Abstract
In this paper, we study the initial boundary value problem for the nonlocal parabolic equation with the Hardy–Littlewood–Sobolev critical exponent on a bounded domain. We are concerned with the long time behaviors of solutions when the initial energy is low, critical or high. More precisely, by using the modified potential well method, we obtain global existence and blow-up of solutions when the initial energy is low or critical, and it is proved that the global solutions are classical. Moreover, we obtain an upper bound of blow-up time for \(J_{\mu }(u_{0})<0\) and decay rate of \(H^{1}_{0}\) and \(L^{2}\)-norm of the global solutions. When the initial energy is high, we derive some sufficient conditions for global existence and blow-up of solutions. In addition, we are going to consider the asymptotic behavior of global solutions, which is similar to the Palais-Smale (PS for short) sequence of stationary equation.
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1 Introduction and Main Results
In this paper, we consider a nonlocal parabolic initial-boundary value problem:
where \(\Omega \) is a bounded domain in \({\mathbb {R}}^{N} (N\ge 3)\) and \(2^{*}_{\mu }=\frac{2N-\mu }{N-2}\) is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. Depending on suitable properties of the initial value \(u_{0}\), we are interested in the long time behaviors of solutions (global existence, blow-up in finite time) and asymptotic behavior of the global solutions.
The nonlocal parabolic equation of type (P) has important background arising from a variety of physical, chemical and biological problems. For example, problem (P) can be applied to nonlocal heat physics, where \(\left( \vert x\vert ^{-\mu }*|u|^{2^{*}_{\mu }}\right) |u|^{2^{*}_{\mu }-2}u\) represents the nonlocal source, and it can also be applied to the population model with nonlocal competition, where \(\left( \vert x\vert ^{-\mu }*|u|^{2^{*}_{\mu }}\right) |u|^{2^{*}_{\mu }-2}u\) models the individuals are competing not only with others at their own point in space but also with individual at other points in the domain (see [10, 18, 31, 35]), and so forth.
The following parabolic initial-boundary value problem:
has been extensively studied by many author with different methods. For example, critical-point theory by Ambrosetti-Rabinowitz [1], the potential well method which was constructed by Payne and Sattinger [32, 33], semigroup theory by Weissler [40, 41] and classical tools by Hoshino-Yamada [12] in a new functional analytic framework (or the monograph by Henry [11] for more detailed).
In particular, since Sattinger [33] constructed the so called stable set, the method of potential well was applied to study the existence of global solutions far and wide (see [7, 13,14,15,16, 29, 30, 38, 39] e.g.). Furthermore, Levine [19], Payne and Sattinger [32] considered blowing-up properties of solutions. When f(u) is local source term, i.e. \(f(u)=\vert u\vert ^{p-1}u\), it is well known that there exist choices of initial value \(u_{0}\) such that the homologous solutions global existence and the global solution tend to zero as \(t\rightarrow \infty \) and there exist choices of initial value \(u_{0}\) such that the homologous solutions blow-up. When the exponent is subcritical, i.e. \(1<p<\frac{N+2}{N-2}\), with the help of energy functional, Nehari functional and potential well method, there exists two invariant sets W (stable set) and V (unstable set), and the long time behavior of solutions for (1.1) with low energy initial value (the energy of initial value is smaller than the depth of potential well) was described. More detailed, if initial data \(u_{0}\) belongs to the stable set W, the associated solution is global, if initial data \(u_{0}\) belongs to the unstable set V, the associated solution blow-up in a finite time, and blow-up in infinite time does not occur in this case (see [14, 15]). Dickstein et al. [5] generalized the above results to the critical energy level initial data. When the initial data has high energy (the energy of initial value is larger than the depth of potential well), the situation is much more complicated, since the invariance of W and V are invalid and potential well method can not be used. In [8], by using the comparison principle and variational methods, Gazzola and Weth obtained the existence of global solution and blow-up in finite time of solutions with high energy initial value. When the exponent is critical, i.e. \(p=\frac{N+2}{N-2}\), by using the potential well method, Tan [36], Ikehata and Suzuki [14, 34] considered the problem (1.1), they established the existence of global solutions and blow-up of solutions in finite time, which depend on the initial value \(u_{0}\in H^{1}_{0}(\Omega )\). Moreover, the asymptotic behavior of global solutions was studied. In particular, we emphasize the blow-up of case of exponent subcritical is simpler than exponent critical, since the embedding of \(H^{1}_{0}(\Omega )\) into \(L^{p}(\Omega )\) is compact for \(p<\frac{2N}{N-2}\), while is non-compact for \(p=\frac{2N}{N-2}\). From the point of view of critical point theory, the compactness condition is a sufficient and necessary condition for the PS condition to hold. Moreover, it is also a necessary condition for the nontrivial solutions existence of the stationary equation of (1.1) under conditions that do not require the geometry of the domain from the point of view of elliptic problems. In particular, contrary to the subcritical case, global unbounded solutions may exist for critical case (see [28]).
When f(u) is of a nonlocal source, i.e. \(f(u)=(\vert x\vert ^{-(N-2)}*\vert u\vert ^{p})\vert u\vert ^{p-2}u\), Liu et al. considered the global existence and blow-up in finite time of solutions for problem (1.1) with \(1<p<\frac{N+2}{N-2}\) by using the potential well method. In [25], they obtained a sharp threshold for global existence and finite time blow-up of solutions with lower energy initial data. In [26], they extended the results to case of critical energy initial value and obtained the asymptotic behavior of solutions. Later, they also consider the case of high energy initial value and found a criteria for global existence and blow-up in finite time of solutions respectively. Moreover, the asymptotic profile to both solutions vanishing at infinity and blowing up in finite time was established.
However, to the best of our knowledge, the nonlocal parabolic equation with critical exponent has not been studied yet. Therefore, this paper aims to study the global existence and blow-up of solution on initial value with lower energy, critical energy and high energy for problem (P). The energy functional of problem (P) is defined by
By the Hardy–Littlewood–Sobolev inequality, \(J_{\mu }(u)\) is well defined in the Sobolev space \(H^{1}_{0}(\Omega )\). The equation corresponds to the \(L^{2}\) gradient flow associated of this energy functional. Then, along the flow generated by problem (P), we have
For more details, see Lemma 2.4 below.
For the Hardy–Littlewood–Sobolev critical exponent case, the corresponding functional \(J_{\mu }\) does not satisfy the Palais-Smale (PS for short) condition (or \((PS)_{c}\) condition). From the critical-point theory point of view, the (PS) condition plays an important role in the proof of the existence of critical points of \(J_{\mu }\), and that the stationary problem has solutions. However, by Brezis and Nirenberg [4], for \(\Omega \) be a bounded domain in \({\mathbb {R}}^{N}\), any \((PS)_{c}\) sequence for \(c<\frac{1}{N}S^{\frac{N}{2}}\) is relatively compact in \(H^{1}_{0}(\Omega )\), where S is the best constant for the Sobolev embedding \(H^{1}_{0}(\Omega )\hookrightarrow L^{2^{*}}(\Omega )\). Importantly, the Brezis-Nirenberg type critical problem for nonlinear Choquard equation was studied by Du, Gao and Yang in [6, 9], \(S_{H,L}\) and the minimax level was estimated in [9] and they classify the positive solutions of this equation in [6], where \(S_{H,L}\) is the best embedding constant in the sense of the Hardy–Littlewood–Sobolev inequality. In [2], Alves et al. study the singularly perturbed critical Choquard equation and establish the existence of ground states with constant coefficients. Moreover, they obtained the multiplicity of solution and the concentration behavior was characterized for perturbed problem.
Let us recall the well-known Hardy–Littlewood–Sobolev inequality, which plays a fundamental role throughout this paper.
Lemma 1.1
(Hardy–Littlewood–Sobolev inequality, see [21].) Let \(t,r>1\) and \(0<\mu <N\) with \(1/t+\mu /N+1/r=2\). For \({\bar{f}}\in L^{t}({\mathbb {R}}^{N})\) and \({\bar{h}}\in L^{r}({\mathbb {R}}^{N})\), there exists a sharp constant \(C(t,N,\mu ,r)\) independent of \({\bar{f}}\) and \({\bar{h}}\), such that
If \(t=r=\frac{2N}{2N-\mu }\), then
In this case, the equality in (1.4) holds if and only if \({\bar{f}}\equiv C{\bar{h}}\) and
for some \(A\in {\mathbb {C}}\), \(0\ne \gamma \in {\mathbb {R}}\) and \(a\in {\mathbb {R}}^{N}\).
From the Hardy–Littlewood–Sobolev inequality, for all \(u\in D^{1,2}({\mathbb R}^N)\), one has
where \(C(N,\mu )\) is defined as in Lemma 1.1 and hence we call \(2_{\mu }^{*}=\frac{2N-\mu }{N-2}\) the upper Hardy–Littlewood–Sobolev critical exponent. Denote best constant by
Lemma 1.2
[9, Lemma 1.2] The constant \(S_{H,L}\) defined in (1.5) is achieved if and only if
where \(C>0\) is a fixed constant, \(a\in {\mathbb {R}}^{N}\) and \(b\in (0,\infty )\) are parameters. Furthermore,
where S is the best Sobolev constant.
As in [42], let \(U(x)=\frac{[N(N-2)]\frac{N-2}{4}}{(1+\vert x\vert ^{2})^{\frac{N-2}{2}}}\) be a minimizer for S, then
is the unique positive minimizer for \(S_{H,L}\) that satisfies
and
Let
For every open subset \(\Omega \) of \({\mathbb {R}}^{N}\),
where \(S_{H,L}(\Omega )\) is never achieved except when \(\Omega ={\mathbb {R}}^{N}\), see [9, Lemma 1.3].
Following [32, 36], we define stable set and unstable set as follows
and
where \(I_{\mu }(u)\) is the Nehari functional for problem (P) defined by
Remark 1.3
-
(i)
If \(0<J_{\mu }(u)<m_{\mu }\) and \(I_{\mu }(u)\ge 0\), then,we have \(I_{\mu }(u)>0\). Indeed, if \(I_{\mu }(u)=0\), by the Hardy–Littlewood–Sobolev inequality, we have
$$\begin{aligned} \int _{\Omega }|\nabla u|^{2}dx=\int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy \le S^{-\frac{2N-\mu }{N-2}}_{H,L}\left( \int _{\Omega }|\nabla u|^{2}dx \right) ^{\frac{2N-\mu }{N-2}}, \end{aligned}$$which implies that \(\int _{\Omega }|\nabla u|^{2}dx\ge S^{\frac{2N-\mu }{N-\mu +2}}_{H,L}\). Furthermore, by the definition of \(J_{\mu }(u)\), we have \(J_{\mu }(u)\ge \frac{N-\mu +2}{2(2N-\mu )}S^{\frac{2N-\mu }{N-\mu +2}}_{H,L}=m_{\mu }\), a contradiction.
-
(ii)
If \(I_{\mu }(u)>0\), then \(J_{\mu }(u)>0\). Indeed, since \(I_{\mu }(u)>0\), we have
$$\begin{aligned} \int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy<\int _{\Omega }\vert \nabla u\vert ^{2}dx. \end{aligned}$$Furthermore, we can derive
$$\begin{aligned} J_{\mu }(u)= & {} \frac{1}{2}\int _{\Omega }\vert \nabla u\vert ^{2}dx -\frac{1}{22^{*}_{\mu }}\int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy\\> & {} \frac{2^{*}_{\mu }-1}{22^{*}_{\mu }}\int _{\Omega }\vert \nabla u\vert ^{2}dx\ge 0. \end{aligned}$$Thus, \(J_{\mu }(u)>0\).
