共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
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
3.
In this paper, we study a fourth order parabolic equation with nonlinear principal part modeling epitaxial thin film growth in two space dimensions. On the basis of the Schauder type estimates and Campanato spaces, we prove the global existence of classical solutions. 相似文献
4.
This paper is concerned with entire solutions of a monostable reaction-advection-diffusion equation in infinite cylinders without the condition f′(u)≤f′(0). By constructing a quasi-invariant manifold, we prove that there exist two classes of entire solutions. Furthermore, we show that one class of such entire solutions is unique up to space and time translation. 相似文献
5.
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
6.
Christian Stinner 《Journal of Differential Equations》2010,248(2):209-228
This paper deals with weak solutions of the one-dimensional viscous Hamilton-Jacobi equation
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.
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
9.
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
10.
In this paper we study some criteria for the full (space-time) regularity of weak
solutions to the Navier-Stokes equations. In particular, we generalize some
classical and very recent criteria involving the velocity, or its
derivatives. More specifically, we show with elementary tools that if a weak
solution, or its vorticity, is small in appropriate Marcinkiewicz spaces, then it
is regular. 相似文献
11.
12.
In this paper we obtain the continuity of attractors for semilinear parabolic problems with Neumann boundary conditions relatively to perturbations of the domain. We show that, if the perturbations on the domain are such that the convergence of eigenvalues and eigenfunctions of the Neumann Laplacian is granted then, we obtain the upper semicontinuity of the attractors. If, moreover, every equilibrium of the unperturbed problem is hyperbolic we also obtain the continuity of attractors. We also give necessary and sufficient conditions for the spectral convergence of Neumann problems under perturbations of the domain. 相似文献
13.
Matteo Bonforte 《Advances in Mathematics》2010,223(2):529-578
We investigate qualitative properties of local solutions u(t,x)?0 to the fast diffusion equation, t∂u=Δ(um)/m with m<1, corresponding to general nonnegative initial data. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form [0,T]×Ω, with Ω⊆Rd. They combine into forms of new Harnack inequalities that are typical of fast diffusion equations. Such results are new for low m in the so-called very fast diffusion range, precisely for all m?mc=(d−2)/d. The boundedness statements are true even for m?0, while the positivity ones cannot be true in that range. 相似文献
14.
Hiroyuki Chihara 《Journal of Differential Equations》2009,246(2):681-723
We discuss gain of analyticity phenomenon of solutions to the initial value problem for semilinear Schrödinger equations with gauge invariant nonlinearity. We prove that if the initial data decays exponentially, then the solution becomes real-analytic in the space variable and a Gevrey function of order 2 in the time variable except in the initial plane. Our proof is based on the energy estimates developed in our previous work and on fine summation formulae concerned with a matrix norm. 相似文献
15.
The existence, uniqueness and regularity of strong solutions for Cauchy problem and periodic problem are studied for the evolution equation:
where is the so-called subdifferential operator from a real Banach space V into its dual V*. The study in the Hilbert space setting (V = V* = H: Hilbert space) is already developed in detail so far. However, the study here is done in the V–V* setting which is not yet fully pursued. Our method of proof relies on approximation arguments in a Hilbert space H. To assure this procedure, it is assumed that the embeddings
are both dense and continuous. 相似文献
16.
Rossella Agliardi 《Bulletin des Sciences Mathématiques》2002,126(6):435-444
We consider p-evolution equations with real characteristics. We give a condition, on the lower order terms, that is sufficient for well-posedness of the Cauchy problem in Sobolev spaces. 相似文献
17.
Mario Bukal Ansgar Jüngel Daniel Matthes 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2013
This paper is concerned with the analysis of a sixth-order nonlinear parabolic equation whose solutions describe the evolution of the particle density in a quantum fluid. We prove the global-in-time existence of weak nonnegative solutions in two and three space dimensions under periodic boundary conditions. Moreover, we show that these solutions are smooth and classical whenever the particle density is strictly positive, and we prove the long-time convergence to the spatial homogeneous equilibrium at a universal exponential rate. Our analysis strongly uses the Lyapunov property of the entropy functional. 相似文献
18.
Jan W. Cholewa 《Journal of Differential Equations》2010,249(3):485-588
We consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors. We then show some applications of the abstract scheme to various evolutionary problems, from ODEs and retarded functional differential equations to parabolic and hyperbolic PDEs. In particular, we exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in RN with nonlinearities depending on the gradient of the solution. We consider as well systems of reaction-diffusion equations in RN and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in RN. We further discuss some nonlocal and quasilinear problems, as well as the fourth order Cahn-Hilliard equation. 相似文献
19.
Manuela Chaves 《Journal of Functional Analysis》2004,215(2):253-270
We study stability of an equilibrium f∗ of autonomous dynamical systems under asymptotically small perturbations of the equation. We show that such stability takes place if the domain of attraction of the equilibrium f∗ contains a one-parametric ordered family . In the stability analysis we need a special S-relation (a kind of “restricted partial ordering”) to be preserved relative to the family . This S-relation is inherited from the Sturmian zero set properties for linear parabolic equations. As main applications, we prove stability of the self-similar blow-up behaviour for the porous medium equation, the p-Laplacian equation and the dual porous medium equation in with nonlinear lower-order perturbations. For such one-dimensional parabolic equations the S-relation is Sturm's Theorem on the nonincrease of the number of intersections between the solutions and particular solutions with initial data in . This Sturmian property plays a key role and is true for the unperturbed PME, but is not true for perturbed equations. 相似文献
20.
Goro Akagi 《Journal of Differential Equations》2006,231(1):32-56
We prove the existence of solutions of the Cauchy problem for the doubly nonlinear evolution equation: dv(t)/dt+V∂φt(u(t))∋f(t), v(t)∈H∂ψ(u(t)), 0<t<T, where H∂ψ (respectively, V∂φt) denotes the subdifferential operator of a proper lower semicontinuous functional ψ (respectively, φt explicitly depending on t) from a Hilbert space H (respectively, reflexive Banach space V) into (−∞,+∞] and f is given. To do so, we suppose that V?H≡H∗?V∗ compactly and densely, and we also assume smoothness in t, boundedness and coercivity of φt in an appropriate sense, but use neither strong monotonicity nor boundedness of H∂ψ. The method of our proof relies on approximation problems in H and a couple of energy inequalities. We also treat the initial-boundary value problem of a non-autonomous degenerate elliptic-parabolic problem. 相似文献