In this paper, we study the initial boundary value problem for a class of fractional p-Laplacian Kirchhoff type diffusion equations with logarithmic nonlinearity. Under suitable assumptions, we obtain the extinction property and accurate decay estimates of solutions by virtue of the logarithmic Sobolev inequality. Moreover, we discuss the blow-up property and global boundedness of solutions.
Citation: Peng Shi, Min Jiang, Fugeng Zeng, Yao Huang. Initial boundary value problem for fractional p-Laplacian Kirchhoff type equations with logarithmic nonlinearity[J]. Mathematical Biosciences and Engineering, 2021, 18(3): 2832-2848. doi: 10.3934/mbe.2021144
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In this paper, we study the initial boundary value problem for a class of fractional p-Laplacian Kirchhoff type diffusion equations with logarithmic nonlinearity. Under suitable assumptions, we obtain the extinction property and accurate decay estimates of solutions by virtue of the logarithmic Sobolev inequality. Moreover, we discuss the blow-up property and global boundedness of solutions.
In this paper, we study the extinction and the blow-up for the following fractional p-Laplacian Kirchhoff type equations with logarithmic nonlinearity.
{ut+M(‖u‖p)(−Δ)spu=λ|u|r−2uln|u|−β|u|q−2u,in Ω×(0,T),u(0)=u0,in Ω,u=0,on ∂Ω×(0,T), | (1.1) |
where
‖u‖=(∬Q|u(x)−u(y)|p|x−y|N+spdxdy)1p, |
Q=R2N∖(CΩ×CΩ), CΩ=RN∖Ω, Ω⊂RN (N>2s) is a bounded domain with Lipschitz boundary, s∈(0,1), 1<p<2, 1<q≤2, r>1, λ, β>0, (−Δ)sp is the fractional p-Laplacian operator and satisfies
(−Δ)spu(x)=2lim |
where u(x)\in C^{\infty} and u(x) has compact support in \Omega , B_\gamma (x)\subset \mathbb{R}^N is the ball with center x and radius \gamma . u_0(x)\in L^\infty(\Omega)\cap W^{s, p}_0(\Omega) is a nonzero non-negative function, where L^\infty(\Omega) and W^{s, p}_0(\Omega) are Lebesgue space and fractional Sobolev space respectively, which will be given in section 2. M(\cdot) is a Kirchhoff function with the following assumptions
( M_1 ) 0 < s < 1, M(\tau): = a+b\theta\tau^{\theta-1} for \tau\in\mathbb{R}^+_0: = [0, +\infty) ( a > 0, b\geq0 are two constants), \theta\geq1 ;
( M_2 ) M: \mathbb{R}^+_0\rightarrow\mathbb{R}^+_0\backslash\{0\} is continuous and there exits m_0\geq0 such M(\tau)\geq m_0 for all \tau\geq0.
It is worth pointing out that the interest in studying problems like (1.1) relies not only on mathematical purposes, but also on their significance in real models. For example, in the study of biological populations, we can use u(x, t) to represent the density of the population at x at time t , the term (-\Delta)^s_pu represents the diffusion of density, \mu|u|^{q-2}u represents the internal source and \lambda|u|^{r-2}u\ln|u| denotes external influencing factors. For more practical applications of problems like (1.1) , please refer to the studies [1,2,3].
Compared with integer-order equations, it is very difficult to study the problem (1.1), which contains both non-local terms (including fractional p -Laplacian operators and Kirchhoff functions) and logarithmic nonlinearity. For the fractional order theory, we refer the readers to the studies [4,5,6]. In [7,8], the authors use Sobolev space and Nehari manifold to study the existence of solutions for fractional equations. In [9,10], the solutions for fractional equations are discussed by virtue of Nehari manifold and fibrillation diagram. By using different methods from above, the properties of the solutions for such partial differential equations are considered by the method of variational principle and topological theory in the the literature [11,12,13]. Moreover, the authors prefer to use potential well theory, Galerkin approximation and Nehari manifold method to prove the existence of solutions, decay estimation and blow-up, we refer the reader to the literature [14,15,16].
Existence, extinction and blow-up of solutions are three important topics which regard parabolic problems; in particular, the study of extinction properties has made great progress in recent years. In [17], Liu considered the following initial boundary value problem for the fractional p -Laplacian equation
\begin{equation} u_t-\text{div}(|\nabla u|^{p-2}\nabla u)+\beta u^q = \lambda u^r, \ \ x\in\Omega,\ \ t > 0, \end{equation} | (1.2) |
where 1 < p < 2 , q\leq1 and r, \lambda, \beta > 0 . By employing the differential inequality and comparison principle, they obtained the extinction and the non-extinction of weak solutions. In [18], Sarra Toualbia et al. considered the following initial boundary value problem of a nonlocal heat equations with logarithmic nonlinearity
\begin{equation} u_t-\text{div}(|\nabla u|^{p-2}\nabla u) = |u|^{p-2}u\log|u|-\oint_\Omega|u|^{p-2}u\log|u|dx, \ x\in\Omega, \ t > 0, \end{equation} | (1.3) |
where p\in(2, +\infty) . By using the logarithmic Sobolev inequality and potential well method, they obtained decay, blow-up and non-extinction of solutions. In [19], Xiang and Yang studied the first initial boundary value problem of the following fractional p -Kirchhoff type
\begin{equation} u_t+M([u]^p_{s,p})(-\Delta)^s_pu = \lambda|u|^{r-2}u-\mu|u|^{q-2}u,\ \ (x,t)\in\Omega\times(0,\infty), \end{equation} | (1.4) |
where M: [0, \infty)\rightarrow(0, \infty) is a continuous function, 0 < s < 1 < p < 2 , 1 < q\leq 2 , r > 1 , \lambda, \mu > 0 . Under suitable assumptions, they proved the extinction and non-extinction of solutions and perfected the Gagliardo-Nirenberg inequality. For more information on the extinction properties of the solution, please refer to the studies [20,21,22,23].
Inspired by the above works, we overcome the research difficulties of logarithmic nonlinearity, p -Laplace operator and Kirchhoff coefficients in problem (1.1) based on the potential well theory, Nehari manifold and differential inequality methods, we give the extinction and the blow-up properties of solutions. In addition, we give the global boundedness of the solution by appropriate assumptions. To the best of our knowledge, it is the first result in the literature to investigate the extinction and blow-up of solutions for fractional p -Laplacian Kirchhoff type with logarithmic nonlinearity.
In order to introduce our main results, we first give some related definitions and sets.
Definition 1.1 (Weak solution). A function u(x, t) is said to be a weak solution of problem (1.1), if (x, t)\in\Omega\times[0, T) , u\in L^p\big(0, T;W^{s, p}_0(\Omega))\cap C(0, T; L^2(\Omega)\big) , u_t\in L^2\big(0, T;L^2(\Omega)\big) , u(x, 0) = u_0(x)\in W^{s, p}_0(\Omega) , for all v\in W^{s, p}_0(\Omega) , t\in(0, T) , the following equation holds
\int_\Omega u_tvdx+M(\|u\|^p)\langle u,v\rangle = \lambda\int_\Omega v|u|^{r-2}u\ln|u|dx-\beta\int_\Omega v|u|^{q-2}udx , |
where
\langle u,v\rangle = \iint_{\mathcal{Q}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+sp}}dxdy. |
Definition 1.2 (Extinction of solutions). Let u(t) be a weak solution of problem (1.1). We call u(t) an extinction if there exists T > 0 such that u(x, t) > 0 for all t\in(t, T) and u(x, t)\equiv0 for all t\in[T, +\infty) .