Inspired by [36], we are going to investigate the critical parabolic equation with nonlocal interaction. By using the modified potential well method, one of the aims is to study the global existence, blow-up of solutions for problem (P) with lower energy initial value and decay rate of \(H^{1}_{0}\) and \(L^{2}\) norm for global solutions. Furthermore, we give an upper bound for blow-up time for \(J_{\mu }(u_{0})<0\). Moreover, we also consider global existence and blow-up of solutions for problem (P) with critical energy initial value and give some sufficient conditions for the existence of global and blow-up solutions with high energy initial value. Finally, we are going to consider the asymptotic behavior of global solutions, we shall prove that there exists a sequence \(\left\{ t_{n}\right\} \) such that the asymptotic behavior of \(u(x,t_{n})\) as \(t_{n}\rightarrow \infty \) is the PS sequence of stationary equation of problem (P).
Firstly, we state our main results about the global existence and blow-up of solutions with lower energy initial value as follows.
Theorem 1.4
If \(u_{0}\in H^{1}_{0}(\Omega )\) with \(J_{\mu }(u_{0})<m_{\mu }\), \(I_{\mu }(u_{0})>0\), then, problem (P) has a global weak solution \(u(x,t)\in L^{\infty }(0,T;H^{1}_{0}(\Omega ))\) with \(u_{t}\in L^{2}(0,T;L^{2}(\Omega ))\) and \(u(x,t)\in W\) for \(0\le t<\infty \).
Theorem 1.5
If \(u_{0}\in H^{1}_{0}(\Omega )\) with \(J_{\mu }(u)<m_{\mu }\), \(I_{\mu }(u)<0\), then, the weak solution u(x, t) of problem (P) blow-up in finite time. In particular, there exists a \(T>0\) such that
Moreover, for \(J_{\mu }(u_{0})<0\), an upper bound for blow-up time T is given by
Furthermore, we have the decay rate of the \(H^{1}_{0}\) and \(L^{2}\)-norm of the global solutions with lower energy initial value. Then, we state our main result of this as follows.
Theorem 1.6
Under the assumption in Theorem 1.4, for the global weak solution u(x, t) of problem (P), there exists \(\alpha _{1},\alpha _{2}>0\) such that
and
Under the existence of global solution, we now consider the regularity of the global weak solutions with lower energy initial value, by applying a nonlocal version of the Brezis-Kato estimate, we prove that the global solutions are classical for \(t\ge t_{0}>0\). The statement for more detailed as follows.
Theorem 1.7
Let u(x, t) be a global solution. Then, \(u\in L^{p}(\Omega \times [t_{0},\infty ))\) for every \(p\in [2,\frac{N}{N-\mu }\frac{2N}{N-2})\). In particular, u is a classical solution for \(t\ge t_{0}>0\).
With the help of modified potential well method as in [24], we further study the global existence and blow-up in finite time for the case of critical energy initial value, i.e. \(J_{\mu }(u_{0})= m_{\mu }\). Before state our main results of global existence and finite time blow-up of solutions, we give a remark.
Remark 1.8
If \(J_{\mu }(u)= m_{\mu }\), \(I_{\mu }(u)\ge 0\), then \(I_{\mu }(u)>0\). Indeed, we can know that the Hardy–Littlewood–Sobolev constant is not attained on a bounded domain, and hence \(E=\left\{ u\in H^{1}_{0}(\Omega )\,\ u \ \text {satisfies}\ -\Delta u=\left( \vert x\vert ^{-\mu }*|u|^{2^{*}_{\mu }}\right) |u|^{2^{*}_{\mu }-2}u\ \text {and}\ J_{\mu }(u)= m_{\mu } \right\} =\emptyset \) (see [9, Lemma 1.3]). However, the case \(J_{\mu }(u)=m_{\mu }\) and \(I_{\mu }(u)=0\) means means \(u\in E\), this is impossible.
Theorem 1.9
Let \(u_{0}(x)\in H^{1}_{0}(\Omega )\), \(J_{\mu }(u_{0})= m_{\mu }\). Then,(i) If \(I_{\mu }(u_{0})>0\), then the problem (P) has a global weak solution \(u\in L^{\infty }(0,T;H^{1}_{0}(\Omega ))\) with \(u_{t}\in L^{2}(0,T;L^{2}(\Omega ))\) and \(u(x,t)\in {\bar{W}}\) for \(0\le t<\infty \). Moreover, there exists \(\alpha _{1},\alpha _{2}>0\) such that
and
(ii) If \(I_{\mu }(u_{0})<0\), then the solutions of problem (P) blows up in finite time. In particular, there exists a \(T>0\) such that
In view of above results, for the case \(J_{\mu }(u_{0})\le m_{\mu }\), whether or not the solution for problem (P) exists globally is totally determined by the Nehari functional, and it is natural to ask what will happen when \(J_{\mu }(u_{0})>m_{\mu }\). However, since the invariance of W and V under the flow of (1.3) is invalid, potential well method can not be used for this case. To this end, we now introduce the following sets as in [8], define
and
Furthermore, for all \(d>m_{\mu }\), set
Next, we also introduce the following sets
Then, we can characterize the sets \({\mathcal {B}}\), \({\mathcal {G}}\) and \({\mathcal {G}}_{0}\), that is, to determine the global existence and blow-up of the solution of (P) whose initial value \(u_{0}\) in \(H^{1}_{0}(\Omega )\). Our main results for \(J_{\mu }(u_{0})>m_{\mu }\) are to show as follows.
Theorem 1.10
Assume that \(J_{\mu }(u_{0})>m_{\mu }\), then the following statements hold
-
(i)
If \(u_{0}\in {\mathcal {N}}_{+}\) and \(\Vert u_{0}\Vert _{2}\le \lambda _{J(u_{0})}\), then \(u_{0}\in {\mathcal {G}}_{0}\);
-
(ii)
If \(u_{0}\in {\mathcal {N}}_{-}\) and \(\Vert u_{0}\Vert _{2}\ge \Lambda _{J(u_{0})}\), then \(u_{0}\in {\mathcal {B}}\).
Theorem 1.11
For \(0<\mu <\min \left\{ 4,N\right\} \). If \(u_{0}\in H^{1}_{0}(\Omega )\) satisfies
where \(r_{\Omega }:=diam(\Omega )=\sup _{x,y\in \Omega }\vert x-y\vert <\infty \). Then, \(u_{0}\in {\mathcal {N}}_{-}\cap {\mathcal {B}}\).
Theorem 1.12
For any \(M>0\), there exists \(u_{M}\in {\mathcal {N}}_{-}\) such that \(J(u_{M})\ge M\) and \(u_{M}\in {\mathcal {B}}\).
Finially, we consider the asymptotic behavior of the global solutions, which is similar to the Palais-Smale (PS for short) sequence of stationary equation.
Theorem 1.13
Let \(u(x,t;u_{0})\) be a global solution of the problem (P) and uniformly bounded in \(H^{1}_{0}(\Omega )\) with respect to t. Then, for any subsequence \(t_{n}\rightarrow \infty \), there exists a stationary solution w such that \(u(x,t_{n};u_{0})\rightarrow w\) in \(H^{1}_{0}(\Omega )\).
Theorem 1.14
Let \(u(x,t;u_{0})\) be a global solution of the problem (P). Then, its \(\omega \)-limit contains a stationary solution w.
The rest of this paper is organized as follows. In Sect. 2, we give some notations and definitions, introduce potential well sets and prove local existence theorem of the problem (P) in subsection of Sect. 2. Next, we will give global existence and blow up of the problem (P) with lower energy initial value, critical energy initial value and high energy initial value in Sects. 3, 4, 5 respectively. In Sect. 6, we prove Theorems 1.13 and 1.14.
2 Preliminaries
In this section, let us first give some definitions of the weak solution, maximal existence time and finite time blow-up. And then we introduce some functions and notations. Final, we give local existence result of solutions for problem (P). Throughout this paper, we denote \(\Vert \cdot \Vert _{L^{q}(\Omega )}\) by \(\Vert \cdot \Vert _{q}\) for \(1\le q\le \infty \) and C is a constant that can change from one line to another.
2.1 Definitions
Definition 2.1
(Weak solution). We say that a function \(u=u(x,t)\) is a weak solution of problem (P) in \(Q_{T}:=\Omega \times (0,T)\) if and only if
and satisfies problem (P) in the distribution sense, that is
where \(u(x,0)=u_{0}(x)\in H^{1}_{0}(\Omega )\) and \((\cdot ,\cdot )\) denote the \(L^{2}(\Omega )\)-inner product.
Definition 2.2
(Maximal existence time). Let u(x, t) be a weak solution of problem (P). We define the maximum existence time \(T_{\max }\) of u(x, t) as follows:
-
(i)
if u(x, t) exists for all \(0\le t<\infty \), then \(T_{\max }=\infty \);
-
(ii)
if there exists \(t^{*}\in (0,\infty )\) such that u(x, t) exists for all \(0\le t<t^{*}\), but does not exist at \(t=t^{*}\), then \(T_{\max }=t^{*}\).
Definition 2.3
(Finite time blow-up). Let u(x, t) be a weak solution of problem (P). We say u(x, t) blow-up in finite time if the maximal existence time \(T_{\max }\) is finite and
Multiplying (P) by u and \(u_{t}\) respectively and then integrating over \(\Omega \), we can get
and
Then, by (2.3), we have the following Lemma.
Lemma 2.4
(Energy identity). For \(0<T\le \infty \) and let u(x, t) be a weak solution of problem (P) on [0, T) with initial value \(u_{0}\in H^{1}_{0}(\Omega )\). Then, \(J_{\mu }(u(t))\) is non-increasing with respect to t. More precisely,
for any \(0\le s\le t<T\).
2.2 Introduction of Potential Well
In this subsection, we shall introduce a class of potential wells for problem (P). Firstly, let us give some properties of \(J_{\mu }(u)\) and \(I_{\mu }(u)\).
Lemma 2.5
Let \(u\in H^{1}_{0}(\Omega )\backslash \{0\}\). Then,
-
(i)
\(\lim _{s\rightarrow 0}J_{\mu }(su)=0\) and \(\lim _{s\rightarrow +\infty }J_{\mu }(su)=-\infty \);
-
(ii)
there exists a unique \({\bar{s}}=s(u)>0\) such that
$$\begin{aligned} \frac{d}{d s}J_{\mu }(su)|_{s={\bar{s}}}=0; \end{aligned}$$(2.5) -
(iii)
\(J_{\mu }(su)\) is increasing on \(0\le s\le {\bar{s}}\), decreasing on \({\bar{s}}\le s\le +\infty \) and takes the maximum at \(s={\bar{s}}\).
-
(iv)
\(I_{\mu }(su)>0\) for \(0<s<{\bar{s}}\), \(I_{\mu }(su)<0\) for \({\bar{s}}<s<+\infty \) and \(I_{\mu }({\bar{s}} u)=0\).
Proof
- (i):
-
By the definition of \(J_{\mu }(u)\) in (1.2), we can get
$$\begin{aligned} J_{\mu }(su)= & {} \frac{1}{2}\int _{\Omega }|\nabla (su)|^{2}dx -\frac{1}{22^{*}_{\mu }} \int _{\Omega }\int _{\Omega }\frac{|su(x)|^{2^{*}_{\mu }}|su(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy\nonumber \\= & {} \frac{s^{2}}{2}\int _{\Omega }|\nabla u|^{2}dx -\frac{s^{22^{*}_{\mu }}}{22^{*}_{\mu }} \int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy, \end{aligned}$$(2.6)which implies that
$$\begin{aligned} \lim _{s\rightarrow 0}J_{\mu }(su)=0\ \text {and}\ \lim _{s\rightarrow +\infty }J_{\mu }(su)=-\infty . \end{aligned}$$Consequently, the proof of (i) is complete.
