \bigr\Vert ^{2} \\ &\qquad {}- \frac{a}{p+1} \bigl\Vert u_{m}(x,t) \bigr\Vert _{p+1}^{p+1} + \int _{0}^{t} \bigl\Vert \nabla ^{M}u_{m\tau }(x,\tau ) \bigr\Vert ^{2}\,d\tau \\ &\quad \leq \int _{0}^{t} \bigl\Vert u_{m\tau }(x, \tau ) \bigr\Vert ^{2}\,d\tau + \frac{1}{2} \bigl\Vert \nabla ^{M}u_{m}(x,t) \bigr\Vert ^{2} \\ &\qquad {}- \int _{\Omega }F\bigl(u_{m}(x,t)\bigr)\,dx + \int _{0}^{t} \bigl\Vert \nabla ^{M}u_{m\tau }(x,\tau ) \bigr\Vert ^{2}\,d\tau \\ &\quad =\frac{1}{2} \bigl\Vert \nabla ^{M}u_{m}(x,0) \bigr\Vert ^{2}- \int _{\Omega }F \bigl(u_{m}(x,0) \bigr)\,dx \\ &\quad =E_{m}(0)< d. \end{aligned}$$
(4.9)
In the above conclusion, inequality (4.7) has been used.
From inequality (4.9) we can get
$$ J \bigl(u_{m}(x,t_{0}) \bigr)< d. $$
This is a contradiction with the definition of d in (1.7). Since \(u_{m}(x,t_{0})\in H_{0}^{M}(\Omega )\), \(I (u_{m}(x,t_{0}) )=0\), \(\|\nabla ^{M}u_{m}(x,t_{0}) \|\neq 0\), the inequality \(J (u_{m}(x,t_{0}) )\geq d\) should hold. Hence \(I (u_{m}(x,t_{0}) )=0\) and \(\|\nabla ^{M}u_{m}(x,t_{0}) \|\neq 0\) do not hold. On the other hand, by inequality (4.9), \(J (u_{m}(x,t_{0}) )=d\) is impossible. Therefore, for sufficiently large m and for \(t\in [0,T)\), \(u_{m}(x,t)\in W\), we obtain that \(I (u_{m}(x,t) )>0\) holds.
Substituting
$$ J(u_{m})=\frac{p-1}{2(p+1)} \bigl\Vert \nabla ^{M}u_{m} \bigr\Vert ^{2} + \frac{1}{p+1}I(u_{m}) $$
into inequality (4.9), we get
$$\begin{aligned} & \int _{0}^{t} \bigl\Vert u_{m\tau }(x, \tau ) \bigr\Vert ^{2}\,d\tau + \frac{p-1}{2(p+1)} \bigl\Vert \nabla ^{M}u_{m} \bigr\Vert ^{2} \\ &\quad {}+\frac{1}{p+1}I(u_{m}) + \int _{0}^{t} \bigl\Vert \nabla ^{M}u_{m\tau }(x, \tau ) \bigr\Vert ^{2}\,d\tau < d,\quad t\in [0,T). \end{aligned}$$
By \(I (u_{m}(x,t) )>0\) for \(t\in [0,T)\), it shows that
$$ \textstyle\begin{cases} \int _{0}^{t} \Vert u_{m\tau }(x, \tau ) \Vert ^{2}\,d\tau < d,\quad t\in [0,T), \\ \frac{p-1}{2(p+1)} \Vert \nabla ^{M}u_{m}(x,t) \Vert ^{2}< d,\quad t\in [0,T), \\ \int _{0}^{t} \Vert \nabla ^{M}u_{m\tau }(x,\tau ) \Vert ^{2}\,d\tau < d,\quad t\in [0,T). \end{cases} $$
(4.10)
Therefore, \(\{u_{m}(x,t) \}_{m=1}^{+\infty }\) is bounded in \(L^{\infty }(0,T;H_{0}^{M}(\Omega )\cap L^{p+1}(\Omega ))\) (p satisfies condition (H)) and \(\{u_{mt}(x,t) \}\) is bounded in \(L^{2}(0,T;H_{0}^{M}(\Omega ))\). By \(f(u)=a|u|^{p-1}u\), it shows that \(\{f(u_{m}) \}_{m=1}^{+\infty }\) is bounded in \(L^{\infty } (0,T;L^{q}(\Omega ) )\) (\(q=\frac{p+1}{p}\)).
