Abstract
In this paper, we study the following parabolic problem of Kirchhoff type with logarithmic nonlinearity:
where \([u]_s\) is the Gagliardo seminorm of u, \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain with Lipschitz boundary, \(0<s<1\), \({\mathcal {L}}_K\) is a nonlocal integro-differential operator defined in (1.2), which generalizes the fractional Laplace operator \((-\Delta )^s\), \(u_0\) is the initial function, and \(M:[0,+\infty )\rightarrow [0,+\infty )\) is continuous. Let \(J(u_0)\) be the initial energy (see (2.1) for the definition of J), \(d>0\) be the mountain-pass level given in (2.4), and \({\widetilde{M}}\in (0,d]\) be the constant defined in (2.6). Firstly, we get the conditions on global existence and finite time blow-up for \(J(u_0)\le d\). Then we study the lower and upper bounds of blow-up time to blow-up solutions under some appropriate conditions. Secondly, for \(J(u_0)\le {\widetilde{M}}\), the growth rate of the solution is got. Moreover, we give some blow-up conditions independent of d and study the upper bound of the blow-up time. Thirdly, the behavior of the energy functional as \(t\rightarrow T\) is also discussed, where T is the blow-up time. In addition, for \(J(u_0)\le d\), we give some equivalent conditions for the solutions blowing up in finite time or existing globally. Finally, we consider the existence of ground state solutions and the asymptotical behavior of the general global solution.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In this paper, we study the global existence and blow-up phenomena for the following fractional Kirchhoff-type parabolic problem with logarithmic nonlinearity:
where
\(\Omega \subset {\mathbb {R}}^N\) is a bounded domain with Lipschitz boundary \(\partial \Omega \), \({\mathcal {L}}_K\) is a nonlocal integro-differential operator, which is defined by
Here, \(K:{\mathbb {R}}^N\setminus \{0\}\rightarrow {\mathbb {R}}^+\) is a function with the following properties:
- \((k_1)\):
-
\(\gamma K\in L^1({\mathbb {R}}^N)\), with \(\gamma (x):=\min \{|x|^2,1\}\);
- \((k_2)\):
-
there exists \(K_0>0\) such that \(K(x)\ge K_0|x|^{-N-2s}\) for all \(x\in {\mathbb {R}}^N\setminus \{0\}\).
Furthermore, we make the following assumptions:
- \((M_1)\):
-
\(0<s<1\), \(M(\tau ):=a+b\tau ^{\theta -1}\) for \(\tau \in {\mathbb {R}}_0^+:=[0,+\infty )\) (\(a\ge 0\), \(b>0\) are two constants), \(\theta \in \left[ 1,{2_s^*}/{2}\right) \), \(p\in (2\theta ,2_s^*)\). Here,
$$\begin{aligned} 2_s^*:=\left\{ \begin{array}{ll} \displaystyle \frac{2N}{N-2s}, &{} \hbox { if }2s<N; \\ \displaystyle \infty , &{} \hbox { if }2s\ge N. \end{array} \right. \end{aligned}$$
In the past few decades, more and more attention has been devoted to the study of Kirchhoff type problems. More specifically, Kirchhoff in 1883 proposed the following Kirchhoff model
which was as a generalization of the well-known D’Alembert wave equation for free vibrations of elastic strings, where the above constants have the following meanings: L is the length of the string, h is the area of the cross-section, E is the Young modulus of the material, \(\rho \) is the mass density and \(P_0\) is the initial tension.
It is worth mentioning that the above equation received much attention after the work of Lions [24], where a functional analysis framework was proposed for the following higher dimension problem in presence of an external force term f:
where \(\Delta \) denotes the Euclidean Laplace operator.
Recently, in [14], Han and Li studied the following initial boundary value problem for a class of Kirchhoff type parabolic equation with a nonlinear term
Here the diffusion coefficient \(M(\tau )=a+b\tau \) with the parameters a, b being positive, \(\Omega \subset {\mathbb {R}}^N (N\ge 1)\) is a bound domain with smooth boundary \(\partial \Omega \), \(3<q\le 2^*-1\), where \(2^*\) is the Sobolev conjugate of 2. By using the potential well theory and variational methods, the authors obtained the global existence and finite time blow-up of solutions when the initial energy was subcritical, critical and supercritical. After this work, in [15], the authors investigated the upper and lower bounds of blow-up time to the blow-up solutions of problem (1.3).
It is well known that many mathematical models involving fractional and nonlocal operators are actively studied in recent years. More precisely, this type of operators arises in a quite natural way in many applications, such as finance, physics, fluid dynamics, population dynamics, image processing, minimal surfaces and game theory. As for the research motivation, we would like to point out that Applebaum in [1] viewed the fractional Laplacian operators of the form \((-\Delta )^s\) as the infinitesimal generators of stable radially symmetric Lévy processes. Laskin in [19] deduced the fractional Schrödinger equation as a result of expanding the Feynman path integral, from the Brownian-like to the Lévy-like quantum mechanical paths. In particular, we would like to point out that \((-\Delta )^s\) can be reduced to the classical Laplace operator \(-\Delta \) as \(s\rightarrow 1^-\), see [9] for more details. For more recent results involving the fractional Laplacian, interested readers may refer to, for example, [2, 3, 9, 11, 18, 26, 36] and the references therein.
In particular, in [32], the authors studied the following parabolic equations of Kirchhoff type involving the fractional Laplacian:
By using the Galerkin method and differential inequality technique, the local existence of weak solutions and the conditions on blow-up were studied.
In recent years, logarithmic nonlinearity appears frequently in partial differential equations which describes important physical phenomena (see [5, 6, 10, 12, 16, 17, 25, 29, 42]) and the references therein). Especially, in the classical case, Chen and Tian [5] studied the following semilinear pseudo-parabolic equation with logarithmic nonlinearity:
in a bounded domain \(\Omega \subset {\mathbb {R}}^N\) \((N\ge 1)\) with zero Dirichlet boundary condition. By using the logarithmic Sobolev inequality (see [7, 8, 21]), they studied the existence of global solution, blow-up at \(\infty \) and behavior of vacuum isolation of the solutions, and they also compared the difference between logarithmic nonlinearity and polynomial nonlinearity.
Inspired by the above works, in the present article we consider model (1.1). To our best knowledge, this is the first attempt to study the properties of the solutions for Kirchhoff-type equation with logarithmic nonlinearity. In this paper, we mainly discuss the properties of global existence and finite time blow-up for the solutions of problem (1.1) when the initial energy is subcritical and critical by potential well method which was established by Payne and Sattinger [27] and the concavity method which was established by Levine [22, 23], see also [12, 13, 33,34,35, 40, 41, 43] and references therein for more applications of these two methods. Furthermore, we also obtain the growth estimates of blow-up solutions. Moreover, the blow-up conditions independent of mountain-pass level are also investigated. In particular, under some appropriate conditions, we obtain the upper and lower bounds of blow-up time to blow-up solutions of problem (1.1). Finally, we consider the ground state solutions for the stationary problem. Here we say the initial energy is subcritical and critical if \(J(u_0)<d\) and \(J(u_0)=d\) are satisfied respectively, where \(J(u_0)\) denotes the initial energy defined in (2.1) and \(d>0\) is the mountain-pass level defined in (2.4). We remark that to handle the logarithmic nonlinear term of problem (1.1), we use some other methods instead of logarithmic Sobolev inequality, which is a key inequality to get the results in [4,5,6, 20, 21, 38].
Throughout this paper, we denote by \((\cdot ,\cdot )\) the \(L^2(\Omega )\)-inner product, i.e.
We also denote the norm of \(L^{\gamma }(\Omega )\) for \(1\le \gamma \le \infty \) by \(\Vert \cdot \Vert _{\gamma }\). That is, for any \(u\in L^{\gamma }(\Omega )\),
Now we recall some necessary properties of fractional Sobolev spaces which will be used later. Let X be the linear space of Lebesgue measurable function \(u:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) whose restrictions to \(\Omega \) belong to \(L^2(\Omega )\) and such that
where \({\mathcal {Q}}={\mathbb {R}}^{2N}\setminus ({\mathcal {C}}\Omega \times {\mathcal {C}}\Omega )\) and \({\mathcal {C}}\Omega ={\mathbb {R}}^N\setminus \Omega \). The space X is endowed with the norm
for all \(\varphi \in X\). We observe that bounded and Lipschitz functions belong to X, thus X is not reduced to \(\{0\}\).
The functional space Z denotes the closure of \(C_0^\infty (\Omega )\) in X. The scalar product defined for any \(\varphi \), \(\psi \in Z\) as
makes Z a Hilbert space. The norm
is equivalent to the usual norm defined in (1.5). Note that in (1.5)–(1.7) the integrals can be extended to \({\mathbb {R}}^{2N}\), since \(u=0\) a.e. in \({\mathcal {C}}\Omega \). By Lemma 6 of [28] and \((k_1)\), the Hilbert space \(Z=(Z,\Vert \cdot \Vert _Z)\) is continuously embedded in \(L^r(\Omega )\) for any \(r\in [1,2_s^*]\). Hence there exists \(C_r>0\) such that
Next, we consider the eigenvalue of the operator \({\mathcal {L}}_K\) with homogeneous Dirichlet boundary data, namely the eigenvalue of the the problem (see [32])
we denote by \(\lambda _1\) the first eigenvalue of problem (1.9), i.e.