- (ii):
-
By (2.6), one can derive that
$$\begin{aligned} \frac{d}{ds}J_{\mu }(su)=s\left( \int _{\Omega }|\nabla u|^{2}dx -s^{22^{*}_{\mu }-2}\int _{\Omega }\int _{\Omega } \frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy\right) . \end{aligned}$$(2.7)Therefore, there exist a unique
$$\begin{aligned} {\bar{s}}:=\left( \frac{\int _{\Omega }|\nabla u|^{2}dx}{\int _{\Omega }\int _{\Omega } \frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy}\right) ^{1/(22^{*}_{\mu }-2)} \end{aligned}$$such that (2.5) is true.
- (iii):
-
By (2.7), we get \(\frac{d}{ds}J_{\mu }(su)\ge 0\) on \(0\le s\le {\bar{s}}\) and \(\frac{d}{ds}J_{\mu }(su)\le 0\) on \({\bar{s}}\le s\le +\infty \). Hence, \(J_{\mu }(su)\) is increasing on \(0\le s\le {\bar{s}}\), decreasing on \({\bar{s}}\le s\le +\infty \) and takes the maximum at \(s={\bar{s}}\).
- (iv):
-
By the definition of \(I_{\mu }(u)\) in (1.8) and (2.7), we have
$$\begin{aligned} I_{\mu }(su)=s^{2}\int _{\Omega }|\nabla u|^{2}dx -s^{22^{*}_{\mu }}\int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy =s\frac{d}{ds}J_{\mu }(su). \end{aligned}$$
Then, by (ii), the proof is complete. \(\square \)
Now, for \(0<\delta <2^{*}_{\mu }\), we define
and
where
Then, as in [24], we define modified potential wells by
and
Let
Then, we have the following results.
Lemma 2.6
Let \(u\in H^{1}_{0}(\Omega )\backslash \{0\}\).
-
(i)
If \(0<\Vert \nabla u\Vert ^{2}_{2}<r(\delta )\), then \(I_{\mu ,\delta }(u)>0\). In particular, if \(0<\Vert \nabla u\Vert ^{2}_{2}<r(1)\), then \(I_{\mu }(u)>0\).
-
(ii)
If \(I_{\mu ,\delta }(u)<0\), then \(\Vert \nabla u\Vert ^{2}_{2}>r(\delta )\). In particular, if \(I_{\mu }(u)<0\), then \(\Vert \nabla u\Vert ^{2}_{2}>r(1)\).
-
(iii)
If \(I_{\mu ,\delta }(u)=0\), then \(\Vert \nabla u\Vert ^{2}_{2}\ge r(\delta )\) or \(\Vert \nabla u\Vert ^{2}_{2}=0\). In particular, if \(I_{\mu }(u)=0\), then \(\Vert \nabla u\Vert ^{2}_{2}\ge r(1)\) or \(\Vert \nabla u\Vert ^{2}_{2}=0\).
-
(iv)
If \(I_{\mu ,\delta }(u)=0\) and \(\Vert \nabla u\Vert ^{2}_{2}\ne 0\), then \(J_{\mu }(u)>0\) for \(0<\delta <2^{*}_{\mu }\), \(J_{\mu }(u)=0\) for \(\delta =2^{*}_{\mu }\), \(J_{\mu }(u)<0\) for \(\delta >2^{*}_{\mu }\).
Proof
- (i):
-
By (2.8) and the Hardy–Littlewood–Sobolev inequality, we have
$$\begin{aligned} I_{\mu ,\delta }(u)= & {} \delta \Vert \nabla u\Vert _{2}^{2} -\int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy\\\ge & {} \Vert \nabla u\Vert _{2}^{2}\left( \delta -S_{H,L}^{-\frac{2N-\mu }{N-2}}(\Vert \nabla u\Vert ^{2}_{2})^{\frac{N-\mu +2}{N-2}}\right) . \end{aligned}$$Hence, we have \(I_{\mu ,\delta }(u)>0\), since \(0<\Vert \nabla u\Vert ^{2}_{2}<r(\delta )\).
- (ii):
-
Since \(I_{\mu ,\delta }(u)<0\), using (2.8) and the Hardy–Littlewood–Sobolev inequality again, we can get
$$\begin{aligned} \delta \Vert \nabla u\Vert _{2}^{2} <\int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy \le S_{H,L}^{-\frac{2N-\mu }{N-2}}(\Vert \nabla u\Vert ^{2}_{2})^{\frac{2N-\mu }{N-2}}. \end{aligned}$$Hence, \(\Vert \nabla u\Vert ^{2}_{2}>r(\delta )\).
- (iii):
-
Obviously, if \(\Vert \nabla u\Vert ^{2}_{2}=0\), then \(I_{\mu ,\delta }(u)=0\). So, we assume that \(I_{\mu ,\delta }(u)=0\) and \(\Vert \nabla u\Vert ^{2}_{2}\ne 0\). By (2.8) and the Hardy–Littlewood–Sobolev inequality, one has
$$\begin{aligned} \delta \Vert \nabla u\Vert _{2}^{2} =\int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy \le S_{H,L}^{-\frac{2N-\mu }{N-2}}(\Vert \nabla u\Vert ^{2}_{2})^{\frac{2N-\mu }{N-2}} \end{aligned}$$which implies that \(\Vert \nabla u\Vert ^{2}_{2}\ge r(\delta )\).
- (iv):
-
Since \(I_{\mu ,\delta }(u)=0\) and \(\Vert \nabla u\Vert ^{2}_{2}\ne 0\), by (iii) above, we have \(\Vert \nabla u\Vert ^{2}_{2}\ge r(\delta )\). Furthermore, it follows from (1.2) that
$$\begin{aligned} J_{\mu }(u)= & {} \frac{1}{2}\int _{\Omega }|\nabla u|^{2}dx -\frac{1}{22^{*}_{\mu }} \int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy\\= & {} \left( \frac{1}{2}-\frac{\delta }{22^{*}_{\mu }}\right) \int _{\Omega }|\nabla u|^{2}dx \ge \left( \frac{1}{2}-\frac{\delta }{22^{*}_{\mu }}\right) r(\delta ). \end{aligned}$$
Consequently, the proof is complete. \(\square \)
Lemma 2.7
\(m_{\mu }(\delta )\) defined in (2.9) satisfies
-
(i)
\(m_{\mu }(\delta )\ge a(\delta )r(\delta )\) for \(0<\delta <2^{*}_{\mu }\), where \(a(\delta ):=\frac{1}{2}-\frac{\delta }{22^{*}_{\mu }}\);
-
(ii)
\(\lim _{\delta \rightarrow 0}m_{\mu }(\delta )=0\), \(m_{\mu }(2^{*}_{\mu })=0\) and \(m_{\mu }(\delta )<0\) for \(\delta >2^{*}_{\mu }\);
-
(iii)
\(m_{\mu }(\delta )\) is increasing on \(0<\delta \le 1\), decreasing in \(1<\delta <2^{*}_{\mu }\) and takes the maximum \(m_{\mu }(\delta )=m_{\mu }(1)\) at \(\delta =1\).
Proof
- (i):
-
For any \(u\in {\mathcal {N}}_{\mu ,\delta }\), we have \(I_{\mu ,\delta }(u)=0\) and \(\Vert \nabla u\Vert ^{2}_{2}\ne 0\). It follows from Lemma 2.6 (iii) that \(\Vert \nabla u\Vert ^{2}_{2}\ge r(\delta )\). Furthermore, we can deduce that
$$\begin{aligned} J_{\mu }(u)= & {} \frac{1}{2}\int _{\Omega }|\nabla u|^{2}dx -\frac{1}{22^{*}_{\mu }} \int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy\\= & {} \left( \frac{1}{2}-\frac{\delta }{22^{*}_{\mu }}\right) \int _{\Omega }|\nabla u|^{2}dx \ge \left( \frac{1}{2}-\frac{\delta }{22^{*}_{\mu }}\right) r(\delta ). \end{aligned}$$Therefore, \(m_{\mu }(\delta )\ge a(\delta )r(\delta )\), where \(a(\delta ):=\frac{1}{2}-\frac{\delta }{22^{*}_{\mu }}\).
- (ii):
-
Fix \(u\in H^{1}_{0}(\Omega )\) and \(\Vert \nabla u\Vert ^{2}_{2}\ne 0\) and let \({\tilde{s}}u\in {\mathcal {N}}_{\mu ,\delta }\), i.e.
$$\begin{aligned} 0=I_{\mu ,\delta }({\tilde{s}}u)=\delta \Vert \nabla ({\tilde{s}}u)\Vert _{2}^{2} -\int _{\Omega }\int _{\Omega }\frac{|{\tilde{s}}u(x)|^{2^{*}_{\mu }}|{\tilde{s}}u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy. \end{aligned}$$Then, we can derive
$$\begin{aligned} {\tilde{s}}:=s(\delta )=\left( \frac{\delta \Vert \nabla u\Vert _{2}^{2}}{\int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy}\right) ^{1/(22^{*}_{\mu }-2)}, \end{aligned}$$(2.10)and
$$\begin{aligned} \lim _{\delta \rightarrow 0}{\tilde{s}}=0. \end{aligned}$$(2.11)Furthermore, by Lemma 2.5(i) and (2.11), we can get
$$\begin{aligned} \lim _{\delta \rightarrow 0}J_{\mu }({\tilde{s}} u)=\lim _{{\tilde{s}}\rightarrow 0}J_{\mu }({\tilde{s}}u)=0. \end{aligned}$$Hence,
$$\begin{aligned} \lim _{\delta \rightarrow 0}m_{\mu }(\delta )=0. \end{aligned}$$Next, by Lemma 2.6 (iv), we can get \(m_{\mu }(2^{*}_{\mu })=0\) and \(m_{\mu }(\delta )<0\) for \(\delta >2^{*}_{\mu }\).
