共查询到20条相似文献,搜索用时 15 毫秒
1.
QunFei Long 《偏微分方程(英文版)》2020,33(3):222-234
We investigate the initial boundary value problem of the pseudo-parabolic equation $u_{t} - \triangle u_{t} - \triangle u = \phi_{u}u + |u|^{p - 1}u,$ where $\phi_{u}$ is the Newtonian potential, which was studied by Zhu et al. (Appl. Math. Comput., 329 (2018) 38-51), and the global existence and the finite time blow-up of the solutions were studied by the potential well method under the subcritical and critical initial energy levels. We in this note determine the upper and lower bounds for the blow-up time. While estimating the upper bound of blow-up time, we also find a sufficient condition of the solution blowing-up in finite time at arbitrary initial energy level. Moreover, we also refine the upper bounds for the blow-up time under the negative initial energy. 相似文献
2.
We consider the critical nonlinear Schrödinger equation $iu_{t} = -\Delta u-|u|^{4/N}$
with initial condition u(0, x) = u0.For u0$\in$H1, local existence in time of solutions on an interval
[0, T) is known, and there exist finite time blow-up solutions, that is
u0 such that $\textrm{lim} _{t\uparrow T <+\infty}|\nabla u(t)|_{L^{2}}=+\infty$.
This is the smallest power in the nonlinearity for which blow-up occurs, and is critical in this sense.The question we address is to control the blow-up rate from above
for small (in a certain sense) blow-up solutions with negative energy.
In a previous paper [MeR], we established some blow-up properties
of (NLS) in the energy space which implied a control
$|\nabla u(t)|_{L^{2}} \leq C \frac{|\ln(T-t)|^{N/4}}{\sqrt{T-t}}$
and removed the rate of the known explicit blow-up solutions which is $\frac{C}{T-t}$.In this paper, we prove the sharp upper bound expected from numerics as$|\nabla u(t)|_{L^{2}} \leq C \left(\frac{\ln|\ln(T-t)|}{T-t} \right)^{1/2}$by exhibiting the exact geometrical structure of dispersion for the
problem. 相似文献
3.
In this paper,the asymptotic behavior of a non-local hyperbolic problem modelling Ohmic heating is studied.It is found that the behavior of the solution of the hyperbolic problem only has three cases:the solution is globally bounded and the unique steady state is globally asymptotically stable;the solution is infinite when t→∞;the solution blows up.If the solution blows up,the blow-up is uniform on any compact subsets of(0,1] and the blow-up rate is lim t → T*-u(x,t)(T*-t)1/α+βp-1=(α+βp-1/1-α)1/1-α-βp,where T* is the blow-up time. 相似文献
4.
Blow-up vs. Global Finiteness for an Evolution $p$-Laplace System with Nonlinear Boundary Conditions
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Xuesong Wu & Wenjie Gao 《数学研究通讯:英文版》2009,25(4):309-317
In this paper, the authors consider the positive solutions of the system of
the evolution $p$-Laplacian equations $$\begin{cases} u_t ={\rmdiv}(| ∇u |^{p−2} ∇u) + f(u, v), & (x, t) ∈ Ω × (0, T ),
& \\ v_t = {\rmdiv}(| ∇v |^{p−2} ∇v) + g(u, v), &(x, t) ∈ Ω × (0, T) \end{cases}$$with nonlinear boundary conditions $$\frac{∂u}{∂η}= h(u, v),
\frac{∂v}{∂η} = s(u, v),$$and the initial data $(u_0, v_0)$, where $Ω$ is a bounded domain in$\boldsymbol{R}^n$with smooth
boundary $∂Ω, p > 2$, $h(· , ·)$ and $s(· , ·)$ are positive $C^1$ functions, nondecreasing
in each variable. The authors find conditions on the functions $f, g, h, s$ that prove
the global existence or finite time blow-up of positive solutions for every $(u_0, v_0)$. 相似文献
5.
In this paper,the higher order asymptotic behaviors of boundary blow-up solutions to the equation■in bounded smooth domain■are systematically investigated for p and q.The second and third order boundary behaviours of the equation are derived.The results show the role of the mean curvature of the boundary■and its gradient in the high order asymptotic expansions of the solutions. 相似文献
6.
Finite time blow-up and global existence of weak solutions for pseudo-parabolic equation with exponential nonlinearity
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This paper is concerned with the initial boundary value problem of a class of pseudo-parabolic equation $u_t - \triangle u - \triangle u_t + u = f(u)$ with an exponential nonlinearity. The eigenfunction method and the Galerkin method are used to prove the blow-up, the local existence and the global existence of weak solutions. Moreover, we also obtain other properties of weak solutions by the eigenfunction method. 相似文献
7.