Define the following two functionals on W^{s, p}_0(\Omega)
\begin{equation} E(u) = \frac{1}{p}a\|u\|^p+\frac{1}{p}b\|u\|^{\theta p}-\lambda\frac{1}{r}\int_\Omega |u|^{r}\ln|u|dx+\lambda\frac{1}{r^2}\int_\Omega|u|^{r}dx+\beta\frac{1}{q}\int_\Omega |u|^qdx, \end{equation} | (1.5) |
\begin{equation} I(u) = a\|u\|^p+\theta b\|u\|^{\theta p}-\lambda\int_\Omega |u|^{r}\ln|u|dx+\beta\|u\|^q_q. \end{equation} | (1.6) |
Let
Z = \Big\{u\in L^p\big(0,T;W^{s,p}_0(\Omega))\cap C(0,T; L^2(\Omega)\big), u_t\in L^2\big(0,T;L^2(\Omega)\big)\Big\}. |
Remark 1 Since u\in Z , 1 < p < 2 , M(\cdot) is a continuous function and
\int_\Omega|u|^{r}\ln|u|dx\leq\frac{1}{\sigma}\|u\|^{r+\sigma}_{r+\sigma}\leq\frac{1}{\sigma}C^{r+\sigma}_{r+\sigma}\|u\|^{r+\sigma}, |
where 0 < \sigma < p^*_s -r , then we can claim that E(u) and I(u) are well-defined in W^{s, p}_0(\Omega) . Further, by arguing essentially as in [24], one can prove the that
u \mapsto \int_\Omega|u|^{r}\ln|u|dx |
is continuous from W^{s, p}_0(\Omega) to \mathbb{R} . It follows that E(u) and I(u) are continuous.
Define some sets as follows
\begin{equation} W: = \{u\in W^{s,p}_0 (\Omega)\ | \ I(u) > 0,\ E(u) < h \}\cup\{ 0\}, \end{equation} | (1.7) |
\begin{equation} V: = \{u\in W^{s,p}_0 (\Omega)\ | \ I(u) < 0,\ E(u) < h \}, \end{equation} | (1.8) |
the mountain pass level
\begin{equation} h: = \inf\limits_{{u\in \mathcal{N}}}E(u), \end{equation} | (1.9) |
the Nehari manifold
\begin{equation} \mathcal{N}: = \{u\in W^{s,p}_0(\Omega)\backslash \{0\} \ | \ I(u) = 0\}. \end{equation} | (1.10) |
Moreover, we define
\begin{equation} \mathcal{N_+}: = \{u\in W^{s,p}_0 (\Omega)\ | \ I(u) > 0\}, \end{equation} | (1.11) |
\begin{equation} \mathcal{N_-}: = \{u\in W^{s,p}_0(\Omega) \ | \ I(u) < 0 \}. \end{equation} | (1.12) |
Let \lambda_1 be the first eigenvalue of the problem
\begin{equation} (-\Delta)^s_pu = \lambda|u|^{p-2}u \ \ \text{in} \ \Omega,\ \ \ u|_{\mathbb{R}^N\backslash\Omega} = 0, \end{equation} | (1.13) |
and \phi(x) > 0 a.e. in \Omega be the eigenfunction corresponding to the eigenvalue \lambda_1 > 0 , \phi(x)\in L^\infty(\Omega) and \|\phi\|_{L^\infty(\Omega)}\leq1.
First, we give some results satisfying I(u_0) > 0 and q = 2.
Theorem 1.1 Assume that I(u_0) > 0 , r = p and q = 2. Let m_0 be as in assumption (M_2) , and let
l: = \frac{2N-(s+N)p}{sp}, \ P_1: = \frac{m_0}{\lambda_1L(p,\Omega)+\ln(R)}, \ P_2: = \frac{\lambda_1m_0lp^{p-1}}{[\lambda_1L(p,\Omega)+\ln(R)](p+l-1)^{p-1}}, |
where L(p, \Omega) and R are given in Lemma 2.1 and Lemma 2.5. Then, there exist positive constants C_1 , C_2 , T_1 and T_2 such that
(ⅰ) If \lambda < \lambda_1P_1 , then the weak solution of (1.1) vanishes in the sense of \|\cdot\|_2 as t\rightarrow+\infty.
(ⅱ) If 2N/(N+2s) < p < 2 and \lambda < \lambda_1P_1 or 1 < p\leq 2N/(N+2s) and \lambda < P_2 , then the nonnegative solutions of (1.1) vanish in finite time, and
\left\{ \begin{array}{@{}l@{\quad}l} \|u\|_2\leq\Big[\Big(\|u_0\|^{2-p}_2+\frac{C_1}{\beta}\Big)e^{(p-2)\beta t}-\frac{C_1}{\beta}\Big]^{\frac{1}{2-p}}, \; \; \; & t\in[0,T_1),\\ \|u\|_2\equiv0,\; \; \; \; & t\in[T_1,\infty),\\ \end{array}\right. |
for 2N/(N+2s) < p < 2 , and
\left\{ \begin{array}{@{}l@{\quad}l} \|u\|_{l+1}\leq\Big[\Big(\|u_0\|^{2-p}_{l+1}+\frac{C_2}{\beta}\Big)e^{(p-2)\beta t}-\frac{C_2}{\beta}\Big]^{\frac{1}{2-p}}, \; \; \; & t\in[0,T_2),\\ \|u\|_{l+1}\equiv0,\; \; \; \; & t\in[T_{2},\infty),\\ \end{array}\right. |
for 1 < p < 2N/(N+2s) .
Theorem 1.2 Assume that I(u_0) > 0 , 0 < \sigma\leq p^*_s-r , r > p and q = 2 . Let m_0 be as in assumption (M_2) , and let
P_3: = \text{max}\Big\{\frac{m_0}{ L(r,\Omega)R^{r-p}+\varepsilon\Phi}, \frac{\beta}{\Phi\varepsilon^{\frac{r(\vartheta_1-1)}{p-r(1-\vartheta_1)}}}\Big\},\ P_4: = \text{max}\Big \{\frac{m_0lp^p}{\varepsilon_1(p+l-1)^p}, \beta\varepsilon^{\frac{r_2(1-\vartheta_2)}{p-r_2(1-\vartheta_2)}}_1\Big\}, |
where
\Phi = \ln(R)|\Omega|^{\frac{s_1-r}{s_1}}C^{r(1-\vartheta_1)}_{p^*_s},\ p^*_s = \frac{Np}{N-sp}, \ s_1 = 2\vartheta_1+p^*_s(1-\vartheta_1),\vartheta_1 = \frac{2(r-p)}{r(2-p)},\ r_2 = \frac{p(l+r+\sigma-1)}{l+p-1}, |
C_{p^*_s} is the best constant of embedding from W^{s, p}_0(\Omega) to L^{p^*_s}(\Omega) , L(p, \Omega) and R are given in Lemma 2.1 and Lemma 2.5, and \varepsilon, \varepsilon_1 > 0 are two constants. Then, there exist positive constants C_4 , C_5 , C_6 , C_7 , T_3 and T_4 such that the non-negative weak solution of problem (1.1) vanishes in finite time and
\left\{ \begin{array}{@{}l@{\quad}l} \|u\|_2\leq\Big[\Big(\|u_0\|^{2-p}_{l+1}+\frac{C_5}{C_4}\Big)e^{(p-2)C_4 t}-\frac{C_5}{C_4}\Big]^{\frac{1}{2-p}}, \; \; \; & t\in[0,T_3),\\ \|u\|_2\equiv0,\; \; \; \; & t\in[T_3,\infty),\\ \end{array}\right. |
for 2N/(N+2s)\leq p < 2 , \lambda < P_3 , and
\left\{ \begin{array}{@{}l@{\quad}l} \|u\|_{l+1}\leq\Big[\Big(\|u_0\|^{2-p}_{l+1}+\frac{C_6}{C_7}\Big)e^{(p-2)C_7t}-\frac{C_6}{C_7}\Big]^{\frac{1}{2-p}}, \; \; \; & t\in[0,T_4),\\ \|u\|_{l+1}\equiv0,\; \; \; \; & t\in[T_4,\infty),\\ \end{array}\right. |
for 1 < p < 2N/(N+2s) , \lambda < P_4 .
Secondly, we give some results satisfying I(u_0) > 0 and q < 2.
Theorem 1.3 Assume that I(u_0) > 0 , p = r , l_1 > l\geq1 and 1 < q < 2 . Let m_0 be as in assumption (M_2) , and let
P_5: = \frac{\lambda_1m_0l_1p^{p-1}}{[\lambda_1L(p,\Omega)+\ln(R)](p+l_1-1)^{p-1}}, |
where L(p, \Omega) and R are given in Lemma 2.1 and Lemma 2.5. If 2N/(N+2s) < p < 2 with \lambda < \lambda_1P_1 or 1 < p\leq2N/(N+2s) with \lambda < P_5 , then the non-negative weak solution of problem (1.1) vanishes in finite time for any non-negative initial data.