- (iii):
-
It is enough to prove that for any \(0<\delta ^{\prime }<\delta ^{\prime \prime }<1\) or \(1<\delta ^{\prime \prime }<\delta ^{\prime }<2^{*}_{\mu }\) and for any \(u\in {\mathcal {N}}_{\mu ,\delta ^{\prime \prime }}\), there exist a \(v\in {\mathcal {N}}_{\mu ,\delta ^{\prime }}\) and a constant \(c(\delta ^{\prime },\delta ^{\prime \prime })\) such that \(J_{\mu }(v)<J_{\mu }(u)-c(\delta ^{\prime },\delta ^{\prime \prime })\). Indeed, for \(u\in {\mathcal {N}}_{\mu ,\delta ^{\prime \prime }}\), we define \(s(\delta )\) as (2.10), then \(I_{\mu ,\delta }(s(\delta )u)=0\) and \(s(\delta ^{\prime \prime })=1\). Let \(h(s)=J_{\mu }(su)\), we can get
$$\begin{aligned} \frac{d}{d s}h(s)= & {} s\Vert \nabla u\Vert _{2}^{2} -s^{22^{*}_{\mu }-1}\int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy\\= & {} \frac{1}{s}\left( (1-\delta )\Vert \nabla (su)\Vert _{2}^{2}+I_{\mu ,\delta }(su)\right) . \end{aligned}$$Take \(v=s(\delta ^{\prime })u\), then \(v\in {\mathcal {N}}_{\mu ,\delta ^{\prime }}\). For \(0<\delta ^{\prime }<\delta ^{\prime \prime }<1\), we have
$$\begin{aligned} J_{\mu }(u)-J_{\mu }(v)= & {} h(1)-h(s(\delta ^{\prime }))\\ {}> & {} (1-\delta ^{\prime \prime })s(\delta ^{\prime })r(\delta ^{\prime \prime })(1-s(\delta ^{\prime })) \equiv c(\delta ^{\prime },\delta ^{\prime \prime }) . \end{aligned}$$For \(1<\delta ^{\prime \prime }<\delta ^{\prime }<2^{*}_{\mu }\), we have
$$\begin{aligned} J_{\mu }(u)-J_{\mu }(v)= & {} {} h(1)-h(s(\delta ^{\prime }))\\> & {} {} (\delta ^{\prime \prime }-1)s(\delta ^{\prime \prime })r(\delta ^{\prime \prime })(s(\delta ^{\prime })-1) \equiv c(\delta ^{\prime },\delta ^{\prime \prime }) \end{aligned}$$Therefore, the proof of (iii) is complete. \(\square \)
2.3 Local Existence
In this subsection, we shall give local existence result in \(H^{1}_{0}(\Omega )\) for problem (P) by applying the method of [12, 14]. Denote \(A=-\Delta \) with the Dirichlet null condition in \(L^{2}(\Omega )\), and define the fractional powers \(A^{\alpha }\) of A and the semigroup \(\left\{ e^{tA}\right\} _{t\ge 0}\) generated by A as in [37]. Before state our main result, we introduce a lemma as follows.
Lemma 2.8
[12, Lemma 2.1]
-
(i)
For each \(\theta \ge 0\), there exist a positive constant \(C_{1}(\theta )\) such that
$$\begin{aligned} \Vert A^{\theta }e^{-tA}u\Vert _{2}\le C_{1}(\theta )t^{-\theta }e^{-\lambda t}\Vert u\Vert _{2} \end{aligned}$$for all \(u\in L^{2}(\Omega )\) and \(t>0\).
-
(ii)
For each \(0\le \theta \le 1\), there exist a positive constant \(C_{2}(\theta )\) such that
$$\begin{aligned} \Vert (e^{-tA}-I)u\Vert _{2}\le C_{2}(\theta )t^{\theta }\Vert A^{\theta }u\Vert _{2} \end{aligned}$$for all \(u\in D(A^{\theta })\) and \(t>0\).
-
(iii)
For each \(\theta >0\) and \(u\in L^{2}(\Omega )\),
$$\begin{aligned} t^{\theta }\Vert A^{\theta }e^{-tA}u\Vert _{2}\rightarrow 0\ \text {as}\ t\rightarrow 0. \end{aligned}$$
Proposition 2.9
Suppose that \(0<\mu <\min \left\{ N,4\right\} \). For each \(u_{0}\in H^{1}_{0}(\Omega )\), there exists a \(T>0\) such that problem (P) has a unique solution \(u(t)\in C([0,T];H^{1}_{0}(\Omega ))\) satisfying:
-
(i)
\(u(t)\in C((0,T];D(A^{\alpha }))\cap C((0,T];D(A^{\beta }))\);
-
(ii)
\(u(t)=e^{-tA}u_{0}+\int _{0}^{t}e^{-(t-s)A}\left[ \left( \vert x\vert ^{-\mu }*|u(s)|^{2^{*}_{\mu }}\right) |u(s)|^{2^{*}_{\mu }-2}u(s)\right] ds\);
-
(iii)
\(\lim _{t\rightarrow 0}t^{\alpha -\frac{1}{2}}\Vert A^{\alpha }u(t)\Vert _{2}=0\) and \(\lim _{t\rightarrow 0}t^{\beta -\frac{1}{2}}\Vert A^{\beta }u(t)\Vert _{2}=0\).
Proof
Note that
where \(f(u)=\left( \vert x\vert ^{-\mu }*|u|^{2^{*}_{\mu }}\right) |u|^{2^{*}_{\mu }-2}u\),
and
By the Hölder inequality, the Hardy–Littlewood–Sobolev inequality and the mean value theorem, we have
where \(\xi (x)\) is a function between \(\vert u(x)\vert \) and \(\vert v(x)\vert \), C are different constants from one line to another, \(q, q^{\prime }\in [1,+\infty ]\) are conjugate and \(r\in (1,+\infty )\) satisfies
Similarly, by the Hölder inequality, the Hardy–Littlewood–Sobolev inequality and the mean value theorem, we also have
where \(q, q^{\prime }\) and r were defined as above and C are different constants from one line to another. Hence, it follows from (2.12)–(2.13) and (2.15) that
Let
where \(\theta >0\) large enough such that \(\alpha >\frac{1}{2}\) and close to \(\frac{1}{2}\). Clearly, \(\alpha ,\beta \in (\frac{1}{2},1)\) provided that \(N>2\) and \(0<\mu <\min \left\{ N,4\right\} \). And taking
Obviously, \(q, q^{\prime }\in [1,+\infty ]\) are conjugate and \(r\in (1,+\infty )\) satisfies (2.14).
Since \(2^{*}_{\mu }r=\frac{2N}{N-4\alpha }\) and \(2(2^{*}_{\mu }-1)q^{\prime }=\frac{2N}{N-4\beta }\), it follows from the Sobolev imbedding theorem (see [11]) that
and
Hence, by (2.16)–(2.18), we can derive that
Next, we apply the contraction mapping argument in the set
endowed with the norm
\((Y_{T,K},\Vert \vert u\Vert \vert _{Y_{T,K}})\) is a Banach space.
Given \(u\in Y_{T,K}\), we set
We shall show that \(\Phi [u]\) is a strict contraction map on \(Y_{T,K}\). By (2.20), we have
By Lemma 2.8 and (2.19) with \(v=0\), we can estimate as follows
where \(\gamma \in (0,1)\) and \(C_{i}>0 (i=1,2)\) are some constants.
By (2.21)–(2.22) with \(\gamma =\alpha \) and \(\gamma =\beta \) respectively, we can derive that
and
Hence, by (2.23) and (2.24), one has
Note that
then, we can derive that \(\Vert \vert \Phi [u]\Vert \vert _{Y_{T,K}}\le K\) provided that
and
which implies that \(\Phi \) maps \(Y_{T,K}\) into itself.
Next, we show that the mapping \(\Phi : Y_{T,K}\rightarrow Y_{T,K}\) is a strictly contraction. For any \(u,v\in Y_{T,K}\), it follows from (2.19) and Lemma 2.8 that
Furthermore, we have
Note that
so, we obtain that
provided that
which implies that \(\Phi : Y_{K,T}\rightarrow Y_{K,T}\) is a strictly contraction mapping.
Now, we prove that there exists \(K,T>0\) such that (2.26)–(2.27) and (2.29) hold. Indeed, by taking \(K>0\) small enough, we can derive that (2.26) and (2.29) hold. By Lemma 2.5(iii), since \(\alpha ,\beta \in (\frac{1}{2},1)\), one has
and
for \(u\in D(A^{\frac{1}{2}})\) as \(t\rightarrow 0\). Hence, we can get there exists \(T>0\) small such that (2.27) holds.
Therefore, by applying Banach’s fixed point theorem, we can show that there exist a unique fixed point u in \(Y_{K,T}\) and u is a mild solution of (2.20). The remainder of proof is similar to [12, 14], so we omit it here. \(\square \)
At the end of this section, we introduce a lemma (see [20]), which plays an important role in the proof of blow-up.
Lemma 2.10
Suppose that \(0<T\le \infty \) and a non-negative function \(f(t)\in C^{2}[0,T)\) satisfying
for some \(\alpha >0\). If \(f(0)>0\) and \(f^{\prime }(0)>0\), then
and \(f(t)\rightarrow +\infty \) as \(t\rightarrow T\).
3 Lower Energy Initial Value
In this section, we establish the global existence and finite time blow-up of solution with lower energy initial value. Moreover, we derive the regularity and decay estimate of global solutions and an upper bound of blow-up time for \(J_{\mu }(u_{0})<0\).
3.1 Global Existence and Blow-up of Solution
In this subsection, we give the global existence and finite time blow-up of solutions.
Proof of Theorem 1.4
From Proposition 2.9, we can derive the local existence result for problem (P) in a more general case of initial value \(u_{0}\in H^{1}_{0}(\Omega )\) and \(u\in C([0,T];H^{1}_{0}(\Omega ))\).
Now, we need to prove that u(t) satisfies \(J_{\mu }(u(t))<m_{\mu }\) and \(I_{\mu }(u(t))>0\) for any \(t>0\). On the contrary, from continuity about time, there exist \(t_{0}\) such that \(u(x,t_{0})\in \partial W\), that is \(J_{\mu }(u(t_{0}))=m_{\mu }\) or \(I_{\mu }(u(t_{0}))=0, \int _{\Omega }\vert \nabla u(t_{0})\vert ^{2}dx\not =0\). From (2.4), we easily know that \(J_{\mu }(u(t_{0}))\not =m_{\mu }\). If \(I_{\mu }(u(t_{0}))=0, \int _{\Omega }\vert \nabla u(t_{0})\vert ^{2}dx\not =0\), by Remark 1.3, we know that \(J_{\mu }(u(t_{0}))\ge m_{\mu }\), a contradiction. Therefore, u(t) satisfies \(I_{\mu }(u(t))>0\) for any \(t>0\).
Furthermore, we have
Furthermore, by (2.4) and (3.1), we have
Therefore, for any \(T>0\),
and
hold, which implies that u(x, t) is a global solution of problem (P).
Next, we employ the classical concavity method to prove finite time blow-up for problem (P) with \(J_{\mu }(u_{0})<m_{\mu }\). The idea was inspired by Levine and Payne [22, 23] and Levine [19], by constructing an auxiliary function. Here, we need the following lemma.
Lemma 3.1
Let u(x, t) is the solution for problem (P) with \(u_{0}\in V\) satisfies \(J_{\mu }(u_{0})>0\). Then, there exist \(\rho >0\) such that
for \(t\in [0,\infty )\).
Proof
Since \(u_{0}\in V\), we have \(u(x,t)\in V\) for all \(t>0\). Indeed, on the contrary, from continuity about time, there exist \(t_{0}\) such that \(u(x,t_{0})\in \partial V\), that is \(J_{\mu }(u(t_{0}))=m_{\mu }\) or \(I_{\mu }(u(t_{0}))=0, \int _{\Omega }\vert \nabla u(t_{0})\vert ^{2}dx\not =0\). From (2.4), we easily know that \(J_{\mu }(u(t_{0}))\not =m_{\mu }\). If \(I_{\mu }(u(t_{0}))=0, \int _{\Omega }\vert \nabla u(t_{0})\vert ^{2}dx\not =0\), by Remark 1.3, we know that \(J_{\mu }(u(t_{0}))\ge m_{\mu }\), a contradiction. Furthermore, we have \(I_{\mu }(u)<0\). Therefore, by (1.7) and the Hardy–Littlewood–Sobolev inequality, we have
Let
we have \(\rho _{0}>0\) since \(J_{\mu }(u_{0})<m_{\mu }\). Furthermore, by (1.2), (1.3) and (1.8), we have
Next, we claim that
where \(\rho :=\frac{(2^{*}_{\mu }-1)\rho _{0}}{2}\). Indeed, since, \(\rho :=\frac{(2^{*}_{\mu }-1)\rho _{0}}{2}\), we have
Therefore, the claim is hold.