The rest of this paper is organized as follows. In Sect. 2, we state the main results of this paper. In Sect. 3, we give some important lemmas, which will be used in the proof of the main results. In Sect. 4, we give the proof of the main results.
2 Main Results
In this section, we will give the main results of this paper and we always assume \((M_1)\) holds. The energy functional J and the Nehari functional I are as follows:
and
where \(\langle \cdot ,\cdot \rangle \) denotes the dual pairing between Z and \(Z'\).
By \((M_1)\), we know that \(2<p<2_s^*\). Let \(\varrho :=2_s^*-p>0\). Since \(\log \left( |u|^\varrho \right) \le |u|^\varrho \), it follows from (1.8) that
and \(\Vert u\Vert _p\le C_p\Vert u\Vert _Z\). So J and I are well-defined for \(u\in Z\).
Obviously, from (2.1) and (2.2), we have
Let
denote the mountain-pass level, where N is the Nehari manifold, which is defined by
By Lemma 5, we know that d satisfies
where \(r_*\) is a positive constant defined in (3.6) of Lemma 4.
Moreover, we define
Finally, the potential well W and its corresponding set V are defined by
To state the main results succinctly, we need the following two definitions.
Definition 1
(Weak solution) A function \(u=u(t)\in L^\infty (0,T;Z)\) is called a weak solution of problem (1.1), if \(u_t\in L^2(0,T;L^2(\Omega ))\) and the following equality holds
for all \(\phi \in Z\). Moreover, the following inequality
holds for a.e. \(t\in (0,T)\).
Definition 2
(Maximal existence time) Let \(u=u(t)\) be a weak solution of problem (1.1). We define the maximal existence time T of u as follows:
-
(1)
If u exists for all \(0\le t<\infty \), then \(T=\infty \);
-
(2)
If there exists a \(t_0\in (0,\infty )\) such that u exists for \(0\le t<t_0\), but doesn’t exist at \(t=t_0\), then \(T=t_0\).
Based on the above preparations, the main results of this paper are as follows. The first result is about global existence.
Theorem 1
Let \((M_1)\) hold, \(u_0\in Z\). Assume that \(J(u_0)<d\) and \(I(u_0)>0\). Then problem (1.1) admits a global weak solution \(u(t)\in L^\infty (0,\infty ;Z)\) with \(u_t\in L^2(0,\infty ;L^2(\Omega ))\) and \(u(t)\in W\) for \(0\le t<\infty \). Furthermore, the weak solution is unique if it is bounded. Moreover, for any \(\varepsilon \in (0,2_s^*-p]\), if \(J(u_0)<d(\varepsilon )\), then
where
Here \(\lambda _1\), \(r(\varepsilon )\) and \(C_*\) are defined in (1.10), (3.2) and (3.3) respectively.
Remark 1
We show that \(d(\varepsilon )\le d\). In fact, for any \(u\in N\). By (2) of Lemma 3, we get \(\Vert u\Vert _Z>r(\varepsilon )\). Then it follows from from (2.3) that
Then by the definition of d in (2.4), we get \(d(\varepsilon )\le d\).
By using Theorem 1, we get the following corollary:
Corollary 1
Let \((M_1)\) hold, \(u_0\in Z\). Assume that \(J(u_0)\le d\) and \(I(u_0)\ge 0\). Then problem (1.1) admits a global weak solution \(u(t)\in L^\infty (0,\infty ;Z)\) with \(u_t\in L^2(0,\infty ;L^2(\Omega ))\) and \(u(t)\in {\overline{W}}\) for \(0\le t<\infty \).
As the other side of the above theorem, we have the following blow-up result.
Theorem 2
Let \((M_1)\) hold, \(u_0\in Z\). If \(J(u_0)\le d\), \(I(u_0)<0\), and \(u=u(t)\) is a corresponding solution of problem (1.1), then u(t) blows up at some finite time T in the sense of
Moreover,
-
1.
if \(J(u_0)<d\), then
$$\begin{aligned} T\le \frac{4(p-1)\Vert u_0\Vert _2^2}{p(d-J(u_0))(p-2)^2}; \end{aligned}$$ -
2.
if \(2\theta<p<2\theta +2-{4\theta }/{2_s^*}\), then for any \(\varepsilon \in \left( 0,2\theta +2-{4\theta }/{2_s^*}-p\right) \), it holds
$$\begin{aligned} T>\frac{1}{2{\widehat{C}}(\zeta -1)\Vert u_0\Vert _2^{2(\zeta -1)}} \end{aligned}$$and
$$\begin{aligned} \Vert u\Vert _2>\left( 2{\widehat{C}}(\zeta -1)(T-t)\right) ^{-\frac{1}{2(\zeta -1)}}, \end{aligned}$$where
$$\begin{aligned} \begin{aligned} \zeta&=\frac{\beta \theta (p+\varepsilon )}{2\theta -(1-\beta )(p+\varepsilon )}{>1}\ \ (\hbox {see Remark}~2),\\ {\widehat{C}}&=\left( \frac{{\widetilde{C}}^{p+\varepsilon }}{\varepsilon b^{\frac{(1-\beta )(p+\varepsilon )}{2\theta }}}\right) ^{\frac{2\theta }{2 \theta -(1-\beta )(p+\varepsilon )}}. \end{aligned} \end{aligned}$$Here,
$$\begin{aligned} \tilde{C}=\sup _{u\in Z\setminus \{0\}}\frac{\Vert u\Vert _{p+\varepsilon }}{\Vert u\Vert _Z^{1-\beta }\Vert u\Vert _2^\beta }\in (0,\infty )~~~(\hbox {see Remak}~2) \end{aligned}$$(2.13)and
$$\begin{aligned} \beta =\frac{2(2_s^*-p-\varepsilon )}{(p+\varepsilon )(2_s^*-2)}{\in (0,1)}~~~(\hbox {see Remak}~2). \end{aligned}$$(2.14)
Remark 2
In this remark, we show that \(\beta \in (0,1)\), \(\tilde{C}\) is well-defined and \(\zeta >1\).
-
1.
Since \(\theta \in \left[ 1,{2_s^*}/{2}\right) \), \(\varepsilon \in \left( 0,2\theta +2-{4\theta }/{2_s^*}-p\right) \) and \(2_s^*>2\), we get
$$\begin{aligned} p+\varepsilon<2\theta +2-\frac{4\theta }{2_s^*}=2\left( 1-\frac{2}{2_s^*}\right) \theta +2<2\left( 1-\frac{2}{2_s^*}\right) \frac{2_s^*}{2}+2=2_s^*,\nonumber \\ \end{aligned}$$(2.15)which implies \(\beta >0\). On the other hand, by \(p+\varepsilon >2\), we obtain \(2\cdot 2_s^*<2_s^*(p+\varepsilon )\), i.e., \(2\cdot 2_s^*-2p-2\varepsilon <2_s^*(p+\varepsilon )-2p-2\varepsilon =(p+\varepsilon )(2_s^*-2)\), thus, we get \(\beta \in (0,1)\).
-
2.
Since \(2<p+\varepsilon <2_s^*\), we get there exists a positive constant such that
$$\begin{aligned} \Vert u\Vert _{p+\varepsilon }\le C\Vert u\Vert _{2_s^*}^{1-\beta }\Vert u\Vert _2^\beta , \end{aligned}$$which, together with (1.8), implies
$$\begin{aligned} \Vert u\Vert _{p+\varepsilon }\le C C_{2_s^*}^{1-\beta }\Vert u\Vert _Z^{1-\beta }\Vert u\Vert _2^\beta . \end{aligned}$$Then \(\tilde{C}\) is well-defined. Here \(\beta \) satisfies
$$\begin{aligned} \frac{1}{p+\varepsilon }=\frac{1-\beta }{2_s^*}+\frac{\beta }{2}, \end{aligned}$$i.e., (2.14) holds.
-
3.
Now, we prove \(\zeta >1\). In fact, from the definitions of \(\zeta \) and \(\beta \), by a direct computation, we have
$$\begin{aligned} \zeta =\frac{2\theta (2_s^*-p-\varepsilon )}{2\theta (2_s^*-2)-2_s^*(p+\varepsilon -2)}. \end{aligned}$$(2.16)Since \(\varepsilon <2\theta +2-{4\theta }/{2_s^*}-p\), we get
$$\begin{aligned} 2_s^*(p+\varepsilon -2)<2_s^*\left( 2\theta -\frac{4\theta }{2_s^*}\right) =2\theta (2_s^*-2), \end{aligned}$$which, together with (2.15) and (2.16), implies
$$\begin{aligned} \zeta>1&\Leftrightarrow 2\theta (2_s^*-p-\varepsilon )>2\theta (2_s^*-2)-2_s^* (p+\varepsilon -2)\\&\Leftrightarrow 2_s^*(p+\varepsilon -2)>2\theta (2_s^*-2)-2\theta (2_s^*-p-\varepsilon ) =2\theta (p+\varepsilon -2). \end{aligned}$$Then by \(p+\varepsilon >2\) and \(\theta <2_s^*/2\), we get \(\zeta >1\).