Theorem 1.4 Assume that I(u_0) > 0 , l_1 > l\geq1 , r\leq2 and 1 < q < 2 . Let m_0 be as in assumption (M_2) , and let
P_6: = \text{max}\Big\{\frac{m_0}{ L(r,\Omega)R^{r-p}},\frac{\beta\varepsilon^{\frac{(1-\vartheta_4)r_4}{p-(1-\vartheta_4)r_4}}_3C^{(\vartheta_4-1)r_4}_{p^*_s}R^{r_4-r}}{\ln(R)|\Omega|^{\frac{2-r}{2}}}\Big\},\ P_7: = \frac{1}{C}\beta \varepsilon^{\frac{(\vartheta_3-1)r_3}{(1-\vartheta_3)-(p+l_1-1)}}_2C^{\frac{(\vartheta_3-1)pr_3}{p+l_1-1}}_{p^*_s}, |
\vartheta_3 = \frac{[(q+l_1-1)p^*_s-(l_1+1)p](p+l_1-1)}{[(q+l_1-1)p^*_s-(p+l_1-1)p](l_1+1)}, \ \vartheta_4 = \frac{(p^*_s-2)p}{(p^*_s-q)2},\ C = \frac{1}{e\sigma}|\Omega|^{1-\frac{r_5}{l_1+1}}C^{r_5}_{l_1+1}R^{r_5-r_3}, |
r_3 = \frac{(q+l_1-1)(p+l_1-1)}{\vartheta_3(p+l_1-1)+(q+l_1-1)(1-\vartheta_3)},\ r_4 = \frac{qp}{q(1-\vartheta_4)+\vartheta_4p},\ r_5 = l_1+r+\sigma-1, |
and \varepsilon_2 , \varepsilon_3 > 0 are two constants. If 2N/(N+2s) < p < 2 with \lambda < P_6 or 1 < p\leq2N/(N+2s) with \lambda < P_7 , then the non-negative weak solution of problem (1.1) vanishes in finite time for any non-negative initial data.
Finally, we discuss the global boundedness and blow up of weak solutions.
Theorem 1.5 Let u(x, t) be the weak solution of problem (1.1) .
(ⅰ) If E(u_0) < 0 , r = p > q and \theta = 1 , then the weak solution u(x, t) blows up at +\infty ;
(ⅱ) If 0 < E(u_0)\leq h and I(u_0)\geq0 , then the weak solution u(x, t) is globally bounded.
The rest of the paper is organized as follows. In Section 2, we give some related spaces and lemmas. In Section 3, we give the proof process for the main results of problem (1.1) .
In order to facilitate the proof of the main results, we start this section by introducing some symbols and Lemmas that will be used throughout the paper.
In this section, we assume that 0 < s < 1 < p < 2 and \Omega\in\mathbb{R}^N\ (N > 2s) is a bounded domain with Lipschitz boundary. We denote by \|u\|_i \ (i\geq1) the norm of Lebesgue space L^{i}(\Omega) . Let W^{s, p}(\Omega) be the linear space of Lebesgue measurable functions u from \mathbb{R}^{N} to \mathbb{R} such that the restriction to \Omega of any function u in W^{s, p}(\Omega) belongs to L^p(\Omega) and
\iint_\mathcal{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dxdy < \infty, |
where \mathcal{Q} = \mathbb{R}^{2N}\backslash(\mathcal C\Omega\times \mathcal C\Omega) , \mathcal C\Omega = \mathbb{R}^N\backslash \Omega . The space W^{s, p}(\Omega) is equipped with the norm
\|u\|_{W^{s,p}(\Omega)} = \Big(\|u\|^p_p+\iint_\mathcal{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dxdy\Big)^{\frac{1}{p}}. |
We further give a closed linear subspace
W^{s,p}_{0}(\Omega) = \big\{u\in W^{s,p}(\Omega)| u(x) = 0 \ \ \text{a.e. in} \ \mathbb{R}^N\backslash\Omega\big\}. |
As shown in [19], it can be concluded that
\|u\| = \Big(\iint_\mathcal{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dxdy\Big)^{\frac{1}{p}} |
is an equivalent norm of W^{s, p}_0(\Omega) .
Next we give the necessary Lemmas.
Lemma 2.1 ([25]) Let u\in W^{s, p}_0(\Omega)\backslash \{0\} . Then
\int_\Omega|u|^{p}\ln|u|dx\leq L(p,\Omega)\|u\|^p+\ln(\|u\|)\int_\Omega|u|^pdx, |
where L(p, \Omega): = \frac{|\Omega|}{ep}+\frac{1}{e(p^*_s-p)}C^{p^*_s}_{p^*_s} , C_{p^*_s} is the best constant of embedding from W^{s, p}_0(\Omega) to L^{p^*_s}(\Omega).
Lemma 2.2 ([26]) Let y(t) be a non-negative absolutely continuous function on [T_0, +\infty) satisfying
\frac{dy}{dt}+\alpha y^k+\beta y\leq0, \ \ \ t\geq0, \ \ \ y(0)\geq0, |
where \alpha, \beta > 0 are constants and k\in(0, 1) . Then
\left\{ \begin{array}{@{}l@{\quad}l} y(t)\leq\Big[\big(y^{1-k}(T_0)+\frac{\alpha}{\beta}\big)e^{(k-1)\beta (t-T_0)}-\frac{\alpha}{\beta}\Big]^{\frac{1}{1-k}}, \; \; \; & t\in[T_0,T_*),\\ y(t)\equiv0,\; \; \; & t\in[T_*,+\infty),\\ \end{array}\right. |
where T_* = \frac{1}{(1-k)\beta}\ln\big(1+\frac{\beta}{\alpha}y^{1-k}(T_0)\big).
Lemma 2.3 ([27]) Suppose that \beta^*\geq0 , N > sp\geq1 , and 1\leq r\leq q \leq(\beta^*+1)p^*_s , then for u such that |u|^{\beta^*}u\in W^{s, p}_0(\Omega) , we have
\|u\|_q\leq C^{\frac{1-\vartheta}{\beta^*+1}}_{p^*_s}\|u\|^{\vartheta}_r\| |u|^{\beta^*} u\|^{\frac{1-\vartheta}{\beta^*+1}}, |
with \vartheta = \frac{[(\beta^*+1)p^*_s-q]r}{[(\beta^*+1)p^*_s-r]q} , where C_{p^*_s} is the embedding constant for W^{s, p}_0(\Omega)\hookrightarrow L^{p^*_s}(\Omega).
Lemma 2.4 ([19]) Let 1 < p < \infty and g: \mathbb{R}\rightarrow\mathbb{R} be an increasing function. Define
G(t) = \int^t_0g'(\tau)^{\frac{1}{p}}d\tau,\ \ \ t\in\mathbb{R} |
then
|a-b|^{p-2}(a-b)(g(a)-g(b))\geq|G(a)-G(b)|^p \ \ \ \text{for all} \ a, b \in\mathbb{R}. |
Lemma 2.5 Assume that ( M_1 ) holds. Let u\in W^{s, p}_0(\Omega)\backslash\{0\} , 0 < \sigma\leq p^*_s-r . We have
(ⅰ) if 0 < \|u\|\leq R , then I(u) > 0 ;
(ⅱ) if I(u)\leq0 , then \|u\| > R ,
where
R = \Big(\frac{a\sigma}{\lambda C^{r+\sigma}_{r+\sigma}}\Big)^{\frac{1}{r+\sigma-p}}, |
\ C_{r+\sigma} is the embedding constant for W^{s, p}_0(\Omega)\hookrightarrow L^{r+\sigma}(\Omega).