Since \(\int _{\Omega } |f(u_{m}) |^{2}\,dx=a^{2}\int _{ \Omega } |u_{m} |^{2p}\,dx\), from the second inequality in (4.10), we can get that \(\|u_{m} \|_{H_{0}^{M}(\Omega )}\) is uniformly bounded with respect to m. By condition (H) and the Sobolev imbedding theorem, \(\int _{\Omega } |u_{m} |^{2p}\,dx\) is uniformly bounded with respect to m, it concludes that \(\|f(u_{m}) \|\) is uniformly bounded with respect to m.
Since \(\{\omega _{j}(x) \}_{j=1}^{+\infty }\) is a complete orthonormal system in \(L^{2}(\Omega )\), \(\|\omega _{j}(x) \|_{L^{2}(\Omega )}= \|\omega _{j}(x) \|=1\) (\(j=1,2,3,\ldots \)), then \(| (f(u_{m}),\omega _{j}(x) ) |\leq \|f(u_{m}) \| \cdot \|\omega _{j}(x) \|= \|f(u_{m}) \|\) is uniformly bounded with respect to m. Hence, for \(t\in [0,T)\), \(| (f(u_{m}),\omega _{j}(x) ) |\) (\(j=1,2,\ldots ,m\)) are uniformly bounded with respect to m.
By the theory of ordinary differential system, there exists a global solution
$$ u_{m}(x,t)=\sum_{j=1}^{m}g_{jm}(t) \omega _{j}(x) $$
to problem (4.2)–(4.3) on \([0,T)\). Therefore, by the compactness principle, there exist functions \(u(x,t)\), \(X(x,t)\) and the subsequence of \(\{u_{m}(x,t) \}_{m=1}^{+\infty }\) (still denoted by \(\{u_{m}(x,t) \}_{m=1}^{+\infty } \)) such that when \(m\rightarrow +\infty \), \(u_{m}(x,t)\rightarrow u(x,t)\) in \(L^{\infty } (0,T;H_{0}^{M}(\Omega )\cap L^{p+1}(\Omega ) )\) weakly-star, \(f (u_{m}(x,t) )\rightarrow X(x,t)\) in \(L^{\infty } (0,T;L^{q}(\Omega ) )\) weakly-star (\(q= \frac{p+1}{p}\)).
By the first inequality of (4.10) it shows that \(\{u_{mt}(x,t) \}_{m=1}^{+\infty }\) is bounded in \(L^{2}(Q_{T})\). Since \(\|\nabla ^{M}u_{m} \|\) is an equivalent norm in \(H_{0}^{M}(\Omega )\), by the second inequality of (4.10) it concludes that \(\{u_{m}(x,t) \}_{m=1}^{+\infty }\) and \(\{\nabla u_{m}(x,t) \}_{m=1}^{+\infty }\) are bounded in \(L^{2}(Q_{T})\). Hence \(\{u_{m}(x,t) \}_{m=1}^{+\infty }\) is bounded in \(H^{1}(Q_{T})\) (\(Q_{T}= \Omega \times [0,T) \)).
Since \(H^{1}(Q_{T})\) can be compactly imbedded into \(L^{2}(Q_{T})\), there exists a subsequence of \(\{u_{m}(x,t) \}_{m=1}^{+\infty }\) (still denoted by \(\{u_{m}(x,t) \}_{m=1}^{+\infty }\)) such that when \(m\rightarrow +\infty \), \(u_{m}(x,t)\) is strongly convergent to \(u(x,t)\) in \(L^{2}(Q_{T})\) and \(u_{m}(x,t)\) is almost everywhere convergent to \(u(x,t)\) in \(Q_{T}\).