The next theorem shows lower bound of the growth rate for the solution got in above theorem under more specific assumptions on \(J(u_0)\) and \(I(u_0)\) (note that, by (2.6), \({\widetilde{M}}\le d\)).
Theorem 3
Let \((M_1)\) hold, \(u_0\in Z\) satisfy \(I(u_0)<0\) and \(J(u_0)\le {\widetilde{M}}\). Then for any \(\gamma \in \left[ 0,{2}/{2_s^*}\right] \), there exists a \(t_\gamma \in (0,T)\) such that the weak solution \(u=u(t)\) of problem (1.1) satisfies
for all \(t\in [t_\gamma ,T)\), where
and
Remark 3
Since \(p<2_s^*\) and \(\gamma \in \left[ 0,{2}/{2_s^*}\right] \), we have \(p\gamma <2\). Then the constant \(C_\gamma \) is well-defined and \(1\le 2/(2-p\gamma )<\infty \).
In view of the above results, one can see they all depend on the mountain-pass level d. Next, we give some blow-up results independent of d but related to \(\lambda _1\), where \(\lambda _1>0\) is the first eigenvalue of problem (1.9).
Theorem 4
Let \((M_1)\) hold and \(u=u(t)\) be a weak solution to problem (1.1). If
then u(t) blows up at some finite time T in the sense of
Moreover, we have
Furthermore, u(t) grows exponentially with \(L^2\)-norm for all \(t\in [0,T)\), that is,
where \(A=\frac{(p-2\theta )b\lambda _1^\theta \Vert u_0\Vert _2^{2\theta -2}}{\theta }\).
The next theorem is about the asymptotic behavior of J(u(t)) as \(t\rightarrow T\), where u(t) is the blow-up solution got from the above theorems.
Theorem 5
Let u(t) be the blow-up solution of problem (1.1) with \(I(u_0)<0\), \(J(u_0)\le d\) or (2.17) holds, and assume T is the maximum existence time of u(t), then
Next, we derive some sufficient and necessary conditions for the solutions blowing up in finite time.
Theorem 6
Let u(t) be a solution of problem (1.1) and \(T\in (0,+\infty ]\) be the maximal existence time of u(t). Then
-
1.
if \(u_0\in Z\setminus \{0\}\) and \(J(u_0)<d\), we have following conclusions:
-
(1)
\(I(u_0)<0\Leftrightarrow T<+\infty \Leftrightarrow \) there exists a \(t_0\in [0,T)\) such that \(J(u(t_0))<0\);
-
(2)
\(I(u_0)>0\Leftrightarrow T=+\infty \Leftrightarrow J(u(t))>0\) for all \(t\in [0,T)\);
-
(1)
-
2.
if \(u_0\in Z\setminus \left\{ N\cup \{0\}\right\} \) and \(J(u_0)=d\), we have following conclusions:
-
(3)
\(I(u_0)<0\Leftrightarrow T<+\infty \Leftrightarrow \) there exists a \(t_0\in [0,T)\) such that \(J(u(t_0))<0\);
-
(4)
\(I(u_0)>0\Leftrightarrow T=+\infty \Leftrightarrow J(u(t))>0\) for all \(t\in [0,T)\),
-
(3)
where N is defined in (2.5).
The next problem we will consider is can the mountain-pass level d defined in (2.4) be achieved by some \(u\in N\)? To this end, we consider the steady-state corresponding to problem (1.1), i.e., the following boundary value problem:
A function \(u\in Z\) is called a weak solution of problem (2.19), if the following equality
holds for all \(\phi \in Z\). Then we introduce the set
where J is defined in (2.1), \(Z'\) is the dual space of Z, and \(\langle \cdot ,\cdot \rangle \) is the dual product between \(Z'\) and Z. We have the following two theorems:
Theorem 7
Assume \((M_1)\) hold. Let N be the set defined in (2.5), then there exists a function \(v_0\in N\) such that
-
(1)
\(J(v_0)=\inf _{u\in N}J(u)=d\);
-
(2)
\(v_0\) is a ground-state solution of problem (2.19), i.e., \(v_0\in \Gamma \setminus \{0\}\) and \(J(v_0)=\inf _{u\in \Gamma \setminus \{0\}}J(u)\).
By Theorem 1, we know that the global solution converges to 0 as \(t\rightarrow \infty \) when \(u_0\) satisfies some special conditions, how about the general global solutions? For this question, we have the following results:
Theorem 8
Assume \((M_1)\) hold. Let \(u=u(t)\) be a global solution to problem (1.1). Then there exists a \(u^*\in \Gamma \) and an increasing sequence \(\{t_k\}_{k=1}^\infty \) with \(t_k\rightarrow \infty \) as \(k\rightarrow \infty \) such that
3 Preliminaries
In this section, we give some lemmas, which will be needed in our proofs. Throughout this section, we denote by \(u=u(t)\) the solution to problem (1.1) with initial value \(u_0\), whose maximal existence time is T.
Let \((M_1)\) hold. For any \(\varepsilon \) satisfying
we define
where \(C_*\) is the optimal embedding constant of \(Z\hookrightarrow L^{p+\varepsilon }(\Omega )\) (see (1.8)), i.e.
The following lemma is used to derive the upper bound of the blow-up time.
Lemma 1
[22, 23] Suppose that \(0<T\le \infty \) and suppose a nonnegative function \(F(t)\in C^2[0,T)\) satisfies
for some constant \(\gamma >0\). If \(F(0)>0\), \(F'(0)>0\), then
and \(F(t)\rightarrow \infty \) as \(t\rightarrow T\).
Lemma 2
[28] For any bounded sequence \(\{v_j\}_{j=1}^\infty \) in Z and any \(m\in [1,2_s^*)\), there exists a \(v\in L^m({\mathbb {R}}^N)\), with \(v=0\) a.e. in \({\mathbb {R}}^N\backslash \Omega \), such that up to a subsequence, still denoted by \(\{v_j\}_{j=1}^\infty \),
Lemma 3
Assume \((M_1)\) hold. Let \(u\in Z\setminus \{0\}\). Then for any \(\varepsilon \) satisfying (3.1) we have
-
(1)
if \(0<\Vert u\Vert _Z\le r(\varepsilon )\), then \(I(u)>0\);
-
(2)
if \(I(u)\le 0\), then \(\Vert u\Vert _Z>r(\varepsilon )\),
where \(r(\varepsilon )\) is defined in (3.2).
Proof
Since \(u\in Z\setminus \{0\}\), we get \(|u(x)|>0\) for a.e. \(x\in \Omega \). By a simple computation, we know (for any \(\varepsilon >0\))
Then by the above inequality and the definition of I(u), we have
For any \(\varepsilon \) satisfying (3.1), it follows from (3.3) that
then by (3.4) we get
(1) If \(0<\Vert u\Vert _Z\le r(\varepsilon )\), then it follows from (3.2) that
so by (3.5) we obtain \(I(u)>0\).
(2) If \(I(u)\le 0\), according to (3.5), we get
which implies
\(\square \)
Lemma 4
Assume \((M_1)\) hold. With the notations in Lemma 3,
exists and
where
and
Here, \(|\Omega |\) is the measure of \(\Omega \), \(\kappa \) is the optimal embedding constant of \(Z\hookrightarrow L^{p}(\Omega )\), i.e.,
Proof
Obviously \(r_*\), if it exists, is positive. So in order to prove the lemma. We only need to prove \(r(\varepsilon )\le \sigma (\varepsilon )\), \(r^*\) exists and \(r^*<\infty \).
Firstly, we prove \(r(\varepsilon )\le \sigma (\varepsilon )\). For any \(u\in Z\), since \((M_1)\) holds and \(\varepsilon \in (0,2_s^*-p]\), we have \(u\in L^p(\Omega )\cap L^{p+\varepsilon }(\Omega )\). By Hölder’s inequality we have
which, together with (3.3) and (3.10), implies
Then it follows from (3.2) that
where \(\sigma (\varepsilon )\) is defined in (3.9).
Secondly, we prove \(r^*\) exists and \(r^*<\infty \). Since \(\varepsilon \in (0,2_s^*-p]\) and \(\sigma (\varepsilon )\) is continuous on \([0,2_s^*-p]\), we have \(r^*\) exists and
\(\square \)
Based on the above two lemmas, we have the following corollary:
Corollary 2
Assume \((M_1)\) hold. Let \(u\in Z\setminus \{0\}\).
-
(1)
if \(0<\Vert u\Vert _Z<r_*\), then \(I(u)>0\);
-
(2)
if \(I(u)\le 0\), then \(\Vert u\Vert _Z\ge r_*\),
where \(r_*\) is defined in (3.6) of Lemma 4.
Proof
We only need to prove (1) since (2) is the direct result of (1). We fix \(u\in Z\setminus \{0\}\) such that \(0<\Vert u\Vert _Z<r_*\). Then by the definition of \(r_*\) in (3.6), there exists a \(\varepsilon _0\) satisfying (3.1) such that \(\Vert u\Vert _Z\le r(\varepsilon _0)\), where \(r(\cdot )\) is defined in (3.2). Then by (1) of Lemma 3, \(I(u)>0\). \(\square \)
Lemma 5
Assume \((M_1)\) hold. Then we have
where d is defined in (2.4) and \(r_*\) is defined in (3.6) of Lemma 4.