Proof. Since u\in W^{s, p}_0(\Omega)\backslash\{0\} , and
\sigma \ln|u(x)| < |u(x)|^\sigma \ \ \ \ \text{for } a.e. x\in\Omega. |
Then by the definition of I(u) , we obtain
\begin{split} I(u) = &a\|u\|^p+\theta b\|u\|^{\theta p}+\beta\|u\|^q_q-\lambda\int_\Omega|u|^r\ln|u|dx\\ & > a\|u\|^p+\theta b\|u\|^{\theta p}+\beta\|u\|^q_q-\lambda\frac{1}{\sigma}\|u\|^{r+\sigma}_{r+\sigma}\\ &\geq a\|u\|^p-\lambda\frac{1}{\sigma}\|u\|^{r+\sigma}_{r+\sigma}\\ &\geq\big(a-\lambda\frac{1}{\sigma}C^{r+\sigma}_{r+\sigma}\|u\|^{r+\sigma-p}\big)\|u\|^p,\\ \end{split} |
where C_{r+\sigma} is the embedding constant for W^{s, p}_0(\Omega)\hookrightarrow L^{r+\sigma}(\Omega).
We can get
\begin{equation} I(u) > \big(a-\lambda\frac{1}{\sigma}C^{r+\sigma}_{r+\sigma}\|u\|^{r+\sigma-p}\big)\|u\|^p. \end{equation} | (2.1) |
If 0 < \|u\|\leq R , then it follows from the definition of R that
a-\lambda\frac{1}{\sigma}C^{r+\sigma}_{r+\sigma}\|u\|^{r+\sigma-p}\geq0, |
thus (ⅰ) holds.
If I(u)\leq0 , by (1.12) , we have
a-\lambda\frac{1}{\sigma}C^{r+\sigma}_{r+\sigma}\|u\|^{r+\sigma-p} < 0, |
thus (ⅱ) holds.
Lemmas 2.6 is similar to [28,Lemmas 9], so we ignore its proof.
Lemma 2.6 ([28]) Assume that E(u_0)\leq h , then the sets \mathcal{N_+} and \mathcal{N_-} are both invariant for u(t) , i.e, if u_0\in \mathcal{N_-} (resp. u_0\in\mathcal{N_+} ), then u(t)\in \mathcal{N_-} (resp. u(t)\in\mathcal{N_+} ) for all t\in[0, T).
Lemma 2.7 ([29]) Let \alpha be positive. Then
t^p\ln(t)\leq\frac{1}{e\alpha}t^{p+\alpha}, \ \ \text{for all} \ p, t > 0. |
In this section, we prove that the main results of problem (1.1) .
Proof of Theorem 1.1
(1) Taking v = u in Definition 1.1, we obtain
\begin{equation} \frac{1}{2}\frac{d}{dt}\int_\Omega u^2dx+M(\|u\|^p)\|u\|^p = \lambda\int_\Omega|u|^p\ln|u|dx-\beta\int_\Omega u^2dx. \end{equation} | (3.1) |
By Lemma 2.1, Lemma 2.5 and Lemma 2.6, we obtain
\frac{1}{2}\frac{d}{dt}\int_\Omega u^2dx+\big(m_0-\lambda L(p,\Omega)-\frac{\lambda}{\lambda_1}\ln(R)\big)\|u\|^p+\beta\int_\Omega u^2dx\leq0. |
Since \lambda < \lambda_1P_1 , we get
\frac{1}{2}\frac{d}{dt}\int_\Omega u^2dx+\beta\int_\Omega u^2dx\leq0. |
thus
\|u(\cdot, t)\|^2_2\leq\|u_0\|^2_2e^{-2\beta t}. |
which implies that \|u(\cdot, t)\|_2\rightarrow0 as t\rightarrow+\infty.
(2) We consider first the case 2N/(N+2s) < p < 2 with \lambda < \lambda_1P_1. By (3.1) and Lemma 2.1, Lemma 2.5 and Lemma 2.6, we have
\begin{equation} \frac{1}{2}\frac{d}{dt}\|u\|^2_2+\Big(m_0-\frac{\lambda}{\lambda_1}\big(\lambda_1L(p,\Omega)+\ln(R)\big)\Big)\|u\|^p+\beta\|u\|^2_2\leq0. \end{equation} | (3.2) |
Using Hölder's inequality and the fractional Sobolev embedding theorem, we have
\begin{equation} \|u\|_2\leq|\Omega|^{\frac{1}{2}-\frac{N-sp}{Np}}\|u\|_{\frac{Np}{N-sp}}\leq C_{p^*_s}|\Omega|^{\frac{1}{2}-\frac{N-sp}{Np}}\|u\|, \end{equation} | (3.3) |
where C_{p^*_s} > 0 is the embedding constant. By (3.2) , (3.3) and \lambda < \lambda_1P_1 , we obtain
\begin{equation} \frac{1}{2}\frac{d}{dt}\|u\|^2_2+C_1\|u\|^p_2+\beta\|u\|^2_2\leq0, \end{equation} | (3.4) |
where
\begin{equation} C_1 = C^{-p}_{p^*_s}|\Omega|^{\frac{N-sp}{N}-\frac{P}{2}}\Big(m_0-\frac{\lambda}{\lambda_1}\big(\lambda_1L(p,\Omega)+\ln(R)\big)\Big) > 0. \end{equation} | (3.5) |
Setting y(t) = \|u(\cdot, t)\|^2_2 , y(0) = \|u_0(\cdot)\|^2_2 , by Lemma 2.2, we obtain
\left\{ \begin{array}{@{}l@{\quad}l} \|u\|_2\leq\Big[\Big(\|u_0\|^{2-p}_2+\frac{C_1}{\beta}\Big)e^{(p-2)\beta t}-\frac{C_1}{\beta}\Big]^{\frac{1}{2-p}}, \; \; \; & t\in[0,T_1),\\ \|u\|_2\equiv0,\; \; \; \; & t\in[T_1,\infty),\\ \end{array}\right. |
where
\begin{equation} T_1 = \frac{1}{(2-p)\beta}\ln\big(1+\frac{\beta}{C_1}\|u_0\|^{2-p}_2\big). \end{equation} | (3.6) |
Next, we consider the case 1 < p\leq2N/(N+2s) and \lambda < \lambda_1P_2. Taking \nu = u^l in Definition 1.1, where l = \frac{2N-(s+N)p}{sp}\geq1 , by Lemma 2.1, Lemma 2.4, Lemma 2.5 and Lemma 2.6, we obtain
\begin{equation} \frac{1}{l+1}\frac{d}{dt}\|u\|^{l+1}_{l+1}+G\|u^{\frac{p+l-1}{p}}\|^p+\beta\|u\|^{l+1}_{l+1}\leq0, \end{equation} | (3.7) |
where G = \Big(\frac{m_0lp^p}{(p+l-1)^p}-\frac{\lambda pL(p, \Omega)}{p+l-1}-\frac{\lambda p\ln(R)}{\lambda_1(p+l-1)}\Big) . By the very choice of l and the fractional Sobolev embedding theorem, we have
\begin{equation} \|u\|^{\frac{p+l-1}{p}}_{l+1} = \Big(\int_\Omega u^{\frac{p+l-1}{p}\cdot\frac{Np}{N-sp}}dx\Big)^{\frac{N-sp}{Np}}\leq C_{p^*_s}\|u^{\frac{p+l-1}{p}}\|. \end{equation} | (3.8) |
Hence,
\begin{equation} \frac{1}{l+1}\frac{d}{dt}\|u\|^{l+1}_{l+1}+C_2\|u\|^{p+l-1}_{l+1}+\beta\|u\|^{l+1}_{l+1}\leq0, \end{equation} | (3.9) |
where
\begin{equation} C_2 = C^{-p}_{p^*_S}\Big(\frac{m_0lp^p}{(p+l-1)^p}-\frac{\lambda pL(p,\Omega)}{p+l-1}-\frac{\lambda p\ln(R)}{\lambda_1(p+l-1)}\Big), \end{equation} | (3.10) |
since \lambda < P_2 , then C_2 > 0 . Setting y(t) = \|u(\cdot, t)\|_{l+1} , y(0) = \|u_0(\cdot)\|_{l+1} , by Lemma 2.2, we obtain
\left\{ \begin{array}{@{}l@{\quad}l} \|u\|_{l+1}\leq\Big[\Big(\|u_0\|^{2-p}_{l+1}+\frac{C_2}{\beta}\Big)e^{(p-2)\beta t}-\frac{C_2}{\beta}\Big]^{\frac{1}{2-p}}, \; \; \; & t\in[0,T_2),\\ \|u\|_{l+1}\equiv0,\; \; \; \; & t\in[T_{2},\infty),\\ \end{array}\right. |
where
\begin{equation} T_2 = \frac{1}{(2-p)\beta}\ln\big(1+\frac{\beta}{C_2}\|u_0\|^{2-p}_{l+1}\big). \end{equation} | (3.11) |
The proof is completed.