Since when \(m\rightarrow +\infty \), \(u_{m}(x,t)\) is almost everywhere convergent to \(u(x,t)\) in \(Q_{T}\) and \(\{f (u_{m}(x,t) ) \}_{m=1}^{+\infty }\) is bounded in \(L^{\infty }(0,T;L^{q}(\Omega ))\) (\(q=\frac{p+1}{p} \)). \(\{f (u_{m}(x,t) ) \}_{m=1}^{+\infty }\) is also bounded in \(L^{q}(Q_{T})\). By Lemma 1.3 in [11], we can get that \(f (u_{m}(x,t) )\) is weakly convergent to \(f (u(x,t) )\) in \(L^{q}(Q_{T})\) as \(m\rightarrow +\infty \).
For \(t\in [0,T)\), integrating (4.2) with respect to t and integrating by parts with respect to x, we get the following equality:
$$\begin{aligned} & \bigl(u_{m}(x,t),\omega _{j}(x) \bigr)+ \bigl(\nabla ^{M}u_{m}(x,t), \nabla ^{M}\omega _{j}(x) \bigr) + \int _{0}^{t} \bigl(\nabla ^{M}u_{m}(x, \tau ),\nabla ^{M}\omega _{j}(x) \bigr)\,d\tau \\ &\quad = \bigl(u_{m}(x,0),\omega _{j}(x) \bigr)+ \bigl( \nabla ^{M}u_{m}(x,0), \nabla ^{M}\omega _{j}(x) \bigr) \\ &\qquad {}+ \int _{0}^{t} \bigl(f\bigl(u_{m}(x, \tau )\bigr), \omega _{j}(x) \bigr)\,d\tau \quad (j=1,2,3,\ldots ). \end{aligned}$$
Let \(m\rightarrow +\infty \), notice that \(u_{m}(x,0)\) is strongly convergent to \(u_{0}(x)\) in \(H_{0}^{M}(\Omega )\), \(u_{m}(x,t)\rightarrow u(x,t)\) in \(L^{\infty } (0,T; H_{0}^{M}(\Omega ) )\) weakly-star, and \(f (u_{m}(x,t) )\) is weakly convergent to \(f (u(x,t) )\) in \(L^{q}(Q_{T})\). Hence, the following equality
$$\begin{aligned} & \bigl(u(x,t),\omega _{j}(x) \bigr)+ \bigl(\nabla ^{M}u(x,t), \nabla ^{M}\omega _{j}(x) \bigr) + \int _{0}^{t} \bigl(\nabla ^{M}u(x, \tau ),\nabla ^{M}\omega _{j}(x) \bigr)\,d\tau \\ &\quad = \bigl(u_{0}(x),\omega _{j}(x) \bigr)+ \bigl(\nabla ^{M}u_{0}(x),\nabla ^{M} \omega _{j}(x) \bigr) + \int _{0}^{t} \bigl(f\bigl(u(x,\tau )\bigr), \omega _{j}(x) \bigr)\,d\tau\quad (j=1,2,3,\ldots ) \end{aligned}$$
holds.
Because \(\{\omega _{j}(x) \}_{j=1}^{+\infty }\) is a complete orthogonal system in \(H_{0}^{M}(\Omega )\), for any \(\upsilon (x)\in H_{0}^{M}(\Omega )\), we get
$$\begin{aligned} & \bigl(u(x,t),\upsilon (x) \bigr)+ \bigl(\nabla ^{M}u(x,t), \nabla ^{M}\upsilon (x) \bigr) + \int _{0}^{t} \bigl(\nabla ^{M}u(x, \tau ), \nabla ^{M}\upsilon (x) \bigr)\,d\tau \\ &\quad = \bigl(u_{0}(x),\upsilon (x) \bigr)+ \bigl(\nabla ^{M}u_{0}(x),\nabla ^{M} \upsilon (x) \bigr) + \int _{0}^{t} \bigl(f\bigl(u(x,\tau )\bigr), \upsilon (x) \bigr)\,d\tau ,\quad t\in [0,T). \end{aligned}$$
In what follows we prove that the initial condition \(u(x,0)=u_{0}(x)\) holds.