Proof
For all \(u\in N\), we have \(u\in Z\setminus \{0\}\) and \(I(u)=0\). Thus by (2) of Corollary 2, we know \(\Vert u\Vert _Z\ge r_*\), and then from (2.3) we get
which gives (3.13). \(\square \)
Lemma 6
Assume \((M_1)\) hold. Let \(u\in Z\) satisfy \(I(u)<0\). Then there exists a \(\lambda ^*\in (0,1)\) such that \(I(\lambda ^*u)=0\).
Proof
We divide the proof into two cases.
Case 1: \(a=0\). Let
Then for any \(\lambda >0\), by the definition of I(u), we have
Since \(I(u)<0\), by (3.14) and (2) of Corollary 2 we get
On the other hand, by the definition of \(\phi (\lambda )\), we have
which, together with \(p>2\theta \), implies
So by (3.15), we get that there exists a \(\lambda ^*\in (0,1)\) such that \(\phi (\lambda ^*)=b\Vert u\Vert _Z^{2\theta }\) and then \(I(\lambda ^*u)=0\).
Case 2: \(a>0\). Let
Then for any \(\lambda >0\), by the definition of I(u), we have
Since \(I(u)<0\), by (3.16) and (2) of Corollary 2 we get
On the other hand, by the definition of \(\phi (\lambda )\), we have
which, together with \(p>2\theta \ge 2\), implies
So by (3.17), we get that there exists a \(\lambda ^*\in (0,1)\) such that \(\phi (\lambda ^*)=a\Vert u\Vert _Z^2\) and then \(I(\lambda ^*u)=0\). \(\square \)
Lemma 7
Assume \((M_1)\) hold. Let \(u\in Z\) satisfy \(I(u)<0\). Then
Proof
First from Lemma 6 we know that there exists a \(\lambda ^*\in (0,1)\) such that \(I(\lambda ^*u)=0\). Set
By a direct computation, we obtain
Then from (2) of Corollary 2, we get
which implies that \(g(\lambda )\) is strictly increasing for \(\lambda >0\), hence according to \(0<\lambda ^*<1\) we get \(g(1)>g(\lambda ^*)\), namely
where the last inequality we have used the \(\lambda ^*u\in N\) and \(d=\inf _{\phi \in N}J(\phi )\), which gives (3.18) immediately. \(\square \)
Lemma 8
Let \((M_1)\) hold and \(u=u(t)\) be the corresponding solution to problem (1.1). Then for all \(t\in [0,T)\) we have
Proof
Let \(\phi =u(t)\) in (2.11) of Definition 1, we get
i.e.
Thus, we have
\(\square \)
Lemma 9
If \(J(u_0)\le d\), then the sets \(N_-\) and \(N_+\) are both invariant for u(t), i.e., if \(u_0\in N_-\) (resp. \(u_0\in N_+\)), then \(u(t)\in N_-\) (resp. \(u(t)\in N_+\)) for all \(t\in [0,T)\).
Proof
We only proof the invariance of \(N_-\) since the proof of the invariance of \(N_+\) is similar.
Firstly, we consider the case \(J(u_0)<d\). If the conclusion is not true, it follows \(J(u(t))\le J(u_0)<d\) for \(t\in [0,T)\) (see the energy inequality (2.12)) that there exists a \(t_0\in (0,T)\) such that
-
\(I(u(t_0))=0\) and \(I(u(t))<0\) for \(t\in [0,t_0)\).
From (2) of Corollary 2 we have \(\Vert u\Vert _Z> r_*>0\) for \(t\in [0,t_0)\), then by the continuity of \(\Vert u\Vert _Z\) with respect to t, we get \(\Vert u(t_0)\Vert _Z\ge r_*>0\), hence \(u(t_0)\in N\). Then it follows from the definition of d in (2.4) that \(J(u(t_0))\ge d\), a contradiction.
Secondly, we consider the case \(J(u_0)=d\). If the conclusion is not true, then by \(I(u_0)<0\), there must be a \(t_1\in (0,T)\) such that \(I(u(t_1))=0\) and \(I(u(t))<0\) for \(t\in [0,t_1)\). On the one hand, we get from (2) of Corollary 2 that \(\Vert u\Vert _Z>r_*>0\) for \(t\in [0,t_1)\), which implies that \(u(t_1)\ne 0\). Then we have \(u(t_1)\in N\) and then it follows from the definition of d in (2.4) that
On the other hand, from \((u_t,u)=-I(u(t))>0\) (see Lemma 8) for \(t\in [0,t_1)\) and \(u(t)|_{\partial {\Omega }}=0\) we deduce \(u_t\ne 0\) and then \(\mathop \int \limits _0^{t_1}\Vert u_\tau \Vert ^2_2d\tau >0\). So by (2.12) we obtain
which conflicts with (3.20). \(\square \)
4 Proof of the Theorems
Proof of Theorem 1
We divide the proof into three steps.
Step 1: Existence of a global weak solution Let \(\omega _j\), \(j=1,2,\ldots \) be the eigenfunctions of the operator \({\mathcal {L}}_k\) subject to the Dirichlet boundary condition (see [32]):
we also normalize \(\omega _j\) such that \(\Vert \omega _j\Vert _2=1\). Then \(\{\omega _j\}^\infty _{j=1}\) is a basis of Z.
First we construct the following approximate solutions \(u_m(t)\) of problem (1.1):
which satisfy
for \(j=1, 2,\ldots , m\), where \((\cdot ,\cdot )\) means the inner product of \(L^2(\Omega )\) and \(\xi _{jm}\) are given constants such that
as \(m\rightarrow \infty \). Existence of such \(\xi _{jm}\) follows from \(u_0\in Z\), and \(\{\omega _j\}^\infty _{j=1}\) is a basis of Z. The standard theory of ODEs, e.g. Peano’s theorem, yields that there exists a \(T>0\) depending only on \(\xi _{jm}\), \(j=1, 2,\ldots , m\), such that in \(g_{jm}\in C^1[0,T]\) and \(g_{jm}(0)=\xi _{jm}\). Thus \(u_m\in C^1\left( [0,T];Z\right) \).
We now try to get a priori estimates for the approximate solution \(u_m(t)\). Multiplying the first equation of (4.2) by \(g'_{jm}(t)\), summing for j from 1 to m and integrating with respect to time from 0 to t, we can obtain
Due to (4.3) and \(g_{jm}(0)=\xi _{jm}\), one has (note that we have assumed that \(J(u_0)<d\) and \(I(u_0)>0\))
and
Therefore, for sufficiently large m, we have
and
which implies that \(u_m(0)\in W\) for sufficiently large m [see the definition of W in (2.9)].
Next, we prove \(u_m(t)\in W\) for sufficiently large m and any \(t\in [0,T]\). Indeed, if it is false, there exists a sufficiently large m and a \(t_0\in (0,T]\) such that \(u_m(t_0)\in \partial W\), which implies that \(u_m(t_0)\in Z\setminus \{0\}\) and \(J(u_m(t_0))=d\) or \(I(u_m(t_0))=0\). From (4.4), \(J(u_m(t_0))=d\) is not true. So \(u_m(t_0)\in N\), then by the definition of d in (2.4), we have \(J(u_m(t_0))\ge d\), which also contradicts (4.4). Hence, \(u_m(t)\in W\) for sufficiently large m and any \(t\in [0,T]\).
By (4.4), \(I(u_m(t))>0\) for sufficiently large m (since \(u_m(t)\in W\) for sufficiently large m) and the fact that (see the definition of J and I in (2.1) and (2.2), respectively)
we obtain
holds for sufficiently large m and any \(t\in [0,T]\), which yields
and
So \(T=\infty \). Then \(u_m(t)\in W\) for all \(t\in [0,\infty )\) and all the above inequalities hold for \(t\in [0,\infty )\).
On the other hand, by a direct calculation, we know
Since
we have
Moreover, since \(\log \tau \le \frac{1}{\mu }\tau ^{\mu }\) for all \(\mu \), \(\tau \in (0,\infty )\), we can choose a positive constant \(\mu \) such that \(\frac{p(p+\mu -1)}{p-1}\in [1,2_s^*]\), then we get from (4.6) that for m sufficiently large,
where \(C_{**}\) is the optimal embedding constant of \(Z\hookrightarrow L^{\frac{p(p+\mu -1)}{p-1}}(\Omega )\).