Proof of Theorem 1.2
We consider first the case p < r < 2 and 2N/(N+2s) < p < 2 . Taking \nu = u in Definition 1.1, we have
\begin{equation} \frac{1}{2}\frac{d}{dt}\|u\|^2_2+M(\|u\|^p)\|u\|^p = \int_\Omega|u|^r\ln|u|dx-\beta\|u\|^2_2. \end{equation} | (3.12) |
By Lemma 2.1, Lemma 2.5 and Lemma 2.6, we obtain
\begin{equation} \frac{1}{2}\frac{d}{dt}\|u\|^2_2+M(\|u\|^p)\|u\|^p\leq\lambda L(r,\Omega)\|u\|^r+\lambda\ln(R)\|u\|^r_r-\beta\|u\|^2_2. \end{equation} | (3.13) |
Using Hölder's inequality and the interpolation inequality, the fractional Sobolev embedding theorem and Young inequality, we can easily obtain (see [19])
\begin{equation} \|u\|^r_r\leq |\Omega|^{\frac{s_1-r}{s_1}}\|u\|^r_{s_1}\leq| \Omega|^{\frac{s_1-r}{s_1}}C^{r(1-\vartheta_1)}_{p^*_s}(\varepsilon\|u\|^p+\varepsilon^{\frac{r(\vartheta_1 -1)}{p-r(1-\vartheta_1)}}\|u\|^2_2), \end{equation} | (3.14) |
where s_1 > r , \vartheta_1\in(0, 1) , \varepsilon > 0 and
s_1 = 2\vartheta_1+p^*_s(1-\vartheta_1), \ \ \vartheta_1 = \frac{2(r-p)}{r(2-p)}. |
By (3.13) and (3.14) , we have
\begin{equation} \frac{1}{2}\frac{d}{dt}\|u\|^2_2+C_3\|u\|^p+C_4\|u\|^2_2\leq0, \end{equation} | (3.15) |
where
\begin{equation} C_3 = m_0-\lambda L(r,\Omega)R^{r-p}-\lambda\varepsilon\ln(R)|\Omega|^{\frac{s_1-r}{s_1}}C^{r(1-\vartheta_1)}_{p^*_s}, \end{equation} | (3.16) |
\begin{equation} C_4 = \beta-\lambda\ln(R)|\Omega|^{\frac{s_1-r}{s_1}}C^{r(1-\vartheta_1)}_{p^*_s}\varepsilon^{\frac{r(\vartheta_1 -1)}{p-r(1-\vartheta_1)}}. \end{equation} | (3.17) |
Since 2N/(N+2s) < p < 2 and \lambda < P_3 , by the fractional embedding theorem and (3.3) , we obtain
\begin{equation} \frac{1}{2}\frac{d}{dt}\|u\|^2_2+C_5\|u\|^p_2+C_4\|u\|^2_2\leq0, \end{equation} | (3.18) |
and C_3 > 0, \; C_4 > 0 , where
\begin{equation} C_5 = C_3C^{-p}_{p^*_s}|\Omega|^{\frac{N-sp}{N}-\frac{P}{2}}. \end{equation} | (3.19) |
Similar to the Theorem 1.1, one can prove the that
\left\{ \begin{array}{@{}l@{\quad}l} \|u\|_2\leq\Big[\Big(\|u_0\|^{2-p}_{l+1}+\frac{C_5}{C_4}\Big)e^{(p-2)C_4t}-\frac{C_5}{C_4}\Big]^{\frac{1}{2-p}}, \; \; \; & t\in[0,T_3),\\ \|u\|_2\equiv0,\; \; \; \; & t\in[T_3,\infty),\\ \end{array}\right. |
where
\begin{equation} T_3 = \frac{1}{(2-p)C_4}\ln\big(1+\frac{C_4}{C_5}\|u_0\|^{2-p}_{2}\big). \end{equation} | (3.20) |
When 1 < p\leq2N/(N+2s) , p < r\leq2 and \lambda < P_4 . Taking \nu = u^l\ (l = \frac{2N-sp-Np}{sp}\geq1) in Definition 1.1, by Lemma 2.4 and Lemma 2.7, we obtain
\begin{equation} \frac{1}{l+1}\frac{d}{dt}\|u\|^{l+1}_{l+1}+\frac{m_0lp^p}{(p+l-1)^p}\|u^{\frac{l+p-1}{p}}\|^p\leq \lambda\frac{1}{e\sigma}\|u\|^{l+r+\sigma-1}_{l+r+\sigma-1}-\beta\|u\|^{l+1}_{l+1} \end{equation} | (3.21) |
further, we have
\begin{equation} \frac{1}{l+1}\frac{d}{dt}\|u^{\frac{l+p-1}{p}}\|^{r_1}_{r_1}+\frac{m_0lp^p}{(p+l-1)^p}\|u^{\frac{l+p-1}{p}}\|^p\leq \lambda\frac{1}{e\sigma}\|u^{\frac{l+p-1}{p}}\|^{r_2}_{r_2}-\beta\|u^{\frac{l+p-1}{p}}\|^{r_1}_{r_1} \end{equation} | (3.22) |
where r_1 = \frac{p(l+1)}{l+p-1} , r_2 = \frac{p(l+r+\sigma-1)}{l+p-1} . Note that, since r_1 < p^*_s , by the Hölder's inequality and the fractional Sobolev embedding theorem, we have
\begin{equation} \|u^{\frac{l+p-1}{p}}\|^p_{r_1}\leq|\Omega|^{\frac{(p^*_s-r_1)p}{r_1p^*_s}}\|u^{\frac{l+p-1}{p}}\|^p_{p^*_s}\leq|\Omega|^{\frac{(p^*_s-r_1)p}{r_1p^*_s}}C^p_{p^*_s}\|u^{\frac{l+p-1}{p}}\|^p. \end{equation} | (3.23) |
Using the same discussion as above, one can conclude that
\begin{equation} \|u^{\frac{l+p-1}{p}}\|^{r_2}_{r_2}\leq| \Omega|^{\frac{s_2-{r_2}}{s_2}}C^{{r_2}(1-\vartheta_2)}_{p^*_s}(\varepsilon_1\|u^{\frac{l+p-1}{p}}\|^p+\varepsilon^{\frac{r_2(\vartheta_2 -1)}{p-r_2(1-\vartheta_2)}}_1\|u^{\frac{l+p-1}{p}}\|^{r_1}_{r_1}), \end{equation} | (3.24) |
where s_2 > r_2 , \vartheta_2\in(0, 1) , \varepsilon_1 > 0 and
s_2 = r_1\vartheta_2+p^*_s(1-\vartheta_2), \ \ \vartheta_2 = \frac{r_1(r_2-p)}{r_2(r_1-p)}. |
Combining (3.22) - (3.24) and \lambda < P_4 , we obtain
\begin{equation} \frac{1}{l+1}\frac{d}{dt}\|u^{\frac{l+p-1}{p}}\|^{r_1}_{r_1}+C_6\|u^{\frac{l+p-1}{p}}\|^p_{r_1}+C_7\|u^{\frac{l+p-1}{p}}\|^{r_1}_{r_1}\leq0, \end{equation} | (3.25) |
and C_6 > 0 , C_7 > 0 , where
\begin{equation} C_6 = \Big(\frac{m_0lp^p}{(l+p-1)^p}-\lambda\varepsilon_1\frac{1}{e\sigma}C^{r_2(1-\vartheta_2)}_{p^*_s}|\Omega|^{\frac{s_2-r_2}{s_2}}\Big)|\Omega|^{\frac{(r_1-p^*_s)p}{r_1p^*_s}}C^{-p}_{p^*_s}, \end{equation} | (3.26) |
\begin{equation} C_7 = \beta-\lambda\frac{1}{e\sigma}|\Omega|^{\frac{s_2-r_2}{s_2}}C^{r_2(1-\vartheta_2)}_{p^*_s}\varepsilon^{\frac{r_2(\vartheta_2 -1)}{p-r_2(1-\vartheta_2)}}_1. \end{equation} | (3.27) |
Using Lemma 2.2 and a direct calculation, we have
\left\{ \begin{array}{@{}l@{\quad}l} \|u\|_{l+1}\leq\Big[\Big(\|u_0\|^{2-p}_{l+1}+\frac{C_6}{C_7}\Big)e^{(p-2)C_7t}-\frac{C_6}{C_7}\Big]^{\frac{1}{2-p}}, \; \; \; & t\in[0,T_4),\\ \|u\|_{l+1}\equiv0,\; \; \; \; & t\in[T_4,\infty),\\ \end{array}\right. |
where
\begin{equation} T_4 = \frac{1}{(2-p)C_7}\ln(\frac{C_7}{C_6}\|u_0\|^{2-p}_{l+1}+1). \end{equation} | (3.28) |
The proof is completed.
