Blow-up of the Solutions to a Parabolic Equation with Fractional Laplace Operator at the Arbitrary Initial Energy Level

Rong-hua JIANG; Jun ZHOU

doi:10.13718/j.cnki.xdzk.2020.05.016

Message Board

Dear readers, authors and reviewers,you can add a message on this page. We will reply to you as soon as possible!

In this paper, we consider a parabolic equation with the fraction Laplace operator. We prove that there exist blow-up solutions with arbitrary initial energy, and then we estimate the upper bound of the blow-up time.

Corresponding author: Jun ZHOU

Received Date: 18/12/2017

Available Online: 01/05/2020

Key words:

Abstract: In this paper, we consider a parabolic equation with the fraction Laplace operator. We prove that there exist blow-up solutions with arbitrary initial energy, and then we estimate the upper bound of the blow-up time.

HTML

本文研究如下带有分数拉普拉斯算子的抛物方程：

其中Ω⊂ $ \mathbb{R}$^N(N≥1)是一个任意有界的开集，0＜s＜1，

其中2^*定义见文献[1]. (-Δ)₂^s是分数拉普拉斯算子，其定义如下：

其中

是标准化的常数，Γ是通常的Gamma函数.

分数次Sobolev空间^[2-3].设Ω⊂ $\mathbb{R} $^N是一个任意开子集，对于s∈(0，1)，我们定义

其范数定义为

对一个任意开集Ω⊂ $\mathbb{R} $^N，我们令

显然W₀^s，2($\overline{\Omega } $)是W^s，2(Ω)的子空间，且通过简单计算可知W₀^s，2($\overline{\Omega } $)存在等价范数$\left| \left\| \cdot \right\| \right| $，其定义为

由文献[4]知存在一个常数C＞0使得对任意u∈W₀^s，2(Ω)有

特别地，如果Ω是有界的，则(4)式对任意的q∈[1，2*]成立.

初始值u₀(x)∈W₀^s，2($\overline{\Omega } $)，W^₀s，2($\overline{\Omega } $)是分数次Sobolev空间，其范数定义为

近期，关于分数拉普拉斯算子的抛物方程被广泛研究^[5-14].在文献[2]中，作者研究了对于问题(1)的弱解(u₀∈L²(Ω))和强解(u₀∈L^∞(Ω))的存在条件.此外，作者还研究了解的动力学行为，如有限维全局吸引子的存在性，平衡态解的全局稳定性等.文献[3]利用势井法研究了问题(1)，并在假设初始能量J(u₀)＜E⁰的条件下得到了解的爆破条件，其中，J定义为

这里C＞0是由(4)式给出的Sobolev常数.

本文将继续研究问题(1)解的爆破条件.为了介绍本文的主要结果，首先介绍文献[2]中的一些定义和结论：

本文的主要结论是如下定理，该定理揭示了问题(1)的解在任意初始能量下都可能发生爆破.

定理1 设q∈(2，2^*]且初始值u₀∈W₀^s，2($\overline{\Omega } $)满足：

其中C＞0是(4)式给出的Sobolev常数，则问题(1)的解u(t)在有限时间T_max爆破且

我们将通过下面引理1来证明定理1.引理1的证明可参见文献[15].

引理1 若F(t)∈C²[0，T)是一个非负函数且满足

其中0＜T≤+∞，r是一个正常数.如果F(0)＞0和F′(0)＞0，则有

且当t→T时，F(t)→+∞.

定理1的证明 定理的证明分为解的爆破及爆破时间的上界估计两个步骤.

第一步(解的爆破)若u(t)是问题(1)的初始值满足不等式(8)的解.如果存在时间t₀使得J(u(t₀))≤0，则由文献[3]的结论可知解在有限时间内爆破.因此在下面的证明中我们始终假设J(u(t))≥0.我们用反证法来证明定理，假设u(t)全局存在，定义函数

根据(7)式、不等式(4)和Hölder不等式以及J(u(t))≥0有

以及

于是由Gronwall不等式可知

另一方面，由(7)式、Hölder不等式以及J(u(t))≥0可得

根据不等式(9)，(10)，我们可以得到

当t足够大时，上述不等式不可能成立，故矛盾.因此u(t)在有限时间T_max内爆破.

第二步(爆破时间的上界) 我们先证明I(u(t))＜0，t∈[0，T_max).我们知道

如果I(u(t))＜0，t∈[0，T_max)不成立，则存在t₀∈[0，T_max)，使得I(u(t₀))=0和I(u(t))＜0，t∈[0，t₀).根据(7)式知‖u(t)‖22在t∈[0，t₀]是单调递增的，则有

根据J(u(t))的单调递减性、(4)式和Hölder不等式有

与(11)式矛盾，故I(u(t))＜0，t∈[0，T_max).