Next, by (3.4) and (3.5), we have
which implies that (3.2) is true. Consequently, the proof is complete. \(\square \)
Proof of Theorem 1.5(Part of finite time blow-up). We shall complete the proof by considering two separate cases.
- (i):
-
For the case \(J_{\mu }(u_{0})\le 0\). Suppose that there existence a global weak solution u(t), i.e. \(T_{\max }=+\infty \), and we define a auxiliary function
$$\begin{aligned} f(t)=\int ^{t}_{0}\int _{\Omega }u(s)^{2}dxds. \end{aligned}$$(3.6)By (2.2), we can derive
$$\begin{aligned} f^{\prime }(t)= & {} \int _{\Omega }|u(t)|^{2}dx \nonumber \\= & {} \int _{\Omega }u_{0}^{2}dx +2\int ^{t}_{0}\left( -\int _{\Omega }|\nabla u|^{2}dx +\int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy\right) ds, \qquad \quad \end{aligned}$$(3.7)and
$$\begin{aligned} f^{\prime \prime }(t)=-2\left( \int _{\Omega }|\nabla u|^{2}dx -\int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy\right) =-2I_{\mu }(u). \end{aligned}$$(3.8)Furthermore, it follows from (2.4) and (3.8) that
$$\begin{aligned} f^{\prime \prime }(t)= & {} -2\int _{\Omega }|\nabla u|^{2}dx +2\int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy\nonumber \\= & {} 2\left( 2^{*}_{\mu }-1\right) \int _{\Omega }|\nabla u|^{2}dx -42^{*}_{\mu }J_{\mu }(u_{0}) +42^{*}_{\mu }\int ^{t}_{0}\int _{\Omega }|u_{s}|^{2}dxds. \end{aligned}$$(3.9)Since \(J_{\mu }(u_{0})\le 0\), we can know that
$$\begin{aligned} 2\left( 2^{*}_{\mu }-1\right) \int _{\Omega }|\nabla u|^{2}dx -42^{*}_{\mu }J_{\mu }(u_{0})>0. \end{aligned}$$(3.10)Furthermore, if \(T_{\max }=+\infty \), by (3.9) and (3.10), we can derive that
$$\begin{aligned} \lim _{t\rightarrow \infty }f^{\prime }(t)=\infty \ \text {and}\ \lim _{t\rightarrow \infty }f(t)=\infty . \end{aligned}$$We also have
$$\begin{aligned} f^{\prime \prime }(t)\ge 42^{*}_{\mu }\int ^{t}_{0}\int _{\Omega }|u_{s}|^{2}dxds. \end{aligned}$$(3.11)By (3.6) and (3.11), making use of the Schwartz inequality, we have
$$\begin{aligned} f(t)f^{\prime \prime }(t)\ge & {} {} 42^{*}_{\mu }\left( \int ^{t}_{{0}}\int _{\Omega }u(s)^{2}dxds\right) \left( \int ^{t}_{0}\int _{\Omega }|u_{s}(s)|^{2}dxds\right) \nonumber \\ {}\ge & {} {} 42^{*}_{\mu }\left( \int ^{t}_{0}\int _{\Omega }uu_{s}dxds\right) ^{2}\nonumber \\= & {} {} 42^{*}_{\mu }\int ^{t}_{0}\int _{\Omega }\left( -\int _{\Omega }|\nabla u|^{2}dx +\int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy \right) dxdt\nonumber \\ {}= & {} {} 42^{*}_{\mu }\left( f^{\prime }(t)-f^{\prime }(0)\right) ^{2}. \end{aligned}$$(3.12)So by (3.12), as \(t\rightarrow \infty \), there exist \(\alpha >0\) such that
$$\begin{aligned} f(t)f^{\prime \prime }(t)\ge (1+\alpha )\left( f^{\prime }(t)\right) ^{2}. \end{aligned}$$(3.13)Then, by Lemma 2.10, there exists a \(T>0\) such that \(\lim _{t\rightarrow T^{-}}f(t)=+\infty \), which contradicts \(T_{\max }=+\infty \).
- (ii):
-
For the case \(0<J_{\mu }(u_{0})<m_{\mu }\). Similar to case (i), suppose that there existence a global weak solution u(t), i.e. \(T_{\max =\infty }\) and let
$$\begin{aligned} f(t)=\int ^{t}_{0}\int _{\Omega }\vert u(s)\vert ^{2}dxds. \end{aligned}$$By Lemma 3.1, there exist \(\rho >0\) such that
$$\begin{aligned} \int _{\Omega }\int _{\Omega }\frac{|u(x,t)|^{2^{*}_{\mu }}|u(y,t)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy \ge (1+\rho )\int _{\Omega }|\nabla u(x,t)|^{2}dx. \end{aligned}$$(3.14)for \(t\in [0,\infty )\). Hence, by (3.8) and (3.14), we have
$$\begin{aligned} f^{\prime \prime }(t)= & {} -2\left( \int _{\Omega }|\nabla u|^{2}dx -\int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy\right) \nonumber \\\ge & {} 2\rho \int _{\Omega }|\nabla u|^{2}dx. \end{aligned}$$(3.15)If \(T_{\max }=\infty \), by (3.15), we can derive that
$$\begin{aligned} \lim _{t\rightarrow \infty }f^{\prime }(t)=\infty \ \text {and}\ \lim _{t\rightarrow \infty }f(t)=\infty . \end{aligned}$$(3.16)Next, similar to (3.9), we also have
$$\begin{aligned} f^{\prime \prime }(t)\ge 2\left( 2^{*}_{\mu }-1\right) \int _{\Omega }|\nabla u|^{2}dx -42^{*}_{\mu }J_{\mu }(u_{0}) +42^{*}_{\mu }\int ^{t}_{0}\int _{\Omega }|u_{s}|^{2}dxds. \end{aligned}$$(3.17)By Lemma 3.1, we have \( \int _{\Omega }|\nabla u|^{2}dx <\int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy\) and since \(J_{\mu }(u_{0})<m_{\mu }\), then, we can derive
$$\begin{aligned}{} & {} 2\left( 2^{*}_{\mu }-1\right) \int _{\Omega }|\nabla u|^{2}dx -42^{*}_{\mu }J_{\mu }(u_{0})\nonumber \\{} & {} \quad \ge 2\left( 2^{*}_{\mu }-1\right) \int _{\Omega }|\nabla u|^{2}dx -2\left( 2^{*}_{\mu }-1\right) S^{\frac{2N-\mu }{N-\mu +2}}_{H,L} \nonumber \\{} & {} \quad >2\left( 2^{*}_{\mu }-1\right) \left[ \int _{\Omega }|\nabla u|^{2}dx-\left( \frac{\int _{\Omega }|\nabla u|^{2}dx}{\left( \int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy\right) ^{\frac{N-2}{2N-\mu }}} \right) ^{^{\frac{2N-\mu }{N-\mu +2}}}\right] \ge 0. \nonumber \\ \end{aligned}$$(3.18)Hence, by (3.17) and (3.18), one has
$$\begin{aligned} f^{\prime \prime }(t)\ge 22^{*}_{\mu }\int ^{t}_{t_{0}}\int _{\Omega }|u_{s}|^{2}dxds. \end{aligned}$$Similar to (3.12), using the Schwartz inequality, we have
$$\begin{aligned} f(t)f^{\prime \prime }(t)\ge 2^{*}_{\mu }\left( f^{\prime }(t)-f^{\prime }(0)\right) ^{2}. \end{aligned}$$(3.19)Therefore, we can derive a contraction as \(J_{\mu }(u_{0})\le 0\). We complete the proof.
Remark 3.2
In Theorem 1.5, we can prove a more general result that if there exists \(t_{0}\) such that \(u(t_{0})\) satisfies \(J_{\mu }(u(t_{0}))\le 0\) or \(0<J_{\mu }(u(t_{0}))\le m_{\mu }\) and \(I_{\mu }(u(t_{0}))>0\), then, the weak solution u(x, t) of problem (P) blow-up in finite time. In fact, we only need to substitute initial time \(t=t_{0}\) for \(t=0\) in above proof.
Next, we give a different proof of Theorem 1.5, by using a modified potential well. Firstly, we give the following lemmas.
Lemma 3.3
Assume that \(u\in H^{1}_{0}(\Omega )\) satisfying \(0<J_{\mu }(u)<m_{\mu }\). Then, the sign of \(I_{\mu ,\delta }(u)\) does not change for \(\delta _{1}<\delta <\delta _{2}\), where \(\delta _{1}<1<\delta _{2}\) be the two roots of \(m_{\mu }(\delta )=J_{\mu }(u)\).
Proof
Obviously, \(J_{\mu }(u)>0\) implies that \(\Vert \nabla u\Vert ^{2}_{2}\ne 0\). On the contrary, if the sign of \(I_{\mu ,\delta }(u)\) is changeable for \(\delta _{1}<\delta <\delta _{2}\), then there exist a \({\bar{\delta }}\in (\delta _{1},\delta _{2})\) such that \(I_{\mu ,{\bar{\delta }}}(u)=0\). Therefore, we can get \(J_{\mu }(u)\ge m_{\mu }({\bar{\delta }})\). On the other hand, by Lemma 2.7(iii), we have \(J_{\mu }(u)=m_{\mu }(\delta _{1})=m_{\mu }(\delta _{2})<m_{\mu }({\bar{\delta }})\), a contradiction. \(\square \)
Proposition 3.4
Assume that \(u\in H^{1}_{0}(\Omega )\) and \(0<\sigma <m_{\mu }\). Let \(\delta _{1}\) and \(\delta _{2}\) with \(\delta _{1}<\delta _{2}\) be the two roots of \(m_{\mu }(\delta )=\sigma \). Then
-
(i)
If \(I_{\mu }(u_{0})>0\), then, all weak solutions u(x, t) of the problem (P) with \(0<J_{\mu }(u_{0})\le \sigma \) belongs to \(W_{\delta }\) for \(\delta _{1}<\delta <\delta _{2}\), \(0\le t<T\);
-
(ii)
If \(I_{\mu }(u_{0})<0\), then, all weak solutions u(x, t) of the problem (P) with \(0<J_{\mu }(u_{0})\le \sigma \) belongs to \(V_{\delta }\) for \(\delta _{1}<\delta <\delta _{2}\), \(0\le t<T\).
Proof
- (i):
-
Let u(x, t) be any weak solution of problem (P) with \(I_{\mu }(u_{0})>0\) and \(0<J_{\mu }(u_{0})\le \sigma <m_{\mu }\). It follows from Lemma 2.7(iii) that \(\delta _{1}<1<\delta _{2}\). Furthermore, by Lemma 3.3 and \(I_{\mu }(u_{0})>0\), we can get \(I_{\mu ,\delta }(u_{0})>0\) for \(\delta _{1}<\delta <\delta _{2}\). Hence, \(u_{0}\in W_{\delta }\) for all \(\delta _{1}<\delta <\delta _{2}\). Next, we prove \(u(t)\in W_{\delta }\) for all \(\delta _{1}<\delta <\delta _{2}\) and \(0<t<T\). Otherwise, there exist a \(t^{**}\in (0,T)\) and a \(\delta ^{*}\in (\delta _{1},\delta _{2})\) such that \(u(t^{**})\in \partial W_{\delta ^{*}}\). Thus, either \(I_{\mu ,\delta ^{*}}(u(t^{**}))=0\), \(\Vert \nabla u(t^{**})\Vert _{2}\ne 0\) or \(J_{\mu }(u(t^{**}))=m_{\mu }(\delta ^{*})\). From (2.4), it follows that
$$\begin{aligned} \int ^{t}_{0}\int _{\Omega }|u_{s}|^{2}dxds+J_{\mu }(u(t)) =J_{\mu }(u_{0})<m_{\mu }(\delta ),\ \delta _{1}<\delta<\delta _{2},\ 0<t<T, \qquad \quad \end{aligned}$$(3.20)which implies that \(J_{\mu }(u(t^{**}))\ne m_{\mu }(\delta ^{*})\). If \(I_{\mu ,\delta ^{*}}(u(t^{**}))=0\), \(\Vert \nabla u(t^{**})\Vert _{2}\ne 0\), then by the definition of \(m_{\mu }(\delta )\), we have \(J_{\mu }(u(t^{**}))\ge m_{\mu }(\delta ^{*})\), which contradicts (3.20).