Proof of Theorem 1.3
We consider first the case 2N/(N+2s) < p < 2 with \lambda < \lambda_1P_1 . Taking v = u in Definition 1.1, we obtain
\begin{equation} \frac{1}{2}\frac{d}{dt}\int_\Omega u^2dx+M(\|u\|^p)\|u\|^p = \lambda\int_\Omega|u|^p\ln|u|dx-\beta\int_\Omega u^qdx. \end{equation} | (3.29) |
Note that \|u\|^p\geq\lambda_1\|u\|^p_p and by Lemma 2.1, Lemma 2.5 and Lemma 2.6, we have
\begin{equation} \frac{1}{2}\frac{d}{dt}\|u\|^2_2+\Big(m_0-\frac{\lambda}{\lambda_1}\big(\lambda_1L(p,\Omega)+\ln(R)\big)\Big)\|u\|^p+\beta\|u\|^q_q\leq0. \end{equation} | (3.30) |
By (3.3) and \beta > 0 and \lambda < \lambda_1P_1 , we obtain
\begin{equation} \frac{1}{2}\frac{d}{dt}\|u\|^2_2+C_8\|u\|^p_2\leq0, \end{equation} | (3.31) |
where
\begin{equation} C_8 = C^{-p}_{p^*_s}|\Omega|^{\frac{2(N-sp)-Np}{2N}}\Big(m_0-\frac{\lambda}{\lambda_1}\big(\lambda_1L(p,\Omega)+\ln(R)\big)\Big) > 0. \end{equation} | (3.32) |
By a direct calculation, we obtain
\left\{ \begin{array}{@{}l@{\quad}l} \|u\|_{2}\leq\Big[\Big(\|u_0\|^{2-p}_{2}+C_8(p-2)t\Big]^{\frac{1}{2-p}}, \; \; \; & t\in[0,T_5),\\ \|u\|_2\equiv0,\; \; \; \; & t\in[T_5,\infty),\\ \end{array}\right. |
where
\begin{equation} T_5 = \frac{1}{(2-p)C_8}\|u_0\|^{2-p}_2. \end{equation} | (3.33) |
Next, we consider the case 1 < p\leq2N(N+2s) with \lambda < P_2 . Multiplying (1.1) by u^{l_1}\ (l_1 > l\geq1) and integrating, by Lemma 2.1, Lemma 2.4, Lemma 2.5 and Lemma 2.6, we obtain
\begin{equation} \frac{1}{l_1+1}\frac{d}{dt}\|u\|^{l_1+1}_{l_1+1}+G_1\|u^{\frac{p+l_1-1}{p}}\|^p+\beta\|u\|^{q+l_1-1}_{q+l_1-1}\leq0, \end{equation} | (3.34) |
where G_1 = \Big(\frac{m_0l_1p^p}{(p+l_1-1)^p}-\frac{\lambda pL(p, \Omega)}{p+l_1-1}-\frac{\lambda p\ln(R)}{\lambda_1(p+l_1-1)}\Big) . By Lemma 2.3, we obtain
\begin{equation} \|u\|_{l_1+1}\leq C^{\frac{(1-\vartheta_3)p}{p+l_1-1}}_{p^*_s}\|u^{\frac{p+l_1-1}{p}}\|^{\frac{(1-\vartheta_3)p}{p+l_1-1}}\|u\|^{\vartheta_3}_{q+l_1-1}, \end{equation} | (3.35) |
where
\vartheta_3 = \frac{[(q+l_1-1)p^*_s-(l_1+1)p](p+l_1-1)}{[(q+l_1-1)p^*_s-(p+l_1-1)p](l_1+1)}. |
By the choice of l_1 , we have 0 < \vartheta_3 < 1 . Hence, using the Young inequality, for every r_3 > 0 and \varepsilon_2 > 0 , we obtain
\begin{equation} \|u\|^{r_3}_{l_1+1}\leq C^{\frac{(1-\vartheta_3)pr_3}{p+l_1-1}}_{p^*_s}\Big(\varepsilon_2\|u^{\frac{p+l_1-1}{p}}\|^p+\varepsilon_2^{\frac{(1-\vartheta_3)r_3}{(1-\vartheta_3)r_3-(p+l_1-1)}}\|u\|^{\frac{\vartheta_3r_3(p+l_1-1)}{(p+l_1-1)-(1-\vartheta_3)r_3}}_{q+l_1-1}\Big). \end{equation} | (3.36) |
We now choose
r_3 = \frac{(q+l_1-1)(p+l_1-1)}{\vartheta_3(p+l_1-1)+(q+l_1-1)(1-\vartheta_3)}, |
and we notice that \frac{\vartheta_3r_3(p+l_1-1)}{(p+l_1-1)-(1-\vartheta_3)r_3} = q+l_1-1 . That means
\begin{equation} \|u\|^{r_3}_{l_1+1}\leq C^{\frac{(1-\vartheta_3)pr_3}{p+l_1-1}}_{p^*_s}\Big(\varepsilon_2\|u^{\frac{p+l_1-1}{p}}\|^p+\varepsilon_2^{\frac{(1-\vartheta_3)r_3}{(1-\vartheta_3)r_3-(p+l_1-1)}}\;\|u\|^{q+l_1-1}_{q+l_1-1}\Big). \end{equation} | (3.37) |
We choose
\varepsilon_2 = \Big[\frac{1}{\beta}\Big(\frac{m_0l_1p^p}{(p+l_1-1)^p}-\frac{\lambda pL(p,\Omega)}{p+l_1-1}-\frac{\lambda p\ln(R)}{\lambda_1(p+l_1-1)}\Big)\Big]^{\frac{(1-\vartheta_3)r_3-(p+l_1-1)}{-(p+l_1-1)}}. |
By (3.34) and (3.37) and \lambda < P_5 , we obtain
\frac{1}{l_1+1}\frac{d}{dt}\|u\|^{l_1+1}_{l_1+1}+C_9\|u\|^{r_3}_{l_1+1}\leq0, |
where
C_9 = C^{\frac{(\vartheta_3-1)pr_3}{p+l_1-1}}_{p^*_s}\beta\Big[\frac{1}{\beta}\Big(\frac{m_0l_1p^p}{(p+l_1-1)^p}-\frac{\lambda pL(p,\Omega)}{p+l_1-1}-\frac{\lambda p\ln(R)}{\lambda_1(p+l_1-1)}\Big)\Big]^{\frac{(\vartheta_3-1)r_3}{-(p+l_1-1)}}, |
and C_9 > 0, which implies that
\left\{ \begin{array}{@{}l@{\quad}l} \|u\|_{l_1+1}\leq\Big[\|u_0\|^{l_1+1-r_3}_{l_1+1}+(r_3-l_1-1)C_9t\Big]^{\frac{1}{l_1+1-r_3}}, \; \; \; & t\in[0,T_6),\\ \|u\|_{l_1+1}\equiv0,\; \; \; \; & t\in[T_6,\infty),\\ \end{array}\right. |
where
T_6 = \frac{1}{(l_1+1-r_3)C_9}\|u_0\|^{l_1+1-r_3}_{l_1+1}. |
The proof is completed.