Then it follows from (4.8), (4.9) and (4.10) that, for m large enough and \(t\in [0,\infty )\),
Therefore, by (4.5), (4.6) and (4.11), there is a function \(u=u(t)\in L^\infty (0,\infty ;Z)\) with \(u_t\in L^2(0,\infty ;L^2(\Omega ))\), \(\chi =\chi (t)\in L^2\left( 0,\infty ;L^{\frac{p}{p-1}}(\Omega )\right) \) and a subsequence of \(\{u_m\}^\infty _{m=1}\) (still denoted by \(\{u_m\}^\infty _{m=1}\)) such that for each \({\widetilde{T}}>0\), as \(m\rightarrow \infty \),
Since \(Z\hookrightarrow L^p(\Omega )\) compactly, by [39] we know that
compactly. So, in view of (4.12) and (4.14), we can assume
which implies \(u_m\rightarrow u\) a.e. in \(\Omega \times (0,{\widetilde{T}})\), and then \(|u_m|^{p-2}u_m\log |u_m|\rightarrow |u|^{p-2}u\log |u|\) a.e. in \(\Omega \times (0,{\widetilde{T}})\). Therefore, it follows from [39] that
To show that the limit function u(t) obtained above is a weak solution to problem (1.1), we fix a positive integer k and choose a function \(v\in C^1([0,{\widetilde{T}}];Z)\) of the following form
where \(\{l_j(t)\}_{j=1}^k\) are arbitrary given \(C^1\) functions. Taking \(m\ge k\) in the first equation of (4.2), multiplying the first equation of (4.2) by \(l_j(t)\), summing for j from 1 to k, and integrating with respect to t from 0 to \({\widetilde{T}}\), we obtain
Letting \(m\rightarrow \infty \) in (4.20) and recalling (4.12), (4.14), (4.16) and (4.18) yield
Since the functions of the form in (4.19) are dense in \(L^2(0,{\widetilde{T}};Z)\), (4.21) also holds for all \(v\in L^2(0,{\widetilde{T}};Z)\). By arbitrariness of \({\widetilde{T}}>0\), we know that
holds for a.e. \(t\in (0,\infty )\) and any \(\phi \in Z\).
In view of (4.12) and (4.14), we get \(u_m(0)\rightharpoonup u(0)\) weakly in \(L^2(\Omega )\). Then by (4.1), (4.3) and \(g_{jm}(0)=\xi _{jm}\), we get \(u(0)=u_0\in Z\).
In view of Definition 1 and the above discussions, to show the limit function u(t) got above is indeed a global weak solution to problem (1.1), we only need to prove (2.12) holds for a.e. \(0<t<\infty \). In fact, for a.e. \(0<t<\infty \), we choose \({\widetilde{T}}>t\). Then it follows from (4.17) that \(u_m(t)\rightarrow u(t)\) strongly in \(L^p(\Omega )\). So by (4.11) and (4.16), we have
as \(m\rightarrow \infty \).
From the convergence of (4.12), (4.14), (4.17), the definition of J in (2.1), (4.1), (4.3), (4.4), (4.22) and \(g_{jm}(0)=\xi _{jm}\), we obtain
which implies (2.12) holds for a.e. \(t\in (0,\infty )\). So the limit function u(t) got above is a global weak solution to problem (1.1). Furthermore, by using \(u_0\in W\) and (2.12), one can get \(u(t)\in W\) for \(0\le t<\infty \) and the proof is same as the proof of \(u_m(t)\in W\).
Step 2: Uniqueness of bounded global weak solution To show the uniqueness of bounded global weak solution, we assume that u, \(v\in L^\infty (0,\infty ;L^\infty (\Omega ))\) are two global weak solutions to problem (1.1). Then for any \(\phi \in Z\), we have
and
Subtracting the above two inequalities, taking \(\phi =u-v\in Z\), we obtain
Moreover, by using \(\langle u,v\rangle _Z\le \frac{\Vert u\Vert _Z^2+\Vert v\Vert _Z^2}{2}\), we have
Then combining (4.23) and (4.24) we have
where
Then we have
which implies \(\Vert \phi \Vert _2^2=0\) for \(t\ge 0\). Thus \(\phi (t)(x)=0\) a.e. in \(\Omega \times (0,\infty )\) and the uniqueness of bounded global weak solution follows.
Step 3: Decay estimates Since \(d(\varepsilon )\le d\) (see (2.4)), by step 1 we know problem (1.1) admits a global solution \(u\in L^\infty (0,\infty ;Z)\) with \(u_t\in L^2(0,\infty ;L^2(\Omega ))\) and \(u(t)\in W\) for \(0\le t<\infty \). So by the definition of W in (2.9), we have \(I(u)\ge 0\). Then it follows from (2.3) and (2.12) that
which, together with (3.3), implies
In view of (3.3) and (4.26), we obtain
By Lemma 8, we have
Then for \(J(u_0)<d(\varepsilon )\), it follows from (1.10), (4.27) and \(\log |u|<\frac{1}{\varepsilon }|u|^\varepsilon \) (for any \(\varepsilon >0\)) that
which implies
where
\(\square \)
Proof of Corollary 1
If \(u_0=0\), then problem (1.1) admits a global solution \(u(t)\equiv 0\), and the proof is complete. So in the following, we assume \(u_0\in Z\setminus \{0\}\) and the proof is divided into three cases.
Case 1: \(I(u_0)>0\) and \(J(u_0)<d\). The conclusion follows from Theorem 1.
Case 2: \(I(u_0)=0\) and \(J(u_0)<d\). This case does not happen because in this case \(u_0\in N\), then it follows from the definition of d in (2.4) that \(J(u_0)\ge d\).
Case 3: \(I(u_0)\ge 0\) and \(J(u_0)=d\). Let \(\lambda _m=1-\frac{1}{m}\) and \(m=2,3,\ldots \). Consider the following approximate problem:
Since \(u_0\in Z\setminus \{0\}\), \(\lambda _m\in (0,1)\) and \(I(u_0)\ge 0\) (i.e. \(a\Vert u_0\Vert _Z^2+b\Vert u_0\Vert _Z^{2\theta }\ge \mathop \int \limits _\Omega |u_0|^p\log |u_0|dx\)), then we have
Next, we will discuss the sign of \(I(u_{0m})\) on two aspects: \(\mathop \int \limits _\Omega |u_0|^p\log |u_0|dx\le 0\) and \(\mathop \int \limits _\Omega |u_0|^p\log |u_0|dx>0\).
(1) When \(\mathop \int \limits _\Omega |u_0|^p\log |u_0|dx\le 0\), from (4.32) we get
(2) When \(\mathop \int \limits _\Omega |u_0|^p\log |u_0|dx>0\), from (4.32) we get
On the other hand, by a simply computation, we obtain
Then by (4.33), (4.34) and (4.35), we have
which implies that \(J(\lambda _mu_0)\) is strictly increasing with respect to \(\lambda _m\). So we can get
From Theorem 1, it follows that for each \(m=2,3,\ldots \), problem (4.31) admits a global weak solution \(u_m(t)\in L^\infty (0,\infty ;Z)\) with \(u_{mt}\in L^2(0,\infty ;L^2(\Omega ))\), which satisfies \(u_m(t)\in W\) for \(0\le t<\infty \) and
holds for any \(\phi \in Z\) and a.e. \(t>0\). Moreover,
From (4.36) and the fact that
we obtain
Then the remainder of the proof is similar to that in the proof of Theorem 1. \(\square \)
Proof of Theorem 2
We divide the proof into three steps.
Step 1: Blow-up in finite time
We divide the proof into two cases.
Case 1: \(J(u_0)<d\). Let \(u=u(t)\), \(t\in [0,T)\) be a weak solution of problem (1.1) with \(J(u_0)<d\) and \(I(u_0)<0\), where T is the maximal existence time. Then from Lemma 9, we have \(u(t)\in V\). Next let us prove that u(t) blows up in finite time. Arguing by contradiction, we suppose that \(T=+\infty \) and define
Then we have
and
hence
so by (1.8) and the above inequality, we have
In addition, from
we obtain
Hence, by (4.39), (4.40) and the Schwartz’s inequality we deduce that
Moreover, since \(M''(t)=-2I(u(t))>0\) (note that \(u(t)\in V\) for \(t\in [0,T)\)), so we have \(M'(t)>M'(0)=\Vert u_0\Vert _2^2>0\). Then by (4.41) we obtain
From Lemma 7 one has
Then we can obtain
Therefore,
Hence there exists a \(t_0\ge 0\) such that
which combined with (4.42) give the inequality
Then we get from Lemma 1 that the maximal existence time \(T_1\) of M(t) satisfying \(T_1<\infty \) and
which contradicts \(T=\infty \).
Case 2: \(J(u_0)=d\) Since \(I(u(t))<0\) for \(t\ge 0\) (see Lemma 9), it follows that
Hence we can get \(\Vert u_t\Vert ^2_2>0\) for \(t\ge 0\). Thus by (2.12), there exists a \(t_1>0\) such that
If we take \(t_1\) as the initial time, then similar to the Case 1 in the proof of this section, we can obtain the finite time blow up result. The proof of Step 1 is complete.
Step 2: Upper bound estimate of the blow-up time
Let \(u=u(t)\) be a solution of problem (1.1) with initial value \(u_0\) satisfying \(I(u_0)<0\) and \(J(u_0)<d\). By Step 1, the maximal existence time \(T<+\infty \). Let
By (2.12), Lemmas 8 and 9, we have
- (R1):
-
\(J(u(t))+\nu ^2(t)\le J(u_0)\), \(\forall t\in [0,T)\);
- (R2):
-
\(\frac{d}{dt}\Vert u\Vert _2^2=-2I(u(t))\), \(\forall t\in [0,T)\);
- (R3):
-
\(u(t)\in N_-\), i.e., \(I(u(t))<0\), \(\forall t\in [0,T)\).