By the second inequality in (4.10), when \(m\rightarrow +\infty \), \(u_{m}(x,t)\rightarrow u(x,t)\) in \(L^{\infty } (0,T; H_{0}^{M}(\Omega ) )\subset L^{2} (0,T;H_{0}^{M}( \Omega ) )\) weakly-star. By the third inequality in (4.10), when \(m\rightarrow +\infty \), \(u_{mt}(x,t)\rightarrow u_{t}(x,t)\) in \(L^{2} (0,T;H_{0}^{M}(\Omega ) )\) weakly-star. According to Lemma 1.2 in [11] it shows that
$$ u_{m}(x,t),\quad u(x,t)\in C \bigl(0,T;H_{0}^{M}( \Omega ) \bigr). $$
Therefore, when \(m\rightarrow +\infty \),
$$ u_{m}(x,0)\rightarrow u(x,0) \quad \text{in } H_{0}^{M}( \Omega ) \text{ weakly-star}. $$
From (4.4) we know that when \(m\rightarrow +\infty \), \(u_{m}(x,0)\) is strongly convergent to \(u_{0}(x)\) in \(H_{0}^{M}(\Omega )\), we can get that \(u(x,0)=u_{0}(x)\). According to the definition of global weak solution, \(u(x,t)\) is the global weak solution to problem (1.1)–(1.3) in \(Q_{T}\).
Lastly, we will prove that, for \(t\in [0,T)\), \(u(x,t)\in W\).
In what follows we use the method of contradiction to prove that, for \(t\in [0,T)\), \(u(x,t)\in W\). If there exists \(t_{0}\in (0,T)\) such that if \(t\in [0,t_{0})\), \(u(x,t)\in W= \{u(x) | u(x)\in H_{0}^{M}(\Omega ), I(u)>0, J(u)< d \}\cup \{0 \}\), but \(u(x,t_{0})\in \partial W\). From \(u(x,t_{0})\in \partial W\), we see that \(u(x,t_{0})\in H_{0}^{M}(\Omega )\), \(I (u(x,t_{0}) )=0\), and \(\|\nabla ^{M}u(x,t_{0}) \| \neq 0\) or \(J (u(x,t_{0}) )=d\).
Making \(L^{2}(\Omega )\) inner product by \(u_{t}(x,t)\) in both sides of equation
$$ u_{t}+(-1)^{M}\Delta ^{M}u_{t}+(-1)^{M} \Delta ^{M}u=f(u), $$
we get
$$ (u_{t},u_{t} )+ \bigl(u_{t},(-1)^{M} \Delta ^{M}u_{t} \bigr)+ \bigl(u_{t},(-1)^{M} \Delta ^{M}u \bigr) = \bigl(u_{t},f(u) \bigr). $$
Integrating by parts with respect to x, it concludes that
$$\begin{aligned} & (u_{t},u_{t} ) + \bigl(\nabla ^{M}u_{t}, \nabla ^{M}u_{t} \bigr) +\frac{1}{2} \frac{d}{dt} \bigl(\nabla ^{M}u,\nabla ^{M}u \bigr) \\ &\quad =\frac{d}{dt} \int _{\Omega }F(u)\,dx \quad \biggl(F(u)= \int _{0}^{u}f(s)\,ds \biggr). \end{aligned}$$
For all \(t\in [0,T)\), integrating from 0 to t with respect to t, we obtain
$$\begin{aligned} & [u_{t},u_{t} ] + \bigl[\nabla ^{M}u_{t}, \nabla ^{M}u_{t} \bigr] +\frac{1}{2} \bigl( \nabla ^{M}u(x,t),\nabla ^{M}u(x,t) \bigr) - \frac{1}{2} \bigl(\nabla ^{M}u(x,0),\nabla ^{M}u(x,0) \bigr) \\ &\quad = \int _{\Omega }F\bigl(u(x,t)\bigr)\,dx- \int _{\Omega }F\bigl(u(x,0)\bigr)\,dx. \end{aligned}$$
That is,
$$\begin{aligned} & [u_{t},u_{t} ] + \bigl[\nabla ^{M}u_{t}, \nabla ^{M}u_{t} \bigr] +\frac{1}{2} \bigl\Vert \nabla ^{M}u(x,t) \bigr\Vert ^{2} - \frac{1}{2} \bigl\Vert \nabla ^{M}u_{0}(x) \bigr\Vert ^{2} \\ &\quad = \int _{\Omega }F\bigl(u(x,t)\bigr)\,dx- \int _{\Omega }F\bigl(u_{0}(x)\bigr)\,dx, \end{aligned}$$
where \([u_{t},u_{t}]=\int _{0}^{t}(u_{\tau }(x,\tau ),u_{\tau }(x, \tau ))\,d\tau \), \([\nabla ^{M}u_{t},\nabla ^{M}u_{t}]=\int _{0}^{t}( \nabla ^{M}u_{\tau }(x,\tau ),\nabla ^{M}u_{\tau }(x,\tau ))\,d\tau \).