- (ii):
-
Let u(x, t) be any weak solution of the problem (P) with \(I_{\mu }(u_{0})<0\) and \(0<J_{\mu }(u_{0})\le \sigma <m_{\mu }\). Similar to the argument of the proof of (i), by Lemma 2.7(iii), Lemma 3.3 and \(I_{\mu }(u_{0})<0\), we can get \(I_{\mu ,\delta }(u_{0})<0\) for \(\delta _{1}<\delta <\delta _{2}\). Hence, \(u_{0}\in V_{\delta }\) for all \(\delta _{1}<\delta <\delta _{2}\). Next, we prove \(u(t)\in V_{\delta }\) for all \(\delta _{1}<\delta <\delta _{2}\) and \(0<t<T\). Otherwise, there exist a \(t^{**}\in (0,T)\) and a \(\delta ^{*}\in (\delta _{1},\delta _{2})\) such that \(u(t^{**})\in \partial V_{\delta ^{*}}\). Thus, either \(I_{\mu ,\delta ^{*}}(u(t^{**}))=0\) or \(J_{\mu }(u(t^{**}))=m_{\mu }(\delta ^{*})\). By (3.20), we can get \(J_{\mu }(u(t^{**}))\ne m_{\mu }(\delta ^{*})\), hence \(I_{\mu ,\delta ^{*}}(u(t^{**}))=0\). We assume that \(t^{**}\) is the first time such that \(I_{\mu ,\delta ^{*}}(u(t))=0\), then \(I_{\mu ,\delta ^{*}}(u(t))<0\) for \(0\le t<t^{**}\). By Lemma 2.6 (ii), we have \(\Vert \nabla u\Vert ^{2}_{2}>r(\delta ^{*})\) for \(0\le t<t^{**}\). Hence, \(\Vert \nabla u(t^{**})\Vert ^{2}_{2}>r(\delta ^{*})\), which implies that \(u(t^{**})\in {\mathcal {N}}_{\delta ^{*}}\) provided that \(I_{\mu ,\delta ^{*}}(u_{t^{**}})=0\). By the definition of \(m_{\mu }(\delta ^{*})\), we can also obtain \(J_{\mu }(u(t^{**}))\ge m_{\mu }(\delta ^{*})\), a contradiction to (3.20). Consequently, the proof is complete.\(\square \)
Proof of Theorem 1.5
Suppose that there existence a global weak solution u(t), i.e. \(T_{\max }=+\infty \), we define a auxiliary function
and
where \(\lambda _{1}\) is the first eigenvalue of problem \(\Delta \varphi +\lambda \varphi =0, \varphi |_{\partial \Omega }=0\).
Note that
then, we can get
Furthermore, by (3.24)–(3.25), and making use of the Schwartz inequality, we have
In the following we shall complete the proof by considering two separate cases.
- (i):
-
If \(J_{\mu }(u_{0})\le 0\), then
$$\begin{aligned} f^{\prime \prime }(t)f(t)-2^{*}_{\mu }\left( f^{\prime }(t)\right) ^{2} \ge 2\left( 2^{*}_{\mu }-1\right) \lambda _{1}f^{\prime }(t)f(t) -22^{*}_{\mu }f^{\prime }(t)\Vert u_{0}\Vert _{2}^{2}, \end{aligned}$$(3.27)for all \(t>0\). Now, we claim that \(I_{\mu }(u)<0\) for all \(t>0\). Otherwise, there exist \(t_{0}>0\) such that \(I_{\mu }(u(t_{0}))=0\) and \(I_{\mu }(u(t))<0\) for \(0\le t<t_{0}\). Then, by Remark 1.3, we have \(J_{\mu }(u(t_{0}))\ge m_{\mu }\), which contradicts (2.4). Hence, by (3.23), we can get \(f^{\prime \prime }(t)>0\) for \(t\ge 0\). And since \(f^{\prime }(0)=\int _{\Omega }|u_{0}|^{2}dx\ge 0\), then there exists a \(t_{0}\ge 0\) such that \(f^{\prime }(t_{0})>0\). For \(t\ge t_{0}\), we derive that
$$\begin{aligned} f(t)\ge f^{\prime }(t_{0})(t-t_{0})>f^{\prime }(0)(t-t_{0}). \end{aligned}$$Therefore, for t large enough, we can get
$$\begin{aligned} \left( 2^{*}_{\mu }-1\right) \lambda _{1}f(t) >2^{*}_{\mu }\Vert u_{0}\Vert _{2}^{2}. \end{aligned}$$Furthermore, by (3.27), we have
$$\begin{aligned} f^{\prime \prime }(t)f(t)-2^{*}_{\mu }\left( f^{\prime }(t)\right) ^{2} >0. \end{aligned}$$ - (ii):
-
If \(0<J_{\mu }(u_{0})<m_{\mu }\), then it follows from Proposition 3.4 that \(u(t)\in V_{\delta }\) for \(1<\delta <\delta _{2}\) and \(t\ge 0\), where \(\delta _{2}\) is the large root of \(m_{\mu }(\delta )=J_{\mu }(u_{0})\). Hence, \(I_{\mu ,\delta }(u)<0\) for \(1<\delta <\delta _{2}\) and \(t\ge 0\). Furthermore, by Lemma 2.6 (ii), we have \(\Vert \nabla u\Vert ^{2}_{2}>r(\delta )\) for \(1<\delta <\delta _{2}\) and \(t\ge 0\). Hence, \(I_{\mu ,\delta _{2}}(u)\le 0\) and \(\Vert \nabla u\Vert ^{2}_{2}\ge r(\delta _{2})\) for \(t\ge 0\). By (3.23), we can get
$$\begin{aligned} f^{\prime \prime }(t)=-2I_{\mu }(u)= & {} 2(\delta _{2}-1)\Vert \nabla u\Vert ^{2}_{2}-2I_{\mu ,\delta _{2}}(u)\\\ge & {} 2(\delta _{2}-1)r(\delta _{2}),\ t\ge 0. \end{aligned}$$Furthermore, we have
$$\begin{aligned} f^{\prime }(t)\ge (\delta _{2}-1)r(\delta _{2})t+f^{\prime }(0) \ge (\delta _{2}-1)r(\delta _{2})t,\ t\ge 0, \end{aligned}$$and
$$\begin{aligned} f(t)\ge \frac{1}{2}(\delta _{2}-1)r(\delta _{2})t^{2},\ t\ge 0. \end{aligned}$$Therefore, for t large enough, we deduce that
$$\begin{aligned} (2^{*}_{\mu }-1)\lambda _{1}f(t)>22^{*}_{\mu }\Vert u_{0}\Vert _{2}^{2}\ \text {and}\ (2^{*}_{\mu }-1)\lambda _{1}f^{\prime }(t) >42^{*}_{\mu }J(u_{0}). \end{aligned}$$Then, from (3.26) it follows that
$$\begin{aligned} f^{\prime \prime }(t)f(t)-2^{*}_{\mu }\left( f^{\prime }(t)\right) ^{2}\ge & {} 2\left( 2^{*}_{\mu }-1\right) \lambda _{1}f^{\prime }(t)f(t) -22^{*}_{\mu }f^{\prime }(t)\Vert u_{0}\Vert _{2}^{2}\\{} & {} -42^{*}_{\mu }J_{\mu }(u_{0})f(t)\\= & {} \left( (2^{*}_{\mu }-1)\lambda _{1}f(t) -22^{*}_{\mu }\Vert u_{0}\Vert _{2}^{2}\right) f^{\prime }(t)\\{} & {} +\left( (2^{*}_{\mu }-1)\lambda _{1}f(t) -42^{*}_{\mu }J(u_{0})\right) f(t)\\> & {} 0. \end{aligned}$$Then, by Lemma 2.10, there exists a \(T>0\) such that \(\lim _{t\rightarrow T^{-}}f(t)=+\infty \), which contradicts \(T_{\max }=+\infty \).
Proof of Theorem 1.5(Upper bound estimate of blow-up time). Next, we prove an upper bound for blow-up of \(J_{\mu }(u_{0})<0\). Define \(g(t)=\int _{\Omega }\vert u\vert ^{2}dx\), by (2.2), we have
where
By (2.3) and standard computation, we have
which implies that \(h(t)\ge h(0)=-42^{*}_{\mu }J_{\mu }(u_{0})>0\) for all \(t\ge 0\) provided that \(J_{\mu }(u_{0})<0\).
Next, by (3.28) and Schwarz’s inequality, we obtain
Integrating (3.30) from 0 to t and by (3.28), we have
Integrating from 0 to t again, we can derive
Then, let t tends to T, one has
Consequently, the proof is complete.
Next, by (1.9) and the Poincaré inequality \(\lambda _{1}\Vert u\Vert ^{2}_{2}\le \Vert \nabla u\Vert ^{2}_{2}\), we further have the following corollary.
Corollary 3.5
Under the conditions of Theorem 1.5, we also have
3.2 Decay Estimate of Global Solutions
In this section, we prove decay rate of the \(H^{1}_{0}\) and \(L^{2}\) norm of the global solutions for problem (P). Firstly, we give the following lemma.
Lemma 3.6
Let u(x, t) is the solution for problem (P) with \(u_{0}\in H^{1}_{0}(\Omega )\) satisfies \(J_{\mu }(u_{0})<m_{\mu }\) and \(I_{\mu }(u_{0})>0\). Then
for \(t\in [0,\infty )\), where \(\kappa \in (0,1)\).
Proof
Since \(u_{0}\in H^{1}_{0}(\Omega )\) satisfies \(J_{\mu }(u_{0})<m_{\mu }\) and \(I_{\mu }(u_{0})>0\), as in the proof of Theorem 1.4, we have \(I_{\mu }(u(t))>0\) for all \(t>0\). Furthermore, we can get
By (2.4), (3.32) and the Hardy–Littlewood–Sobolev inequality
we can derive
Let \(\kappa :=1-S^{-\frac{2N-\mu }{N-2}}_{H,L}\left( \frac{2(2N-\mu )}{N-\mu +2}J_{\mu }(u_{0})\right) ^{\frac{N-\mu +2}{N-2}}\), then we complete the proof of (3.31).
Next, since
we can derive that \(\kappa \in (0,1)\). Consequently, the proof is complete. \(\square \)
Proof of Theorem 1.6
Under the condition in Theorem 1.4, let u be a global solution. By Lemma 3.6, we have
and so, we ca derive
Furthermore, by (1.2), (1.8) and (3.34), we have
By (3.34) and (3.35), we can also derive
Let \(T>0\) be an arbitrary number but fixed, by (2.2) and Poincare’s inequality, we have
where \(\lambda _{1}\) is the first eigenvalue of \(-\Delta \) with homogeneous Dirichlet boundary condition.