New Class of Kirchhoff Type Equations with Kelvin-Voigt Damping and General Nonlinearity: Local Existence and Blow-up in Solutions
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Hanni Dridi & Khaled Zennir 《偏微分方程(英文版)》2021,34(4):313-347
In this paper, we consider a class of Kirchhoff equation, in the presence of a Kelvin-Voigt type damping and a source term of general nonlinearity forms. Where the studied equation is given as follows\begin{equation*}u_{tt} -\mathcal{K}\left( \mathcal{N}u(t)\right)\left[ \Delta_{p(x)}u +\Delta_{r(x)}u_{t}\right]=\mathcal{F}(x, t, u).\end{equation*}Here, $\mathcal{K}\left( \mathcal{N}u(t)\right)$ is a Kirchhoff function, $\Delta_{r(x)}u_{t}$ represent a Kelvin-Voigt strong damping term, and $\mathcal{F}(x, t, u)$ is a source term. According to an appropriate assumption, we obtain the local existence of the weak solutions by applying the Galerkin's approximation method. Furthermore, we prove a non-global existence result for certain solutions with negative/positive initial energy. More precisely, our aim is to find a sufficient conditions for $p(x), q(x), r(x), \mathcal{F}(x,t,u)$ and the initial data for which the blow-up occurs. 相似文献
8.
Pascal Bé gout Ana Vargas 《Transactions of the American Mathematical Society》2007,359(11):5257-5282
In this paper, we show that any solution of the nonlinear Schrödinger equation which blows up in finite time, satisfies a mass concentration phenomena near the blow-up time. Our proof is essentially based on Bourgain's (1998), which has established this result in the bidimensional spatial case, and on a generalization of Strichartz's inequality, where the bidimensional spatial case was proved by Moyua, Vargas and Vega (1999). We also generalize to higher dimensions the results in Keraani (2006) and Merle and Vega (1998).
9.
Peter Y. H. Pang Hongyan Tang Youde Wang 《Calculus of Variations and Partial Differential Equations》2006,26(2):137-169
In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schrödinger equation $ \partial_t u = i ( f(x) \Delta u + \nabla f(x) \cdot \nabla u +k(x)|u|^2u) $ on ${\mathbb{R}}^2In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schr?dinger equation
on
. We present existence and non-existence results and investigate qualitative properties of the solutions when they exist.
Mathematics Subject Classification (2000) 35Q55, 35G25
Dedicated respectfully to Professor Weiyue Ding on the occasion of his sixtieth birthday. 相似文献
10.
P. A. Chubenko 《Differential Equations》2009,45(2):217-225
We consider an initial-boundary value problem for the equation and prove a local existence theorem. By using the energy inequality method, we derive necessary and sufficient conditions for the blow-up of a solution in finite time.
相似文献
$\frac{\partial }{{\partial t}}( - \Delta ^2 u + \Delta u + \Delta _p u) + \Delta u - \left\| {\nabla u} \right\|_2^{2q} \Delta u = 0$
11.
Quoc Hung Phan 《Journal of Evolution Equations》2013,13(2):411-442
We consider the Hardy–Hénon parabolic equation ${u_t-\Delta u =|x|^a |u|^{p-1}u}$ with p > 1 and ${a\in \mathbb{R}}$ . We establish the space-time singularity and decay estimates, and Liouville-type theorems for radial and nonradial solutions. As applications, we study universal and a priori bound of global solutions as well as the blow-up estimates for the corresponding initial-boundary value problem. 相似文献
12.