Proof of Theorem 1.4
We consider first the case r < 2 . When 2N/(N+2) < p < 2 , multiplying (1.1) by u , we obtain
\begin{equation} \frac{1}{2}\frac{d}{dt}\int_\Omega u^2dx+M(\|u\|^p)\|u\|^p = \lambda\int_\Omega|u|^r\ln|u|dx-\beta\int_\Omega u^qdx. \end{equation} | (3.38) |
Similar to the Theorem 1.2, we obtain
\begin{equation} \frac{1}{2}\frac{d}{dt}\|u\|^2_2+\big(m_0-\lambda L(r,\Omega)R^{r-p}\big)\|u\|^p\leq\lambda\ln(R)|\Omega|^{\frac{2-r}{2}}\|u\|^r_2-\beta\|u\|^q_q. \end{equation} | (3.39) |
Taking \beta^* = 0 in Lemma 2.3, we have
\begin{equation} \|u\|_2\leq\|u\|^{(1-\vartheta_4)}_{p^*_s}\|u\|^{\vartheta_4}_q\leq C^{(1-\vartheta_4)}_{p^*_S}\|u\|^{(1-\vartheta_4)}\|u\|^{\vartheta_4}_q, \end{equation} | (3.40) |
where \vartheta_4 = \frac{(p^*_s-2)p}{(p^*_s-q)2} . Then, taking into account that \vartheta_4\in(0, 1) , we can apply Young's inequality: for every r_4 > 0 and \varepsilon_3 > 0 , we have
\begin{equation} \|u\|^{r_4}_2\leq C^{(1-\vartheta_4)r_4}_{p^*_s}\Big(\varepsilon_3\|u\|^p+\varepsilon_3^{\frac{(\vartheta_4-1)r_4}{p-(1-\vartheta_4)r_4}}\|u\|^{\frac{\vartheta_4r_4p}{p-r_4(1-\vartheta_4)}}_q\Big). \end{equation} | (3.41) |
Taking
r_4 = \frac{qp}{q(1-\vartheta_4)+\vartheta_4p}, |
then \frac{\vartheta_4r_4p}{p-k_2(1-\vartheta_4)} = q . By (3.39) , (3.41) , Lemma 2.5 and \lambda < P_6 , and let \varepsilon_3 = \Big(\frac{m_0-\lambda L(r, \Omega)R^{r-p}}{\beta}\Big)^{\frac{p-(1-\vartheta_4)r_4}{p}} , we obtain
\begin{equation} \frac{1}{2}\frac{d}{dt}\|u\|^2_2+C_{10}\|u\|^{r}_2\leq0, \end{equation} | (3.42) |
where
C_{10} = \beta\varepsilon^{\frac{(1-\vartheta_4)r_4}{p-(1-\vartheta_4)r_4}}_{3}C^{(\vartheta_4-1)r_4}_{p^*_s}R^{r_4-r}-\lambda\ln(R)|\Omega|^{\frac{2-r}{2}} > 0, |
which implies that our result holds.
When 1 < p\leq2N/(N+2s) , we multiply (1.1) by u^{l_1}\ (l_1 > l\geq1) , by Lemma 2.4 and Lemma 2.7, we obtain
\begin{equation} \frac{1}{l_1+1}\frac{d}{dt}\|u\|^{l_1+1}_{l_1+1}+\frac{m_0l_1p^p}{(p+l_1-1)^p}\|u^{\frac{l_1+p-1}{p}}\|^p\leq \lambda\frac{1}{e\sigma}\|u\|^{r_5}_{r_5}-\beta\|u\|^{q+l_1-1}_{q+l_1-1}, \end{equation} | (3.43) |
where r_5 = l_1+r+\sigma-1 . Using the Holder inequality and (3.37) , and we choose
\varepsilon_2 = \Big[\frac{m_0l_1p^p}{\beta(p+l_1-1)^p}\Big]^{\frac{p+l_1-1+(1-\vartheta_3)r_3}{p+l_1-1}}, |
then, we have
\begin{equation} \frac{1}{l_1+1}\frac{d}{dt}\|u\|^{l_1+1}_{l_1+1}+\beta\varepsilon_2^{\frac{(\vartheta_3 -1)r_3}{(1-\vartheta_3)r_3-(p+l_1-1)}}C^{\frac{(\vartheta_3-1)pr_3}{p+l_1-1}}_{p^*_s}\|u\|^{r_3}_{l_1+1}\leq\lambda\frac{1}{e\sigma}|\Omega|^{1-\frac{r_5}{l_1+1}}\|u\|^{r_5}_{l_1+1}, \end{equation} | (3.44) |
for 0 < \sigma < 2-r . After performing some simple calculations, we finally obtain
\begin{equation} \frac{1}{l_1+1}\frac{d}{dt}\|u\|^{l_1+1}_{l_1+1}+C_{11}\|u\|^{r_3}_{l_1+1}\leq0, \end{equation} | (3.45) |
where
C_{11} = \beta\varepsilon_2^{\frac{(\vartheta_3 -1)r_3}{(1-\vartheta_3)r_3-(p+l_1-1)}}C^{\frac{(\vartheta_3-1)pr_3}{p+l_1-1}}_{p^*_s}-\lambda C,\ \ C = \frac{1}{e\sigma}|\Omega|^{1-\frac{r_5}{l_1+1}}C^{r_5}_{l_1+1}R^{r_5-r_3}. |
Note that \lambda < P_7 , then C_{11} > 0, which implies that
\left\{ \begin{array}{@{}l@{\quad}l} \|u\|_{l_1+1}\leq\Big[\|u_0\|^{l_1+1-r_3}_{l_1+1}+(r_3-l_1-1)C_{11}t\Big]^{\frac{1}{l_1+1-r_3}}, \; \; \; & t\in[0,T_7),\\ \|u\|_{l_1+1}\equiv0,\; \; \; \; & t\in[T_7,\infty),\\ \end{array}\right. |
where
T_7 = \frac{1}{(l_1+1-r_3)C_{11}}\|u_0\|^{l_1+1-r_3}_{l_1+1}. |
The proof is completed.
Proof of Theorem 1.5
(ⅰ) By the definition of E(u) and I(u) , we obtain
\begin{equation} E(u) = \frac{1}{p}I(u)+\frac{1}{p}b(1-\theta)\|u\|^{\theta p}+\frac{\beta (p-q)}{qp}\|u\|^q_q+\lambda\frac{1}{p^2}\|u\|^p_p. \end{equation} | (3.46) |
Choosing \nu = u_t in Definition 1.1, we have
\begin{equation} \int_\Omega u_tu_tdx = -M(\|u\|^p)\langle u,u_t\rangle+\lambda\int_\Omega u_t|u|^{r-2}u\ln|u|dx-\beta\int_\Omega u_t|u|^{q-2}udx. \end{equation} | (3.47) |
Note that
\begin{equation} \frac{d}{dt}E(u) = \frac{1}{p}\frac{d}{dt}(a\|u\|^p+b\|u\|^{\theta p})-\lambda\int_\Omega|u|^{r-2}uu_t\ln|u|dx+\beta\int_\Omega|u|^{q-2}uu_tdx. \end{equation} | (3.48) |
By (3.47) and (3.48) , we obtain
\frac{d}{dt}E(u)+\int_\Omega u_tu_tdx = 0, |
which implies that
\begin{equation} E(u) = E(u_0)-\int^t_0\|u_\tau\|^2_2d\tau. \end{equation} | (3.49) |
Setting \Gamma(t) = \frac{1}{2}\int_\Omega|u(x, t)|^2dx , then we have
\begin{equation} \Gamma'(t) = \int_\Omega u_tudx = -I(u). \end{equation} | (3.50) |
By (3.46) and (3.49) and (3.50) , we obtain
\begin{split} \Gamma'(t) = &-pE(u)+b(1-\theta)\|u\|^{\theta p}+\frac{1}{q}\beta(p-q)\|u\|^q_q+\lambda\frac{1}{p}\|u\|^p_p\\ = &-pE(u_0)+b(1-\theta)\|u\|^{\theta p}+\frac{1}{q}\beta(p-q)\|u\|^q_q\\ &+\lambda\frac{1}{p}\|u\|^p_p+p\int^t_0\|u_\tau\|^2_2d\tau. \end{split} |
Since p > q , then \frac{1}{q}\mu(p-q)\|u\|^q_q > 0 , we obtain
\Gamma'(t)\geq-pE(u_0) > 0. |
By a simple calculation, we get
\|u\|^2_2\geq-2pE(u_0)t+2\|u_0\|^2_2, \ \text{for all} \ t > 0, |
which implies that our result holds.