Consider the following functional:
where \(\alpha \) and \(\beta \) are two positive constants to be determined later. Then by (R2) and (R3), we have
which implies
and (by (R1), (R2) and Lemma 7)
By Schwartz’s inequality, we have
which, together with the definition of F(t), implies
Then it follows from (4.45) and the above inequality that
Combining (4.46), (4.47) and (4.48), we get
which is nonnegative if we take \(\beta \) small enough such that
Then it follows from Lemma 1 that
By taking \(\alpha \) large enough such that
we get from (4.50) that
The above analysis shows that (\(\rho :=\alpha \beta \))
where
It is easily to find that \(f(\rho ,\alpha )\) is increasing with \(\alpha \). Then
Step 3: Lower bound estimate of the blow-up time
From Step 1 we know that the weak solution \(u(t)=u(x,t)\) of problem (1.1) blows up at finite time T. Now, we estimate the lower bound of T and blow-up rate. To this end, we define a function
then we have
According to Lemma 8, we have
Moreover, by Lemma 9, we know \(I(u)<0\). Thus, by (2.13) and the inequality \(\log |u(x)|<\frac{|u(x)|^\varepsilon }{\varepsilon }\) (for any \(\varepsilon >0\)), we deduce
Since \(0<\varepsilon <2\theta +2-\frac{4\theta }{2_s^*}-p\), \(2\theta<p<2\theta +2-\frac{4\theta }{2_s^*}\) and \(\beta =\frac{2\cdot 2_s^*-2p-2\varepsilon }{(p+\varepsilon )(2_s^*-2)}\in (0,1)\), we have
Thus, by (4.56), we can get
Moreover, by remark 2, we know
Then combining (4.55) and (4.57) we have
where \({\widehat{C}}=\left( \frac{{\widetilde{C}}^{p+\varepsilon }}{\varepsilon b^{\frac{(1-\beta )(p+\varepsilon )}{2\theta }}}\right) ^{\frac{2\theta }{2\theta -(1-\beta )(p+\varepsilon )}}\).
We can prove by contradiction that for any \(t\in [0,T)\), \(f(t)>0\). If not, there exists a \(t_1\ge 0\) such that \(\Vert u(t_1)\Vert _2^2=0\), which contradicts (4.57). Then by (4.58) we have
Integrating the above inequality from 0 to t, we have
letting \(t\rightarrow T\) in (4.60) and using (4.54) we can conclude that
Similarly, integrating the inequality (4.59) from t to T, by (4.54) we have
then by the definition of f(t) we can see that
\(\square \)
Proof of Theorem 3
Let \(u=u(t)\) be the weak solution of problem (1.1) with \(I(u_0)<0\) and \(J(u_0)\le {\widetilde{M}}\). Then according to Lemma 9 we have \(I(u)<0\) for all \(t\in [0,T)\).
Let
then
and
Then by (2.3), (2.6), (2.12), (4.61) and Corollary 2, we get
Since
so we have
Combining (4.62) and (4.63), and using the Schwartz inequality, we get
Then for any \(\gamma \in \left[ 0,\frac{2}{2_s^*}\right] \), we have
Moreover, by Theorem 2, we know that u(t) blows up at some finite time, so we have
Then it follows from (4.64) that there exists a \(t_\gamma \in (0,T)\) such that for all \(t\in [t_\gamma ,T)\)
Since
it follows from (4.65) that for all \(t\in [t_\gamma ,T)\),
Then by \(2-p\gamma \ge 2-\frac{2p}{2_s^*}>0\) and \(G(t_\gamma )>0\), we have
where
Since \(G''(t)>0\) for all \(t\in [0,T)\), thus we have
i.e. (for all \(t\in [0,T)\)),
Combining with (4.66) and the above inequality, we can deduce that for any \(0\le \gamma \le \frac{2}{2_s^*}\) and \(t\in [t_\gamma ,T)\),
\(\square \)
Proof of Theorem 4
To complete this proof, we use some ideas from [30, 31] and we divide the proof into three steps.
Step 1: Blow-up in finite time
First, it follows from the definition of (2.3) and the assumption that
Actually, we may claim that \(I(u(t))<0\) for all \(t\in [0,T)\). Otherwise, there exists a \(t_0\in (0,T)\) such that \(I(u(t_0))=0\) and \(I(u(t))<0\) for all \(t\in [0,t_0)\). By Lemma 8 we know that \(\Vert u\Vert _2^2\) is strictly increasing with respect to t for \(t\in [0,t_0)\), and therefore
On the other hand, by (2.3) and (2.12), we can get
which contradicts (4.67), so we obtain \(I(u(t))<0\) for all \(t\in [0,T)\).
Next, we are going to prove the blow-up of the solution u(t) by contradiction. Fix a \({\widetilde{T}}=\frac{(4(p-1)\Vert u_0\Vert _2^2+1)^2+1}{\varrho (p-2)^2}\), where \(\varrho :=\frac{(p-2\theta )b\lambda _1^\theta }{\theta }\Vert u_0\Vert _2^{2\theta }-2pJ(u_0)>0\), then we suppose that u(t) exists globally on \([0,{\widetilde{T}}]\) and let
where \(\sigma \), \(\epsilon \) are two positive constants which will be specified later.
Then, for any \(t\in [0,{\widetilde{T}}]\), by a simply computation, we obtain
and
thus, we can get \(G(t)\ge G(0)>0\) for all \(t\in [0,{\widetilde{T}}]\).
Let
By using Hölder’s inequality, we have
Then we obtain
The above calculations show that
We choose \(\sigma =\frac{\varrho }{4(p-1)}\), then it follows that \(G(t)G''(t)-\frac{p}{2}\left( G'(t)\right) ^2\ge 0\). Therefore, by Lemma 1 it is seen that
Next, we choose \(\epsilon =\frac{4(p-1)\Vert u_0\Vert _2^2+1}{\varrho (p-2)}\), then we have \(T<{\widetilde{T}}\), which is a contraction. Hence, u(t) will blow-up at some finite time T.
Step 2: Upper bound estimate of the blow-up time
For any \(T_1\in (0,T)\), let
where \(\sigma \), \(\epsilon \) are two positive constants which will be specified later.
Then similar to Step 1 we can get
We choose \(\sigma \) small enough, such that
then it follows that \(F(t)F''(t)-\frac{p}{2}\left( F'(t)\right) ^2\ge 0\). Therefore, by Lemma 1 we obtain
Hence, letting \(T_1\rightarrow T\), we have
Let \(\epsilon \) be large enough such that
then by (4.69), we can get
In view of (4.68) and (4.70), we define
then
Let \(\varsigma =\sigma \epsilon \) and
Since \(f(\epsilon ,\varsigma )\) is decreasing with \(\varsigma \) and we obtain
Hence, by the definition of \(\varrho \) and the above inequality, we have
Step 3: Growth estimates
First, similar to Step 1, we can get \(I(u)<0\) for all \(t\in [0,T)\), so by Lemma 8, we know that \(\Vert u\Vert _2^2\) is strictly increasing with respect to t. Then by Lemma 8 and (2.3) we know
According to (2.12), we know \(J(u(t))\le J(u_0)\) for all \(t\in [0,T)\). So we can deduce from (1.10) and the above equality that
which implies
where \(A=\frac{(p-2\theta )b\lambda _1^\theta \Vert u_0\Vert _2^{2\theta -2}}{\theta }\). \(\square \)
Proof of Theorem 5
Let u(t) be the blow-up solution with \(I(u_0)<0\), \(J(u_0)\le d\) or (2.17) holds, and assume T is the maximal existence time, then by Theorem 2 and Theorem 4 we know
Moreover, by Lemma 9 and Step 1 of Theorem 4, we can get \(I(u)<0\) for all \(t\in [0,T)\), so by Lemma 8, we can infer that \(\Vert u\Vert _2^2\) is strictly increasing for all \(t\in [0,T)\).
By Hölder’s inequality, we obtain
By [37, page 75, Proposition 3.3], we know
Since \(\Vert u\Vert _2^2\) is strictly increasing on [0, T), we know \(\Vert u\Vert _2-\Vert u_0\Vert _2\ge 0\), then the above inequalities imply
So it follows from (2.12) that
Let \(t\rightarrow T\) in the above inequality and using (4.71), we get
\(\square \)
Proof of Theorem 6
We divide the proof into two cases.
Case 1: \(u_0\in Z\setminus \{0\}\) and \(J(u_0)<d\).
First, we claim that \(I(u_0)\ne 0\). Indeed, since \(u_0\ne 0\), if \(I(u_0)=0\), then by the definition of d, we get \(J(u_0)\ge d\), which contradicts \(J(u_0)<d\).
(1) If \(I(u_0)<0\), then together with \(J(u_0)<d\) and Theorem 2, we know the solution blows up in finite time, so we get \(T<+\infty \). Next we claim that if \(T<+\infty \), then we have \(I(u_0)<0\). Indeed, if \(I(u_0)>0\), then together with \(J(u_0)<d\) and Theorem 1, we get \(T=+\infty \), which contradicts \(T<+\infty \). Since \(I(u_0)\ne 0\), so the claim is true. Moreover, if \(I(u_0)<0\), then according to \(J(u_0)<d\) and Theorem 5, we can get there must exist a \(t_0\in [0,T)\) such that \(J(u(t_0))<0\). Hence, in order to complete this proof, we now only need to show
Since there exists a \(t_0\in [0,T)\) such that \(J(u(t_0))<0\), so by (2.3), we have \(I(u(t_0))<0\), then taking \(t_0\) as the initial time, by Theorem 2, we see the solution blows up in finite time, so by Theorem 1 we know that \(I(u_0)>0\) cannot happen. Since \(I(u_0)\ne 0\), so there must be \(I(u_0)<0\).