下面我将利用引理1估计T_max的上界.取

并构建一个新函数

由(7)式和$ \left\| u\left( t \right) \right\|_{2}^{2}$在t∈[0，T_max)上的严格单调递增性有

取

根据Hölder不等式有

于是由引理1可得

Reference (15)

Name
	Name cannot be empty!
E-mail
	Mailbox cannot be empty! Mailbox cannot be empty!
Telephone
	Mobile number cannot be empty! Please enter a valid mobile number!
Title

Content
Verification Code

[1]	CAPONI M, PUCCI P.Existence Theorems for Entire Solutions of Stationary Kirchhoff Fractional p-Laplacian Equations[J].Annali di Matematica Pura ed Applicata, 2016, 195(6):2099-2129. doi: 10.1007/s10231-016-0555-x CrossRef Google Scholar
[2]	FISCELLA A, SERVADEI R, VALDINOCI E.Density Properties for Fractional Sobolev Spaces[J].Annales-Academiae Scientiarum Fennicae Mathematica, 2015, 40(1):235-253. Google Scholar
[3]	GAL C G, WARMA M.Reaction-Diffusion Equations with Fractional Diffusion on Non-SmoothDomains with various Boudary Conditions[J].Discrete and Continuous Dynamical Systems, 2016, 36(3). Google Scholar
[4]	LIONS J L, MAGENES E.Non-Homogeneous Boundary Value Problems and Applications[M].Berlin:Springer-Verlag, 1972. Google Scholar
[5]	ADAMS D R, HEDBERG L I.Function Spaces and Potential Theory[M].Berlin:Springer-Verlag, 1996, 265(2):249-263. Google Scholar
[6]	DIPIERRO S, PALATUCCI G, VALDINOCI E.Existence and Symmetry Results for a SchrödingerType Problem Involving the Fractional Laplacian[J].Le Matematiche (Catania), 2013, 68(1):201-216. Google Scholar
[7]	FELMER P, QUAAS A, TAN J.Positive Solutions of the Nonlinear Schrödinger Equation with theFractional Laplacian[J].Proceedings of the Royal Society of Edinburgh, 2012, 142(6):1237-1262. doi: 10.1017/S0308210511000746 CrossRef Google Scholar
[8]	FISCELLA A, PUCCI P.p-Fractional Kirchhoff Equations Involving Critical Nonlinearities[J].Nonlinear Analysis:Real World Applications, 2017, 35:350-378. doi: 10.1016/j.nonrwa.2016.11.004 CrossRef Google Scholar
[9]	赵文波, 李中平.一类指数边界非局部扩散方程的爆破[J].贵州师范大学学报(自然科学版), 2017, 35(3):69-73. doi: 10.3969/j.issn.1004-5570.2017.03.011 CrossRef Google Scholar
[10]	GAL C G, WARMA M.On some Degenerate Non-Local Parabolic Equation Associated with theFractionalp-Laplacian[J].Dynamics of Partial Differential Equations, 2016, 14(1). Google Scholar
[11]	LEVINE H A.Instability and Nonexistence of Global Solutions to Nonlinear Wave Equation of the Form Pu_tt=-Au+F(u)[J].Transactions of the American Mathematical Society, 1974, 192:1-21. Google Scholar
[12]	XIANG M Q, GIOVANNI M B, TIAN G H, et al.Infinitely Many Solutions for theStationary Kirchhoff Problems Involving the Fractional p-Laplacian[J].Nonlinearity, 2016, 29(2):357. doi: 10.1088/0951-7715/29/2/357 CrossRef Google Scholar
[13]	NEZZA E D, PALATUCCI G, VALDINOCI E.Hitchhiker's Guide to the Fractional Sobolev Spaces[J].Bulletin Des Sciences Mathématiques, 2012, 136(5):521-573. doi: 10.1016/j.bulsci.2011.12.004 CrossRef Google Scholar
[14]	WARMA M.The Fractional Relative Capacity and the Fractional Laplacian with Neumann and Robin Boundary Conditions on Open Sets[J].Potential Analysis, 2015, 42(2):499-547. doi: 10.1007/s11118-014-9443-4 CrossRef Google Scholar
[15]	GRISVARD P.Elliptic Problems in Nonsmooth Domains[M].Marshfield:Pitman Publishing Inc, 1985. Google Scholar

Message Board

Blow-up of the Solutions to a Parabolic Equation with Fractional Laplace Operator at the Arbitrary Initial Energy Level

Abstract

References

Access History

通讯作者: 陈斌, bchen63@163.com

Article Metrics

Access History

Other Articles By Authors