Xiaomeng Li 《偏微分方程(英文版)》2020,33(2):171-192
Let $\Omega\subset \mathbb{R}^4$ be a smooth bounded domain, $W_0^{2,2}(\Omega)$ be the usual Sobolev space. For any positive integer $\ell$, $\lambda_{\ell}(\Omega)$ is the $\ell$-th eigenvalue of the bi-Laplacian operator. Define $E_{\ell}=E_{\lambda_1(\Omega)}\oplus E_{\lambda_2(\Omega)}\oplus\cdots\oplus E_{\lambda_{\ell}(\Omega)}$, where $E_{\lambda_i(\Omega)}$ is eigenfunction space associated with $\lambda_i(\Omega)$. $E^{\bot}_{\ell}$ denotes the orthogonal complement of $E_\ell$ in $W_0^{2,2}(\Omega)$. For $0\leq\alpha<\lambda_{\ell+1}(\Omega)$, we define a norm by $\|u\|_{2,\alpha}^{2}=\|\Delta u\|^2_2-\alpha \|u\|^2_2$ for $u\in E^\bot_{\ell}$. In this paper, using the blow-up analysis, we prove the following Adams inequalities$$\sup_{u\in E_{\ell}^{\bot},\,\| u\|_{2,\alpha}\leq 1}\int_{\Omega}e^{32\pi^2u^2}{\rm d}x<+\infty;$$moreover, the above supremum can be attained by a function $u_0\in E_{\ell}^{\bot}\cap C^4(\overline{\Omega})$ with $\|u_0\|_{2,\alpha}=1$. This result extends that of Yang (J. Differential Equations, 2015), and complements that of Lu and Yang (Adv. Math. 2009) and Nguyen (arXiv: 1701.08249, 2017). 相似文献
13.
Gongwei Liu & Shuying Tian 《分析论及其应用》2022,38(4):451-466
We investigate the initial boundary value problem of some semilinear pseudo-parabolic equations with Newtonian nonlocal term. We establish a lower bound for the blow-up time if blow-up does occur. Also both the upper bound for $T$ and blow up rate of the solution are given when $J(u_0)<0$. Moreover, we establish the blow up result for arbitrary initial energy and the upper bound for $T$. As a product, we refine the lifespan when $J(u_0)<0.$ 相似文献
14.
Lawrence C. Evans Charles K. Smart 《Calculus of Variations and Partial Differential Equations》2011,42(1-2):289-299
We show that an infinity harmonic function, that is, a viscosity solution of the nonlinear PDE ${- \Delta_\infty u = -u_{x_i}u_{x_j}u_{x_ix_j} = 0}$ , is everywhere differentiable. Our new innovation is proving the uniqueness of appropriately rescaled blow-up limits around an arbitrary point. 相似文献
15.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1999,328(4):297-302
In this Note, we are interested in the possible continuation after the blow-up time Tm of radially symmetric positive classical solutions u of the heat equation with nonlinearity f(u) = up, where p > 1. We say that u blows up completely after Tm if u can not be extended beyond Tm (even in the weak sense). We obtain a complete blow up criterion which relies on the asymptotic behaviour of u around the blow-up singularity x = 0. 相似文献
16.
主要研究了关于R~2中一类带有幂型非线性的广义Zakharov方程组的Cauchy问题的有限时间爆破解的爆破率的下界估计.在α≤0和p≥3条件下,对于Cauchy问题任意给定的属于能量空间H~1(R~2)×L~2(R~2)×L~2(R~2)的有限时间的爆破解,得到了对于t靠近有限爆破时间T时的爆破率的最优下界估计.此外,给出了Cauchy问题维里等式的一个应用. 相似文献
17.
主要研究了一类带Robin边界条件的拟线性抛物方程解的整体存在性与爆破问题,利用微分不等式技术,获得了方程的解发生爆破时的爆破时间的下界.然后给出了方程解整体存在的充分条件,最后得到了方程的解发生爆破时发生爆破时间的上界. 相似文献
18.
詹华税 《数学年刊A辑(中文版)》2012,33(4):449-460
对来自金融数学领域的方程xxu+uyu-tu=c(x,y,t,u),(x,y,t)∈QT=R2×[0,T)的Cauchy问题,给出了一种新的熵解的定义,得到了其适定性结果.可以证明所得到的解还是强解,即方程中所出现的各阶偏导数几乎处处连续.最后讨论了解的爆破性质以及与解的间断点相关的几何性质. 相似文献
19.
This article deals with a nonlocal heat system subject to null Dirichlet boundary conditions,where the coupling nonlocal sources consist of mixed type asymmetric nonlinearities.We at first give the cri... 相似文献
20.
我们研究初始值问题(e)u1/(e)t2=(e)2u1/(e)x2+‖u2(·,t)‖p,
(e)2u2/(e)t2=(e)2u2/(e)x2+‖u1(·,t)‖q,-∞<x<∞,t>0,u1(x,0)=f1(x),
(e)u1/(e)t(x,0)=g1(x),u2(x,0)=f2(x), (e)u2/(e)t(x,0)=g2(x),- ∞<x<∞,where‖ui(·,t)‖=∫∞-∞(4)i(x)|ui(x,t)|dx
with (4)i(x)≥0 and ∫∞-∞(4)i(x)dx=1,i=1,2.然后建立解的全局存在和爆破的标准,提出爆破增长率. 相似文献