(2) If \(I(u_0)>0\), together with \(J(u_0)<d\) and Theorem 1, we get \(T=+\infty \). Next, we claim that if \(T=+\infty \), then we have \(I(u_0)>0\). Indeed, if \(I(u_0)<0\), then by (1), we have \(T<+\infty \), which is a contradiction. Since \(I(u_0)\ne 0\), so the claim is true. Moreover, if \(I(u_0)>0\), then according to \(J(u_0)<d\) and Lemma 9, we know \(I(u(t))>0\) for all \(t\in [0,+\infty )\). Thus, by (2.3) we deduce that \(J(u(t))>0\) for all \(t\in [0,+\infty )\). So we only need to prove
Since \(I(u_0)\ne 0\), if \(I(u_0)<0\), then together with \(J(u_0)<d\) and Theorem 5, we have \(\lim _{t\rightarrow T}J(u(t))=-\infty \). By the continuity of J(u(t)) with respect to t, we know there exists a \(t_0\) such that \(J(u(t_0))<0\), which contradicts \(J(u(t))>0\) for all \(t\in [0,T)\) and our proof is complete.
Case 2: \(u_0\in Z\setminus \left\{ N\cup \{0\}\right\} \) and \(J(u_0)=d\).
First, since \(u_0\in Z\setminus \left\{ N\cup \{0\}\right\} \), so we have \(I(u_0)\ne 0\).
(3) If \(I(u_0)<0\), then together with \(J(u_0)=d\) and Theorem 2, we know the solution blows up in finite time, so we get \(T<+\infty \). Next we claim that if \(T<+\infty \), then we have \(I(u_0)<0\). Indeed, if \(I(u_0)>0\), then together with \(J(u_0)=d\) and Corollary 1, we get \(T=+\infty \), which contradicts \(T<+\infty \). Since \(I(u_0)\ne 0\), so the claim is true. Moreover, if \(I(u_0)<0\), then according to \(J(u_0)=d\) and Theorem 5, we can get there must exist a \(t_0\in [0,T)\) such that \(J(u(t_0))<0\). Hence, in order to complete this proof, we now only need to show
Since there exists a \(t_0\in [0,T)\) such that \(J(u(t_0))<0\), so by (2.3), we have \(I(u(t_0))<0\), then taking \(t_0\) as the initial time, by Theorem 2, we see the solution blows up in finite time, so by Corollary 1 we know that \(I(u_0)>0\) cannot happen. Since \(I(u_0)\ne 0\), so there must be \(I(u_0)<0\).
(4) If \(I(u_0)>0\), together with \(J(u_0)=d\) and Corollary 1, we get \(T=+\infty \). Next, we claim that if \(T=+\infty \), then we have \(I(u_0)>0\). Indeed, if \(I(u_0)<0\), then by (3), we have \(T<+\infty \), which is a contradiction. Since \(I(u_0)\ne 0\), so the claim is true. Moreover, if \(I(u_0)>0\), then according to \(J(u_0)=d\) and Lemma 9, we know \(I(u(t))>0\) for all \(t\in [0,+\infty )\). Thus, by (2.3) we deduce that \(J(u(t))>0\) for all \(t\in [0,+\infty )\). So we only need to prove
Since \(I(u_0)\ne 0\), if \(I(u_0)<0\), then together with \(J(u_0)=d\) and Theorem 5, we have \(\lim _{t\rightarrow T}J(u(t))=-\infty \). By the continuity of J(u(t)) with respect to t, we know there exists a \(t_0\) such that \(J(u(t_0))<0\), which contradicts \(J(u(t))>0\) for all \(t\in [0,T)\) and our proof is complete. \(\square \)
Proof of Theorem 7
(1) By the definition of d in (2.4), N in (2.5) and (2.3), we get
Then a minimizing sequence \(\{u_k\}_{k=1}^{\infty }\subset N\) exists such that
Since \(p\in (2\theta ,2_s^*)\), (4.72) ensures that \(\{u_k\}_{k=1}^{\infty }\) is bounded in Z, i.e., there exists a constant \(\vartheta \) independent of k such that
which, together with Z is reflexive, implies there exists a subsequence of \(\{u_k\}_{k=1}^{\infty }\), still denoted by \(\{u_k\}_{k=1}^{\infty }\), and a \(v_0\) such that
Moreover, by Lemma 2, we have
Since (4.73) holds, similar to the proof (4.11), there exists a positive constant \(C_\chi \) independent of k such that
In view of (4.75) and (4.76), similar to get (4.18), there exists a subsequence of \(\{u_k\}_{k=1}^{\infty }\), still denoted by \(\{u_k\}_{k=1}^{\infty }\) such that
Similar to the proof of (4.22), by (4.75) and (4.77), we have, as \(k\rightarrow \infty \),
Since \(\{u_k\}_{k=1}^{\infty }\in N\), by the definition of N in (2.5), we get
which, together with \(\Vert \cdot \Vert _Z\) is weakly lower semi-continuous, (4.74) and (4.78), implies
Next, we claim that \(I(v_0)=a\Vert v_0\Vert _Z^2+b\Vert v_0\Vert _Z^{2\theta }-\mathop \int \limits _\Omega |v_0|^p\log |v_0|dx=0\). Indeed, if the claim is not true, then by (4.79), we get \(a\Vert v_0\Vert _Z^2+b\Vert v_0\Vert _Z^{2\theta }<\mathop \int \limits _\Omega |v_0|^p\log |v_0|dx\). Obviously, we have \(v_0\ne 0\). Then by Lemma 6, there exists a \(\lambda ^*\in (0,1)\) such that \(\lambda ^*v_0\in N\).
By (4.72), the first inequality of (4.79) and (4.75), we get
Then by \(I(\lambda ^*v_0)=0\), \(\lambda ^*\in (0,1)\), we obtain
However, since \(\lambda ^*v_0\in N\), it follows from the definition of d in (2.4) that \(J(\lambda ^*v_0)\ge d\), a contradiction. So the claim holds and we get from (4.79) that
which, together with Z is uniformly convex and (4.74), implies
Then by (2.3) and (4.72), we get
which implies \(v_0\ne 0\). Then by \(I(v_0)=0\), we get \(v_0\in N\) and \(d=J(v_0)=\inf _{u\in N}J(u)\).
(2) Finally, we prove \(v_0\) is a ground-state solution of problem (2.19). That is, \(v_0\in \Gamma \setminus \{0\}\) and
where \(\Gamma \) is defined in (2.20).
In fact, by conclusion (1) and the definition of N in (2.5) we know that
and
Therefore, \(v_0\ne 0\), and by the theory of Lagrange multipliers, there exists a constant \(\mu \in {\mathbb {R}}\) such that
Then
On the other hand, for any \(u\in Z\), we can deduce
which implies
Since \(I(v_0)=0\), we get from (2.2) that \(a\Vert v_0\Vert _Z^2+b\Vert v_0\Vert _Z^{2\theta }=\mathop \int \limits _\Omega |v_0|^p\log |v_0|dx\). Then it follows from the above equality, \(v_0\ne 0\), and \(p\in (2\theta ,2_s^*)\) that
which, together with (4.83), implies \(\mu =0\). Then we get from (4.82) that \(J'(v_0)=0\), so \(v_0\in \Gamma \setminus \{0\}\). Furthermore, by (4.81) and \(\Gamma \setminus \{0\}\subset N\), we get (4.80). \(\square \)
Proof of Theorem 8
Let \(u=u(t)\) be a global solution of problem (1.1). Without loss of generality, we may assume that
Indeed, the second inequality follows from (2.12). Now we prove the first inequality by contradiction argument. If there is a \(t_0\in [0,\infty )\) such that \(J(u(t_0))<0\), then by (2.3) we have \(I(u(t_0))<0\), so it follows from Theorem 2 that u(t) blows up in finite time, which contradicts the assumption that u(t) is global.
Since \(J(u(t))\in [0,J(u_0)]\), so there must exist a subsequence \(\{t_m\}_{m=1}^\infty \) and a constant \(c\in [0,J(u_0)]\) such that
By (2.12), we have
Letting \(t_m\rightarrow \infty \) in the above inequality, we get
which implies there is an increasing sequence \(\{t_k\}_{k=1}^\infty \) with \(t_k\rightarrow \infty \) as \(k\rightarrow \infty \) satisfying
By the definition of J in (2.1) and (2.11), for any \(\psi \in Z\), we have
which, together with (4.85), implies
as \(k\rightarrow \infty \), where \(\lambda _1\) is the first eigenvalue of problem (1.9).