By some computations, we get
$$\begin{aligned} & [u_{t},u_{t} ] + \bigl[\nabla ^{M}u_{t}, \nabla ^{M}u_{t} \bigr] +\frac{1}{2} \bigl\Vert \nabla ^{M}u(x,t) \bigr\Vert ^{2} - \int _{\Omega }F\bigl(u(x,t)\bigr)\,dx \\ &\quad =\frac{1}{2} \bigl\Vert \nabla ^{M}u_{0}(x) \bigr\Vert ^{2}- \int _{\Omega }F\bigl(u_{0}(x)\bigr)\,dx. \end{aligned}$$
Combining
$$ E(t)=\frac{1}{2} \bigl\Vert \nabla ^{M}u(x,t) \bigr\Vert ^{2} - \int _{\Omega }F \bigl(u(x,t) \bigr)\,dx $$
with
$$ E(0)=\frac{1}{2} \bigl\Vert \nabla ^{M}u_{0}(x) \bigr\Vert ^{2} - \int _{\Omega }F \bigl(u_{0}(x) \bigr)\,dx, $$
we obtain
$$ [u_{t},u_{t} ] + \bigl[\nabla ^{M}u_{t}, \nabla ^{M}u_{t} \bigr] +E(t)=E(0),\quad t\in [0,T), $$
it concludes that
$$ E(t)\leq E(0),\quad t\in [0,T). $$
By
$$\begin{aligned}& E(t)=\frac{1}{2} \bigl\Vert \nabla ^{M}u(x,t) \bigr\Vert ^{2} - \int _{\Omega }F \bigl(u(x,t) \bigr)\,dx, \\& J \bigl(u(x,t) \bigr)=\frac{1}{2} \bigl\Vert \nabla ^{M}u(x,t) \bigr\Vert ^{2} - \frac{a}{p+1} \bigl\Vert u(x,t) \bigr\Vert _{p+1}^{p+1}, \end{aligned}$$
and inequality (4.7), we arrive at
$$ \frac{1}{2} \bigl\Vert \nabla ^{M}u \bigr\Vert ^{2} -\frac{a}{p+1} \Vert u \Vert _{p+1}^{p+1} \leq \frac{1}{2} \bigl\Vert \nabla ^{M}u \bigr\Vert ^{2} - \int _{\Omega }F(u)\,dx. $$
Therefore, for \(t\in [0,T)\), we have
$$\begin{aligned} J \bigl(u(x,t) \bigr)&= \frac{1}{2} \bigl\Vert \nabla ^{M}u(x,t) \bigr\Vert ^{2} -\frac{a}{p+1} \bigl\Vert u(x,t) \bigr\Vert _{p+1}^{p+1} \\ &\leq \frac{1}{2} \bigl\Vert \nabla ^{M}u(x,t) \bigr\Vert ^{2} - \int _{\Omega }F \bigl(u(x,t) \bigr)\,dx \\ &=E(t)\leq E(0)< d. \end{aligned}$$
(4.11)
By inequality (4.11) it shows that \(J (u(x,t_{0}) )=d\) is impossible. From the definition of d in (1.7), by \(u(x,t_{0})\in H_{0}^{M}(\Omega )\), \(I (u(x,t_{0}) )=0\), and \(\|\nabla ^{M}u(x,t_{0}) \|\neq 0\), we know that \(J (u(x,t_{0}) )\geq d\). On the other hand, from inequality (4.11), we see that \(J (u(x,t_{0}) )< d\). This is a contradiction! Therefore, \(I (u(x,t_{0}) )=0\) and \(\|\nabla ^{M}u(x,t_{0}) \|\neq 0\) do not hold. Summarizing the above discussion, we have proved that, for \(t\in [0,T)\), \(u(x,t)\in W\).
The proof of Theorem 4.1 is completed. □