It follows from (3.35) and (3.37) that
Therefore, by (3.36) and (3.38), we can derive
Since the arbitrariness of \(T>0\), we can have
where \({\bar{C}}:=\left( \frac{2^{*}_{\mu }-1}{22^{*}_{\mu }\kappa }+\frac{1}{22^{*}_{\mu }}\right) \frac{2^{*}_{\mu }}{\lambda _{1}(2^{*}_{\mu }-1)}\).
Next,taking \(T_{0}>0\) large enough such that \({\bar{C}}\le T_{0}\), then we have
Let
Then, \(F^{\prime }(t)=-J_{\mu }(u(t))\). By (3.35), we have \(J_{\mu }(u(t))>0\) for \(t\ge 0\). Integrating (3.39) from \(T_{0}\) to t, we have
for all \(t>T_{0}\). That is
Next, by (1.3) and (3.39), for \(t>T_{0}\), we have
Therefore, by (3.40) and (3.41), we can derive
for all \(t>T_{0}\).
On the other hand, using (1.3) again, we obtain
It follows from (3.42) and (3.43) that
Furthermore, by (3.35), we can get
which implies the decay of global solution \(\Vert \nabla u(t)\Vert ^{2}_{2}\le Ce^{-\frac{t}{T_{0}}}\) for some \(C>0\) and \(t>T_{0}\) large enough. So, we complete the proof of (1.10).
Next, multiplying (2.1) by any \(d(t)\in [0,\infty )\), we can get
and
Letting \(w=u\), by (3.44), one has
It follows from Lemma 3.6 and (3.45) that
where \(\lambda _{1}\) is the first eigenvalue of \(-\Delta \) with homogeneous Dirichlet boundary condition.
Integrating (3.46) on (0, t), we obtain
So, we complete the proof of (1.11).
3.3 Regularity of Global Solutions
In this section, we shall prove the regularity of global solutions with lower energy initial value by applying a nonlocal version of the Brezis-Kato estimate (see [3, 27]).
Proposition 3.7
If \(H, K\in L^{\frac{2N}{N-\mu }}(\Omega )+L^{\frac{2N}{N-\mu +2}}(\Omega )\) and \(u\in L^{\infty }(0,\infty ; H^{1}_{0}(\Omega ))\) with \(u_{t}\in L^{2}(0,\infty ; L^{2}(\Omega ))\) be a global solution of
Then, \(u\in L^{p}(\Omega \times [t_{0},\infty ))\) for every \(p\in [2,\frac{N}{N-\mu }\frac{2N}{N-2})\).
Proof
For \(M>0\), we define the function \(u_{M}\) by
And for any fixed \(t_{0}>0\) and \(T>0\), choosing \(\eta \in C^{\infty }(0,T)\) with \(0\le \eta \le 1\) in (0, T), \(\eta =1\) in \([t_{0},T]\), \(\eta =0\) in \([0,\frac{t_{0}}{2}]\) and \(|\eta _{t}|<\frac{1}{t_{0}}\).
Taking \(\varphi (x,t)=\vert u_{M}\vert ^{p-2}u_{M}\eta \) as a test function to (3.47), we can obtain
By manipulation, we have
and
If \(p<\frac{2N}{N-\mu }\), by [27, Lemma 3.2] with \(\theta =\frac{2}{p}\), one can get
for some \(C>0\), where \(A_{M}:=\left\{ x\in \Omega \,\ \vert u(x)\vert >M\right\} \).
By the Hardy–Littlewood–Sobolev inequality, one has
Next, by the Sobolev inequality, we have
By iterating over p a finite number of times, we cover the range \(p\in [2,\frac{N}{N-\mu }\frac{2N}{N-2})\). \(\square \)
Proof of Theorem 1.7
Let \(H=K=\vert u\vert ^{2^{*}_{\mu }-2}u\). By Proposition 3.7, \(u(x,t)\in L^{p}(\Omega \times [t_{0},\infty ))\) for every \(p\in [2,\frac{N}{N-\mu }\frac{2N}{N-2})\). Then \(\vert u\vert ^{2^{*}_{\mu }}\in L^{q}(\Omega \times [t_{0},\infty ))\) for every \(q\in [\frac{2(N-2)}{2N-\mu },\frac{N}{N-\mu }\frac{2N}{2N-\mu })\). Since \(\frac{2(N-2)}{2N-\mu }<\frac{N}{N-\mu }<\frac{N}{N-\mu }\frac{2N}{2N-\mu }\), we have \(\vert x\vert ^{-\mu }*\vert u\vert ^{2^{*}_{\mu }}\in L^{\infty }(\Omega \times [t_{0},\infty ))\). By the classical bootstrap method, we have \(u(x,t)\in W_{r}^{2,1}(\Omega \times [t_{0},\infty ))\) for every \(r>1\). Applying the Schauder estimate in [17], we can derive \(u(x,t)\in C^{(2,\alpha )(1,\alpha )}(\Omega \times [t_{0},\infty ))\). Therefore, u(x, t) is a classical solution for all \(t\ge t_{0}>0\).
4 Critical Energy Initial Value
In this section, we consider the global existence and blow-up of solution with critical energy initial value, i.e. \(J_{\mu }(u_{0})=m_{\mu }\) and the decay estimate of global solutions.
Proof of Theorem 1.9(i)
Since \(J_{\mu }(u_{0})=m_{\mu }\), we can see that \(\Vert u_{0}\Vert ^{2}_{2}\ne 0\). Choose a sequence \(\{\beta _{k}\}\) such that \(0<\beta _{k}<1\), \(k=1,2,\cdots \) and \(\beta _{k}\rightarrow 1\) as \(k\rightarrow \infty \), and let \(u_{0,k}(x)=\beta _{k}u_{0}(x)\). Consider the initial and boundary value problem as follows:
Since \(I_{\mu }(u_{0})>0\ \mathrm{by\ Lemma\ } \) 2.5, there exist a \({\bar{s}}:=s(u_{0})>1\) such that \(I_{\mu }({\bar{s}}u_{0})=0\). Thus, since \(\beta _{k}<1< {\bar{s}}\), we can deduce that \(I_{\mu }(u_{0,k})=I_{\mu }(\beta _{k}u_{0})>0\) and \(J_{\mu }(u_{0,k})=J_{\mu }(\beta _{k}u_{0})<J_{\mu }(u_{0})=m_{\mu }\). It follows from Theorem 1.4 that for each k, Eq. (4.1) admits a global weak solution \(u_{k}(t)\in L^{\infty }(0,T;H^{1}_{0}(\Omega ))\) with \((u_{k})_{t}\in L^{2}(\Omega _{T})=L^{2}(0,T;L^{2}(\Omega ))\) and \(u_{k}(t)\in W\) for \(0\le t<\infty \) satisfying
and
From (4.3) and
we can get
This implies that
and by the Hardy–Littlewood–Sobolev inequality, we have
Therefore, there exist a u and a subsequence \(\{u^{\nu }\}\) such that
In (4.2), we fixed s and letting \(k=\nu =\infty \), we can get
and
Moreover, (4.2) gives \(u(x,0)=u_{0}(x)\) in \(H^{1}_{0}(\Omega )\). The reminder proof is similar to the case of \(J_{\mu }(u_{0})<m_{\mu }\), here we omit it. Consequently, the proof is complete.
Proof of Theorem 1.9(ii)
Let u(t) be any weak solution of the problem (P) with \(J_{\mu }(u_{0})=m_{\mu }\) and \(I_{\mu }(u_{0})<0\), we shall prove \(T_{\max }<\infty \), where \(T_{\max }\) be the existence time of u(t). On the contrary, we suppose \(T_{\max }=\infty \), and we define a auxiliary function
By (2.2) and standard manipulation, we have
and
Note that
then, we can get
Furthermore, by (4.10)–(4.11) and the Schwartz inequality, we have
Since \(J_{\mu }(u_{0})=m_{\mu }\) and \(I_{\mu }(u_{0})<0\), and by the continuity of \(J_{\mu }\) and \(I_{\mu }\) with respect to t, there exists a sufficiently small \(t_{1}>0\) such that \(J_{\mu }(u(t))>0\) and \(I_{\mu }(u(t))<0\) for \(0\le t\le t_{1}\). Then, \((u_{t},u)=-I_{\mu }(u)>0\), and \(\Vert u_{t}\Vert ^{2}_{2}>0\) for \(0\le t\le t_{1}\). Hence
Thus, we choose \(t=t_{1}\) as the initial time and by Proposition 3.4, we have \(u(t)\in V_{\delta }\) for \(\delta _{1}<\delta <\delta _{2}\) and \(t_{1}\le t<\infty \), where \(\delta _{1}\) and \(\delta _{2}\) are two roots of \(m_{\mu }(\delta )={\tilde{m}}_{\mu }\). Hence, \(I_{\mu ,\delta }(u)<0\) and \(\Vert \nabla u\Vert ^{2}_{2}>r(\delta )\) for \(\delta _{1}<\delta <\delta _{2}\) and \(t_{1}\le t<\infty \). Hence, \(I_{\mu ,\delta _{2}}(u)<0\) and \(\Vert \nabla u\Vert ^{2}_{2}>r(\delta _{2})\) for \(t_{1}\le t<\infty \).
By (4.9), we can get
Furthermore, we have
and
Therefore, for t large enough, we can get that
Then, from (4.12) it follows that
Then, by Lemma 2.10, there exists a \(T>0\) such that \(\lim _{t\rightarrow T^{-}}f(t)=+\infty \), which contradicts \(T_{\max }=+\infty \).
5 High Energy Initial Value
In this section, we investigate the conditions to ensure the existence of global or finite time blow-up of solutions to problem (P) with high energy initial value, i.e. \(J_{\mu }(u_{0})>m_{\mu }\). As mentioned in Introduction, we define
and the level set of \(J_{\mu }\) as follows:
Obviously, we have
Furthermore, for all \(d>m_{\mu }\), we set
It is clear that \(\lambda _{d}\) is nonincreasing and \(\Lambda _{d}\) is nondecreasing in d.
If \(T_{\max }=\infty \), we denote by
the \(\omega \)-limit set of \(u_{0}\in H^{1}_{0}(\Omega )\). Finally, we introduce the following sets
Clearly, \(H^{1}_{0}(\Omega )={\mathcal {G}}\cup {\mathcal {B}}\). Now, we give two lemmas, which play important roles in the proof of the main results.
Lemma 5.1
We have
-
(i)
0 is away from both \({\mathcal {N}}\) and \({\mathcal {N}}_{-}\), i.e. \(dist(0,{\mathcal {N}})>0\) and \(dist(0,{\mathcal {N}}_{-})>0\);
-
(ii)
For any \(d>0\), the set \(J^{d}\cap {\mathcal {N}}_{+}\) is bounded in \(H^{1}_{0}(\Omega )\).