By (2.2) and (4.86), there exists a positive constant \(\sigma \) such that
Then it follows from (2.1), (2.2), (4.84) and \(p\in (2\theta ,2_s^*)\) that
which implies there exists a positive constant L independent of k such that
Indeed, if \(\Vert u(t_k)\Vert _Z\) is unbounded, then there must be a \({\tilde{t}}_k\) such that \(\Vert u({\tilde{t}}_k)\Vert _Z\rightarrow \infty \), which, together with \(\theta \in \left[ 1,\frac{2_s^*}{2}\right) \) and \(p>2\theta \), implies
a contradiction.
Then by similar arguments as to get (4.74) and (4.75), there exists an increasing subsequence of the sequence \(\{t_k\}_{k=1}^\infty \), still denoted by \(\{t_k\}_{k=1}^\infty \), and a \(u^*\in Z\) such that \(u_k:=u(t_k)\) satisfies
and
By the definition of J in (2.1) and (1.7) we obtain
Similarly, we have
So, we can get
where \(C_{\theta ,b}\) is a positive constant independent of k and
According to (4.87), then similar to the proof of (4.11), we have
where \(C_L\) is a positive constant independent of k.
Therefore, by Hölder’s inequality, (4.89) and (4.93), we have
as \(k\rightarrow \infty \). By (4.88), (4.89) and (4.91), we get
as \(k\rightarrow \infty \). By (4.86) and (4.87), we have
as \(k\rightarrow \infty \).
Then it follows from (4.92), (4.94), (4.95) and (4.96) that
as \(k\rightarrow \infty \). Then we get
in \(Z'\), which, together with (4.86), implies \(J'(u^*)=0\), i.e., \(u^*\in \Gamma \). \(\square \)
References
Applebaum, D.: Lévy processes-from probability to finance and quantum groups. Notices Am. Math. Soc. 51(11), 1336–1347 (2004)
Bisci, G.M., Radulescu, V.D., Servadei, R.: Variational Methods for Nonlocal Fractional Problems, Volume 162 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2016)
Bucur, C., Valdinoci, E.: Nonlocal diffusion and applications, vol. 20 of Lecture Notes of the Unione Matematica Italiana. Springer, [Cham]; Unione Matematica Italiana, Bologna (2016)
Cao, Y., Liu, C.: Initial boundary value problem for a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic nonlinearity. Electron. J. Differ. Equ., pages Paper No. 116, 19 (2018)
Chen, H., Tian, S.: Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity. J. Differ. Equ. 258(12), 4424–4442 (2015)
Chen, H., Luo, P., Liu, G.: Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity. J. Math. Anal. Appl. 422(1), 84–98 (2015)
Del Pino, M., Dolbeault, J.: Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the $p$-Laplacian. C. R. Math. Acad. Sci. Paris 334(5), 365–370 (2002)
Del Pino, M., Dolbeault, J., Gentil, I.: Nonlinear diffusions, hypercontractivity and the optimal $L^p$-Euclidean logarithmic Sobolev inequality. J. Math. Anal. Appl. 293(2), 375–388 (2004)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Ding, H., Zhou, J.: Global existence and blow-up for a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic nonlinearity. J. Math. Anal. Appl. 478(2), 393–420 (2019)
Dipierro, S., Medina, M., Valdinoci, E.: Fractional elliptic problems with critical growth in the whole of ${\mathbb{R}}^n$, vol. 15 of Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale, Pisa (2017)
Dong, Z., Zhou, J.: Global existence and finite time blow-up for a class of thin-film equation. Z. Angew. Math. Phys. 68(4), 89 (2017)
Feng, M., Zhou, J.: Global existence and blow-up of solutions to a nonlocal parabolic equation with singular potential. J. Math. Anal. Appl. 464(2), 1213–1242 (2018)
Han, Y., Li, Q.: Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy. Comput. Math. Appl. 75(9), 3283–3297 (2018)
Han, Y., Gao, W., Sun, Z., Li, H.: Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy. Comput. Math. Appl. 76(10), 2477–2483 (2018)
Ji, C., Szulkin, A.: A logarithmic Schrödinger equation with asymptotic conditions on the potential. J. Math. Anal. Appl. 437(1), 241–254 (2016)
Ji, S., Yin, J., Cao, Y.: Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity. J. Differ. Equ. 261(10), 5446–5464 (2016)
Jiang, R., Zhou, J.: Blow-up and global existence of solutions to a parabolic equation associated with the fraction $p$-Laplacian. Commun. Pure Appl. Anal. 18(3), 1205–1226 (2019)
Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E (3) 66(5), 056108 (2002)
Le, C.N., Le, X.T.: Global solution and blow-up for a class of $p$-Laplacian evolution equations with logarithmic nonlinearity. Acta Appl. Math. 151, 149–169 (2017)
Le, C.N., Le, X.T.: Global solution and blow-up for a class of pseudo $p$-Laplacian evolution equations with logarithmic nonlinearity. Comput. Math. Appl. 73(9), 2076–2091 (2017)
Levine, H.A.: Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt}=-Au+{{{\cal{F}}}}(u)$. Trans. Am. Math. Soc. 192, 1–21 (1974)
Levine, H.A.: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. Anal. 5, 138–146 (1974)
Lions, J.-L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proceedings of International Symposium, Institute of Mathematics, Universidade Federal Rio de Janeiro, Rio de Janeiro, 1977), vol. 30 of North-Holland Math. Stud., pp. 284–346. North-Holland, Amsterdam, New York (1978)
Liu, H., Liu, Z., Xiao, Q.: Ground state solution for a fourth-order nonlinear elliptic problem with logarithmic nonlinearity. Appl. Math. Lett. 79, 176–181 (2018)
Pan, N., Zhang, B., Cao, J.: Degenerate Kirchhoff-type diffusion problems involving the fractional $p$-Laplacian. Nonlinear Anal. Real World Appl. 37, 56–70 (2017)
Payne, L.E., Sattinger, D.H.: Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math. 22(3–4), 273–303 (1975)
Servadei, R., Valdinoci, E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389(2), 887–898 (2012)
Squassina, M., Szulkin, A.: Multiple solutions to logarithmic Schrödinger equations with periodic potential. Calc. Var. Partial Differ. Equ. 54(1), 585–597 (2015)
Sun, F., Liu, L., Wu, Y.H.: Finite time blow-up for a class of parabolic or pseudo-parabolic equations. Comput. Math. Appl. 75(10), 3685–3701 (2018)
Sun, F., Liu, L., Wu, Y.H.: Finite time blow-up for a thin-film equation with initial data at arbitrary energy level. J. Math. Anal. Appl. 458(1), 9–20 (2018)
Xiang, M.Q., Rădulescu, V.D., Zhang, B.: Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions. Nonlinearity 31(7), 3228–3250 (2018)
Xu, G., Zhou, J.: Global existence and blow-up for a fourth order parabolic equation involving the Hessian. NoDEA Nonlinear Differ. Equ. Appl 24(4), 41 (2017)
Xu, G.Y., Zhou, J.: Global existence and blow-up of solutions to a singular non-Newton polytropic filtration equation with critical and supercritical initial energy. Commun. Pure Appl. Anal. 17(5), 1805–1820 (2018)
Xu, G.Y., Zhou, J.: Global existence and finite time blow-up of the solution for a thin-film equation with high initial energ. J. Math. Anal. Appl. 458(1), 521–535 (2018)
Yang, Y., Tian, X., Zhang, M., Shen, J.: Blowup of solutions to degenerate Kirchhoff-type diffusion problems involving the fractional $p$-Laplacian. Electron. J. Differ. Equ. 2018(155), 1–22 (2018)
Zeidler, E.: Nonlinear functional analysis and its applications. I. Springer, New York (1986)
Zhang, H., Liu, G., Hu, Q.Y.: Exponential decay of energy for a logarithmic wave equation. J. Partial Differ. Equ. 28(3), 269–277 (2015)
Zheng, S.: Nonlinear evolution equations, vol. 133 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. Chapman & Hall/CRC, Boca Raton (2004)
Zhou, J.: $L^2$-norm blow-up of solutions to a fourth order parabolic PDE involving the Hessian. J. Differ. Equ. 265(9), 4632–4641 (2018)
Zhou, J.: Global asymptotical behavior and some new blow-up conditions of solutions to a thin-film equation. J. Math. Anal. Appl. 464(2), 1290–1312 (2018)
Zhou, J.: Ground state solution for a fourth-order elliptic equation with logarithmic nonlinearity modeling epitaxial growth. Comput. Math. Appl. 78(6), 1878–1886 (2019)
Zhou, J.: Global asymptotical behavior of solutions to a class of fourth order parabolic equation modeling epitaxial growth. Nonlinear Anal. Real World Appl. 48, 54–70 (2019)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by Graduate Student Scientific Research Innovation Projects in Chongqing (No. CYS19087) and NSFC (Grant No. 11201380).
Rights and permissions
About this article
Cite this article
Ding, H., Zhou, J. Global Existence and Blow-Up for a Parabolic Problem of Kirchhoff Type with Logarithmic Nonlinearity. Appl Math Optim 83, 1651–1707 (2021). https://doi.org/10.1007/s00245-019-09603-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00245-019-09603-z
Keywords
- Parabolic problem of Kirchhoff type
- Logarithmic nonlinearity
- Global existence
- Blow-up
- Ground-state solution