(ⅱ) Here, we only prove the case of E(u_0) < h , and the proof of E(u_0) = h is similar. Choosing \nu = u in Definition 1.1, we have
\begin{equation} \frac{1}{2}\frac{d}{dt}\|u\|^2_2+M(\|u\|^p)\|u\|^p = \int_\Omega|u|^r\ln|u|dx-\beta\|u\|^q_q, \end{equation} | (3.51) |
namely
\begin{equation} \frac{1}{2}\frac{d}{dt}\|u\|^2_2+I(u) = 0. \end{equation} | (3.52) |
Next, taking \nu = u_t in Definition 1.1 and integrating with respect to time from 0 to t , we have
\begin{equation} \int^t_0\|u_\tau\|^2_2dx+E(u(t)) = E(u_0) < h, \text{for} \ t > 0. \end{equation} | (3.53) |
We claim that u(x, t)\in W for any t > 0 . If it is false, there exists a t_0\in\mathbb{R}^+_0\backslash\{0\} such that u(t_0)\in\partial W , which implies
I(u(x,t_0)) = 0 \ \text{or} \ E(u(x,t_0)) = h. |
From (3.53) , E(u(t_0)) = h is not true. So u(t_0)\in\mathcal{N} , then by the definition of h in (1.9) , we have E(u(t_0))\geq h , which also contradicts with (3.53) . Hence, u(t_0)\in W . By (3.52) and u(t)\in W for all t > 0 , we obtain
\|u\|^2_2\leq\|u_0\|^2_2. |
Remark 2 Compared with problem (1.4), we not only discuss the extinction of weak solutions of problem (1.1) with logarithmic nonlinearity, but also prove that the weak solutions are globally bounded and blow up at infinity.
This research was supported by the Project for Young Talents Growth of Guizhou Provincial Department of Education under (Grant No.Ky[2017]133), and by the project of Guizhou Minzu University under (Grant No.16yjrcxm002 and No.GZMU[2019]YB04).
The authors declare no potential conflict of interests.
[1] |
M. Xiang, V. D. RaDulescu, B. Zhang, Nonlocal Kirchhoff diffusion problems: Local existence and blow-up of solutions, Nonlinearity, 31 (2018), 3228–3250. doi: 10.1088/1361-6544/aaba35
![]() |
[2] | M. Kirkilionis, S. KraMker, R. Rannacher, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends Nonlinear Anal., 3 (2003), 153–191. |
[3] |
L. Caffarelli, Nonlocal diffusions, drifts and games, Nonlinear Partial Differ. Equ., 7 (2012), 37–52. doi: 10.1007/978-3-642-25361-4_3
![]() |
[4] | E. Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 5 (2012), 521–573. |
[5] |
E. Lindgren, P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differ. Equ., 49 (2014), 795–826. doi: 10.1007/s00526-013-0600-1
![]() |
[6] | H. Brezis, P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evolut. Equ., 4 (2001), 387–404. |
[7] | E. Azroul, A. Benkirane, A. Boumazourh, M. Shimi, Exstence results for fractional p(x, \cdot)-Laplacian problem via the nehari manifold approach, Appl. Math. Optim., 50 (2020), 968–1007. |
[8] |
M. Xiang, V. D. RaDulescu, B. Zhang, A critical fractional Choquard-Kirchhoff problem with magnetic field, Commun. Contemp. Math., 21 (2019), 1850004. doi: 10.1142/S0219199718500049
![]() |
[9] |
L. X. Truong, The nehari manifold for a class of Schrdinger equation involving fractional p-Laplacian and sign-changing logarithmic nonlinearity, J. Math. Phys., 60 (2019), 111505. doi: 10.1063/1.5084062
![]() |
[10] |
L. X. Truong, The nehari manifold for fractional p-Laplacian equation with logarithmic nonlinearity on whole space, Comput. Math. Appl., 78 (2019), 3931–3940. doi: 10.1016/j.camwa.2019.06.024
![]() |
[11] |
M. Xiang, V. D. RaDulescu, B. Zhang, Combined effects for fracttonal Schrodinger-Kirchhoff systems with critical, Contr. Opti. Calc. Vari., 24 (2018), 1249–1273. doi: 10.1051/cocv/2017036
![]() |
[12] | A. Ardila, H. Alex, Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity, Nonlinear Anal. Theor. Methods Appl., 4 (2017), 52–64. |
[13] |
Y. L. Li, D. B. Wang, J. L. Zhang, Sign-changing solutions for a class of p-Laplacian Kirchhoff-type problem with logarithmic nonlinearity, AIMS Math., 5(2020), 2100–2112. doi: 10.3934/math.2020139
![]() |
[14] |
T. Boudjeriou, Global existence and blow-up for the fractional p-Laplacian with logarithmic nonlinearity, Mediterr J. Math., 17 (2020), 1-24. doi: 10.1007/s00009-019-1430-y
![]() |
[15] |
T. Boudjeriou, Stability of solutions for a parabolic problem involving fractional p-Laplacian with logarithmic nonlinearity, Mediterr J. Math., 17 (2020), 27–51. doi: 10.1007/s00009-019-1442-7
![]() |
[16] | P. Dai, C. Mu, G. Xu, Blow-up phenomena for a pseudo-parabolic equation with p-Laplacian and logarithmic nonlinearity terms, J. Math. Anal. Appl., 481 (2019), 123439. |
[17] |
W. Liu, Extinction properties of solutions for a class of fast diffusive p-Laplacian equations, Nonlinear Anal. Theory Meth. Appl., 74 (2011), 4520–4532. doi: 10.1016/j.na.2011.04.016
![]() |
[18] |
S. Toualbia, Z. Abderrahmane, S. Boulaaras, Decay estimate and non-extinction of solutions of p-Laplacian nonlocal heat equations, AIMS Math., 5 (2020), 1663–1679. doi: 10.3934/math.2020112
![]() |
[19] |
M. Xiang, D. Yang, Nonlocal Kirchhoff problems: Extinction and non-extinction of solutions, J. Math. Anal. Appl., 477 (2019), 133–152. doi: 10.1016/j.jmaa.2019.04.020
![]() |
[20] | B. Guo, W. Gao, Non-extinction of solutions to a fast diffusive p-Laplace equation with Neumann boundary conditions, J. Math. Anal. Appl., 2 (2015), 1527–1531. |
[21] |
W. Gao, B. Guo, Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the p(x, t)-Laplace operator and a non-local term, Discrete Contin. Dyn. Syst., 36 (2015), 715–730. doi: 10.3934/dcds.2016.36.715
![]() |
[22] | L. Yan, Z. Yang, Blow-up and non-extinction for a nonlocal parabolic equation with logarithmic nonlinearity, Bound. Value. Probl., 121 (2018), 1–11. |
[23] |
Y. Tian, C. Mu, Extinction and non-extinction for a p-Laplacian equation with nonlinear source, Nonlinear Anal., 69 (2008), 2422–2431. doi: 10.1016/j.na.2007.08.021
![]() |
[24] | Y. Cao, C. Liu, Initial boundary value problem for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity, Electron. J. Differ. Equ., 18 (2018), 1–19. |
[25] |
M. Xiang, D. Hu, D. Yang, Least energy solutions for fractional Kirchhoff problems with logarithmic nonlinearity, Nonlinear Anal., 198 (2020), 111899. doi: 10.1016/j.na.2020.111899
![]() |
[26] | S. Chen, The extinction behavior of solutions for a class of reaction diffusion equations, Appl. Math. Mech., 22 (2001), 1352–1356. |
[27] | M. Xiang, D. Yang, B. Zhang, Degenerate Kirchhoff-type fractional diffusion problem with logarithmic nonlinearity, Asympt. Anal., 188 (2019), 1–17. |
[28] | H. Ding, J. Zhou, Global existence and blow-up for a parabolic problem of Kirchhoff type with logarithmic nonlinearity, Appl. Math. Optim., 478 (2019), 393–420. |
[29] | S. Boulaaras, Some existence results for elliptic Kirchhoff equation with changing sign data and a logarithmic nonlinearity, J. Intell. Fuzzy Syst., 42 (2019), 8335–8344. |