共查询到20条相似文献,搜索用时 15 毫秒
1.
The growth of the Lm-norm, m [1,], of non-negative solutions to the Cauchy problem t u – u = |u| is studied for non-negative initial data decaying at infinity. More precisely, the function
is shown to be bounded from above and from below by positive real numbers. This result indicates an asymptotic behaviour dominated by the hyperbolic Hamilton-Jacobi term of the equation. A one-sided estimate for ln u is also established. 相似文献
2.
Simona Dabuleanu 《Journal of Evolution Equations》2005,5(1):35-60
We study the weak solvability of viscous Hamilton-Jacobi equation:
\,0,\,x\,\in\,\Omega,$" align="middle" border="0">
with Neumann boundary condition and irregular initial data 0. The domain
is a bounded open set and p > 0. The last part deals with the case a convex set and the initial data 0 = in a open set D such that
and
相似文献
3.
We prove the existence and the uniqueness of strong solutions for the viscous Hamilton-Jacobi equation: with Neumann boundary condition, and initial data μ0, a continuous function. The domain Ω is a bounded and convex open set with smooth boundary, a∈R,a≠0 and p>0. Then, we study the large time behavior of the solution and we show that for p∈(0,1), the extinction in finite time of the gradient of the solution occurs, while for p?1 the solution converges uniformly to a constant, as t→∞. 相似文献
4.
Michael Winkler 《Journal of Differential Equations》2003,192(2):445-474
We study nonglobal positive solutions to the Dirichlet problem for ut=up(Δu+u) in bounded domains, where 0<p<2. It is proved that the set of points at which u blows up has positive measure and the blow-up rate is exactly . If either the space dimension is one or p<1, the ω-limit set of consists of continuous functions solving . In one space dimension it is shown that actually as t→T, where w coincides with an element of a one-parameter family of functions inside each component of its positivity set; furthermore, we study the size of the components of {w>0} with the result that this size is uniquely determined by Ω in the case p<1, while for p>1, the positivity set can have the maximum possible size for certain initial data, but it may also be arbitrarily close to the minimal length π. 相似文献
5.
The purpose of the paper is to study properties of solutions of the Cauchy problem for the equation
under the assumption
.
General selfsimilar solutions are constructed. Moreover, for initial data with some decay at infinity, we determine
the leading term of the asymptotics of solutions in
which is described by either solutions of the linear heat equation or by particular selfsimilar solutions of the original equation. 相似文献
6.
Junjie Li 《Mathematische Annalen》2007,339(2):251-285
We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary
value problem of a fourth order degenerate parabolic equation in higher space dimensions 相似文献
7.
We prove a Harnack inequality for a degenerate parabolic equation using proper estimates based on a suitable version of the
Rayleigh quotient.
Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday 相似文献
8.
In this paper, an initial boundary value problem related to the equation
9.
Yaping Wu 《Journal of Differential Equations》2005,213(2):289-340
This paper is concerned with the stability/instability of a class of positive spiky steady states for a quasi-linear cross-diffusion system describing two-species competition. By detailed spectral analysis, it is proved that the spiky steady states for the related shadow system are linearly unstable and the spiky steady states for the original cross-diffusion system are non-linearly unstable. 相似文献
10.
Kin Ming Hui 《Mathematische Annalen》2007,339(2):395-443
We prove the existence of a unique solution of the following Neumann problem , u > 0, in (a, b) × (0, T), u(x, 0) = u
0(x) ≥ 0 in (a, b), and , where if m < 0, if m = 0, and
m≤ 0, , and the case −1 < m ≤ 0, , for some constant p > 1 − m. We also obtain a similar result in higher dimensions. As a corollary we will give a new proof of a result of A. Rodriguez
and J.L. Vazquez on the existence of infinitely many finite mass solutions of the above equation in for any −1 < m ≤ 0. We also obtain the exact decay rate of the solution at infinity. 相似文献
11.
Shu-Yu Hsu 《Mathematische Annalen》2002,323(2):281-318
We will show that if u is the solution of the equation , in is an even function on and is monotone decreasing in on , , where is a monotone increasing function satisfying with being given by and , then the rescaled function , will converge uniformly on every compact subset of to as where .
Received: 25 May 2000 / Revised version: 26 October 2001 / Published online: 28 February 2002 相似文献
12.
带非局部源的退化半线性抛物方程的解的爆破性质 总被引:1,自引:0,他引:1
This paper deals with the blow-up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation
u
t
− (x
a
u
x
)
x
=∫
0
a
f(u)dx in (0,a) × (0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under
appropriate hypotheses, the global existence and blow-up in finite time of positve solutions are obtained. It is also proved
that the blow-up set is almost the whole domain. This differs from the local case. Furthermore, the blow-up rate is precisely
determined for the special case: f(u)=u
p
, p>1. 相似文献
13.
Jiaqing Pan 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(15):5069-5080
This work studies the large time behavior of free boundary and continuous dependence on nonlinearity for the Cauchy problem of a degenerate parabolic partial differential equation with absorption. Our objective is to give an explicit expression of speed of propagation of the solution and to show that the solution depends on the nonlinearity of the equation continuously. 相似文献
14.
The Cauchy problem to an equation arising in modeling the motion of viscous droplets is studied in the present paper. The authors prove that if the initial data has compact support, then there exists a weak solution which has compact support for all the time. 相似文献
15.
We investigate the Cauchy problem and the initial-boundary value problem for multi-dimensional conservation laws with degenerate viscosity in the whole space and in the half-space respectively. We give the optimal decay estimates in the W1,p(1≤p≤∞) norm for the perturbation from the planar viscous rarefaction wave. The analysis based on the new Lp-energy method and L1-estimates. 相似文献
16.
N. I. Karachalios N. B. Zographopoulos 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(1):11-30
We study a real Ginzburg-Landau equation, in a bounded domain of
with a variable, generally non-smooth diffusion coefficient having a finite number of zeroes. By using the compactness of the embeddings of the weighted Sobolev spaces involved in the functional formulation of the problem, and the associated energy equation, we show the existence of a global attractor. The extension of the main result in the case of an unbounded domain is also discussed, where in addition, the diffusion coefficient has to be unbounded. Some remarks for the case of a complex Ginzburg-Landau equation are given.Received: May 6, 2002; revised: October 3, 2002 相似文献
17.
In this paper, we study the properties of solutions to a degenerate parabolic equation with variable density and absorption. We first obtain a critical exponent, which distinguishes the localization of solutions from the positivity of them. When positivity prevails, we obtain the other critical exponent with respect to the decay of the variable density, which separates the global existence of interfaces from the disappearance of them. Moreover, the long time behavior of interfaces is characterized. 相似文献
18.
Philip Broadbridge Joanna M. Goard 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2004,55(3):534-538
An exact solution is given for the evolution of an
initially v-shaped surface by a fully nonlinear diffusion
equation. This is the unique generalized solution that is
continuous but not twice differentiable. Since the profile
velocity decreases faster than the reciprocal of the profile
curvature, the point of infinite curvature persists for a finite
positive time. 相似文献
19.
We consider a one-dimensional semilinear parabolic equation , for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We establish estimates of blowup rate upper and lower bounds. We prove that in this case the blowup rate does not match the one obtained by the rescaling method. 相似文献
20.
M. Chaves V.A. Galaktionov 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(11):4030-4048
As a key example, the sixth-order doubly degenerate parabolic equation from thin film theory,