Proof
- (i):
-
For any \(u\in {\mathcal {N}}\), we have
$$\begin{aligned} m_{\mu }\le J_{\mu }(u)= & {} \frac{1}{2}\int _{\Omega }|\nabla u|^{2}dx -\frac{1}{22^{*}_{\mu }} \int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy\\= & {} \frac{N-\mu +2}{2(2N-\mu )}\int _{\Omega }|\nabla u|^{2}dx, \end{aligned}$$which implies that there exists a constants \(c>0\) such that \(dist(0,{\mathcal {N}})=\inf _{u\in {\mathcal {N}}}\Vert \nabla u\Vert _{2}>0\). For any \(u\in {\mathcal {N}}_{- }\) , that is \(I_{\mu }(u)<0\), we have \(\Vert \nabla u\Vert _{2}\ne 0\). Then, it follows the Hardy–Littlewood–Sobolev inequality that
$$\begin{aligned} \Vert \nabla u\Vert ^{2}_{2} <\int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy \le S^{-\frac{2N-\mu }{N-2}}_{H,L}\left( \Vert \nabla u\Vert ^{2}_{2}\right) ^{\frac{2N-\mu }{N-2}}, \end{aligned}$$which implies that \(\Vert \nabla u\Vert ^{2}_{2}>S^{\frac{2N-\mu }{N-\mu +2}}_{H,L}\). Therefore, \(dist(0,{\mathcal {N}}_{-})=\inf _{u\in {\mathcal {N}}_{-}}\Vert \nabla u\Vert _{2}>0\).
- (ii):
-
For any \(u\in J^{d}\cap {\mathcal {N}}_{+}\), that is \(J_{\mu }(u)<d\) and \(I_{\mu }(u)>0\). Then, it follows from this and the Hardy–Littlewood–Sobolev inequality that
$$\begin{aligned} d> J_{\mu }(u)= & {} \frac{1}{2}\int _{\Omega }|\nabla u|^{2}dx -\frac{1}{22^{*}_{\mu }} \int _{\Omega }\int _{\Omega }\frac{|u(x)|^{2^{*}_{\mu }}|u(y)|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy\\ {}= & {} {} \frac{N-\mu +2}{2(2N-\mu )}\int _{\Omega }|\nabla u|^{2}dx +\frac{1}{22^{*}_{\mu }}I_{\mu }(u)\\ {}> & {} {} \frac{N-\mu +2}{2(2N-\mu )}\int _{\Omega }|\nabla u|^{2}dx, \end{aligned}$$which implies that
$$\begin{aligned} \Vert \nabla u\Vert ^{2}_{2}<\frac{2(2N-\mu )d}{N-\mu +2}. \end{aligned}$$Consequently, the proof is complete.\(\square \)
Lemma 5.2
Let \(u_{0}\in H^{1}_{0}(\Omega )\). Then,
Proof
Multiplying (P) by u(t) and integrating by parts immediately, we can complete the proof. \(\square \)
Now, the proof of Theorem 1.10-1.12 are to show as follows.
Proof of Theorem 1.10
- (i):
-
Assume that \(u_{0}\in {\mathcal {N}}_{+}\) with \(\Vert u_{0}\Vert _{2}\le \lambda _{J(u_{0})}\). We claim that \(u(t)\in {\mathcal {N}}_{+}\) for all \(t\in [0,T_{\max })\). On the contrary, there is a \(t_{0}>0\) such that \(u(t)\in {\mathcal {N}}_{+}\) for \(0\le t<t_{0}\) and \(u(t_{0})\in {\mathcal {N}}\), then (1.3) and (5.1) imply that
$$\begin{aligned} \Vert u(t_{0})\Vert _{2}\le \Vert u_{0}\Vert _{2}\le \lambda _{J(u_{0})},\ J_{\mu }(u(t_{0}))\le J_{\mu }(u_{0}). \end{aligned}$$This cintradicts the definition of \(\lambda _{J(u_{0})}\) and proves the claim. By Lemma 5.1 (ii), we have that the orbit \(\left\{ u(t)\right\} \) remains bounded in \(H^{1}_{0}(\Omega )\) for \([0,T_{\max })\), so that \(T_{\max }=+\infty \). Now, for any \(w\in \omega (u_{0})\), by (1.3) and (5.1), we have
$$\begin{aligned} \Vert w\Vert _{2}<\lambda _{J_{\mu }(u_{0})}\ \text {and}\ J_{\mu }(w)\le J_{\mu }(u_{0}). \end{aligned}$$By the definition of \(\lambda _{J_{\mu }(u_{0})}\), we can get that \(\omega (u_{0}) \cap {\mathcal {N}}=\emptyset \), hence \(\omega (u_{0}) =\left\{ 0\right\} \). In other words, \(u_{0}\in {\mathcal {G}}_{0}\).
- (ii):
-
\(u_{0}\in {\mathcal {N}}_{-}\) and \(\Vert u_{0}\Vert _{2}\le \Lambda _{J(u_{0})}\). A similar argument as (i), one can get that \(u(t)\in {\mathcal {N}}_{-}\) for all \(t\in [0,T_{\max })\).
Next, by contradiction, if \(T_{\max }=\infty \), then for every \(w\in \omega (u_{0})\), it follows from (1.3) and (5.1) that
By the definition of \(\Lambda _{J_{\mu }(u_{0})}\), we derive that \(\omega (u_{0}) \cap {\mathcal {N}}=\emptyset \). However, since \(dist(0,{\mathcal {N}}_{-})>0\) in Lemma 5.1 (i), we also have \(0\notin \omega (u_{0})\). This gives \(\omega (u_{0}) =\emptyset \), contrary to the assumption that u(t) is a global solution. Hence, \(T_{\max }<\infty \) and the proof is complete.
Proof of Theorem 1.11
For
Since \(r_{\Omega }=\sup _{x,y\in \Omega }\vert x-y\vert \), for any \(u\in H^{1}_{0}(\Omega )\), there hold
By (1.12) and (5.2) and using the H\(\ddot{o}\)lder inequality, for \(\mu <4\), we get
Then, we readily infer that \(\int _{\Omega }\int _{\Omega }\frac{|u_{0}|^{2^{*}_{\mu }}|u_{0}|^{2^{*}_{\mu }}}{|x-y|^{\mu }}dxdy>\Vert \nabla u_{0}\Vert ^{2}_{2}\), which implies that \(u_{0}\in {\mathcal {N}}_{-}\).
Next, we shall show \(u_{0}\in {\mathcal {B}}\). Since \(u_{0}\in {\mathcal {N}}_{-}\), by Theorem 1.10, we only need to prove that \(\Vert u_{0}\Vert _{2}\ge \Lambda _{J(u_{0})}\). For any \(u\in {\mathcal {N}}_{J(u_{0})}\) i.e. \(u\in {\mathcal {N}}\) and \(J_{\mu }(u)<J_{\mu }(u_{0})\), by the H\(\ddot{o}\)lder inequality, we have
Therefore, taking the supremum over \({\mathcal {N}}_{J(u_{0})}\), we immediately get
Therefore, \(\Vert u_{0}\Vert _{2}\ge \Lambda _{J(u_{0})}\) and we complete the proof by Theorem 1.10.
Proof of Theorem 1.12
Let \(M>0\) and \(\Omega _{1},\Omega _{2}\) be two arbitrary disjoint open subdomain of \(\Omega \). Furthermore, let \(v\in H^{1}_{0}(\Omega _{1})\subset H^{1}_{0}(\Omega )\) be an arbitrary nonzero function. Then, we can choose \(\alpha \) large enough such that \(\Vert \alpha v\Vert ^{22^{*}_{\mu }}_{2}\ge \frac{22^{*}_{\mu }}{2^{*}_{\mu }-1}(r_{\Omega })^{\mu }\vert \Omega \vert ^{2^{*}_{\mu }-2}M\) and \(J(\alpha v)\le 0\). Fix such \(\alpha >0\) and pick a function \(w\in H^{1}_{0}(\Omega )\) with \(J(w)=M-J(\alpha v)\). Then, \(u_{M}:=w+\alpha v\) satisfies \(J(u_{M})=J(w)+J(\alpha v)=M\) and
By Theorem 1.11, it is seen that \(u_{M}\in {\mathcal {N}}_{-}\cap {\mathcal {B}}\). This complete the proof.
6 The Proof of Theorem 1.13 and Theorem 1.14
Proof of Theorem 1.13
Let us denote \(u_{n}:=u(x,t_{n})\). Since \(\left\{ u_{n}\right\} \) is uniformly bounded in \(H^{1}_{0}(\Omega )\), then there exists a subsequence (here we still denote by \(\left\{ u_{n}\right\} \) ) and a function \(w\in H^{1}_{0}(\Omega )\) such that
Let \(U_{n}:=u(t_{n}+s)\) for \(s\in (0,1)\). Clearly, \(U_{n}\) is uniformly bounded in \(H^{1}_{0}(\Omega )\), we show
Indeed, for \(s\in (0,1)\), by (2.4), we have
which means \(u_{t}\in L^{2}(\Omega )\). So
for \(0\le s\le 1\) as \(t_{n}\rightarrow \infty \), which implies that \(\Vert u(s+t_{n})-u(t_{n})\Vert _{2}\rightarrow 0\) as \(t_{n}\rightarrow \infty \) for \(0\le s\le 1\). Therefore, we have
and
Since \(\left\{ U_{n}\right\} \) is uniformly bounded in \(H^{1}_{0}(\Omega )\), by (4.7), we also have \(\left( |x|^{-\mu }*|U_{n}|^{2^{*}_{\mu }}\right) |U_{n}|^{2^{*}_{\mu }-2}U_{n}\) is bounded in \(L^{\frac{2N}{N+2}}\), and
In order to show that w is an equilibrium, we pass to the limit (as \(t_{n}\rightarrow \infty \)) in the identity (2.1) with a suitably chosen test function. Let
where
Take \(\phi \) as test function in (2.1), we have
Furthermore, by transforming about t, we get
From the choice of \(\rho \), we can derive
Consequently, the assertion follows then from (6.1).
Proof of Theorem 1.14
Let \(u=u(t,x)\) be a global solution of problem (P). Then, we have
And hence, there exists a sequence \(\left\{ t_{n}\right\} \) satisfying \(t_{n}\rightarrow \infty \) as \(n\rightarrow \infty \) such that
Indeed, on the contrary, if there exist \(c>0\) such that \(\int _{\Omega }\vert u_{t}(t_{n},x)\vert ^{2}dx>c\) as \(n\rightarrow \infty \), then, we can derive a contradiction with (6.3).
Next, let \(u_{n}:=u(t_{n},x)\). By Theorem 1.4 and Remark 1.3, we have \(J_{\mu }(u(t))>0\) for \(t>0\). Then, by (2.4), we have
Therefore, we have for the sequence \(\left\{ t_{n}\right\} \) hold
Then, (6.4) and (6.5) implies that \(u_{n}:=u(t_{n},x)\) is a PS sequence related to the stationary equation of problem (P). similar to the argument of [9], it is easy to prove that there exists a constant C such that
and then there exists a subsequence (denote still by \(\left\{ u_{n}\right\} \) ) and a function w such that
Furthermore, \(u_{n}\rightarrow w\not = 0\) in \(H^{1}_{0} (\Omega )\), which means that w is a nontrivial stationary solution.
Data Availibility
No new data or materials have been used in the preparation of this paper.
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Funding
Minbo Yang is partially supported by NSFC (11971436, 12011530199) and ZJNSF (LZ22A010001, LD19A010001). Vicentiu D. Rădulescu is partially supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/UEFISCDI, project number PCE 137/2021, within PNCDI III.
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Zhang, J., Rădulescu, V.D., Yang, M. et al. Global Existence and Blow-up Solutions for a Parabolic Equation with Critical Nonlocal Interactions. J Dyn Diff Equat (2023). https://doi.org/10.1007/s10884-023-10278-y
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DOI: https://doi.org/10.1007/s10884-023-10278-y
Keywords
- Nonlocal parabolic equation
- Hardy–Littlewood–Sobolev critical exponent
- Global existence
- Asymptotic behavior
- Finite time blow-up