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
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M. O. Korpusov 《Mathematical Notes》2013,93(1-2):90-101
A new model three-dimensional third-order equation of Hamilton-Jacobi type is derived. For this equation, the initial boundary-value problem in a bounded domain with smooth boundary is studied and local solvability in the strong generalized sense is proved; in addition, sufficient conditions for the blow-up in finite time and sufficient conditions for global (in time) solvability are obtained. 相似文献
5.
The review article of Crandall, Ishii, and Lions [Bull. AMS,27, No. 1, 1–67 (1992)] devoted to viscosity solutions of first- and second-order partial differential equations contains the
exact Lax formula
for a solution to the Hamilton-Jacobi nonlinear partial differential equation
where the Cauchy datav:R
n
→R are chosen as a function properly convex and semicontinuous from below, ‖·‖=<·,·> is the usual norm inR
n
,n ∉Z
+, andt ∉R
+ is a positive evolution parameter. The article also states that there is no exact proof of the Lax formula (1) based on general
properties of the Hamiltonian-Jacobi equation (2). This work presents precisely such an exact proof of the Lax formula (1).
Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv; Courant Institute
of Mathematical Sciences at NYU, New York. Published in Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp.
20–25, 相似文献
((1)) |
((2)) |
6.
Michael B. Tamburro 《Israel Journal of Mathematics》1977,26(3-4):232-264
The theory of nonlinear evolution equations in a Banach space is used to prove the existence of global weak solutions of the Cauchy problem for the general time and space-dependent Hamilton-Jacobi equation. 相似文献
7.
R. Funke 《Journal of Mathematical Sciences》1991,53(4):449-452
We consider a stochastic differential equation of the form $$d\xi (t) = \alpha [\xi (t)]^{1 + \gamma } dt + \beta \xi (t)^{1 + \tfrac{\gamma }{2}} dw(t)$$ with initial conditionξ(0)=x ∈ (0, ∞), whereα: γ ∈R 1,β>0, andw(t) is a standard Wiener process. All possible cases for the behavior of the trajectory of the processξ(t) on (0, ∞) are described as functions of the values of the parameters γ and υ=2α/β 2: the process may become infinite after a finite or infinite time, it may vanish after finite or infinite time, it may be regular or recurrent. 相似文献
8.
Benjamin Seeger 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2021,38(4):1217-1253
We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an enhancement effect, for which we provide a quantitative estimate. As an application, we perform a noise sensitivity analysis for Hamilton-Jacobi equations forced by a noise term with small amplitude, and identify the scaling at which the macroscopic enhancement effect is felt. The results depend on new, probabilistic estimates for the large scale Hölder regularity of the solutions, which are of independent interest. 相似文献
9.
Thomas Strömberg 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(6):1802-1811
We present comparison, uniqueness and existence results for unbounded solutions of a viscous Hamilton-Jacobi or eikonal equation. The equation includes an unbounded potential term V(x) subject to a quadratic upper bound. The results are obtained through a tailor-made change of variables in combination with the Hopf-Cole transformation. An integral representation formula for the solution of the Cauchy problem is derived in the case where V(x)=ω2|x|2/2. 相似文献
10.
Patrick Bernard 《Regular and Chaotic Dynamics》2013,18(6):674-685
We study the Cauchy problem for the Hamilton-Jacobi equation with a semiconcave initial condition. We prove an inequality between two types of weak solutions emanating from such an initial condition (the variational and the viscosity solution).We also give conditions for an explicit semi-concave function to be a viscosity solution. These conditions generalize the entropy inequality characterizing piecewise smooth solutions of scalar conservation laws in dimension one. 相似文献
11.
We investigate the large time behavior of positive solutions with finite mass for the viscous Hamilton-Jacobi equationu
t
= Δu + |Δu|
p
,t>0,x ∈ ℝ
N
, wherep≥1 andu(0,.)=u
0≥0,u
0≢0,u
0∈L
1. DenotingI
∞=lim
t→∞‖u(t)‖1≤∞, we show that the asymptotic behavior of the mass can be classified along three cases as follows:
We also consider a similar question for the equationu
t=Δu+u
p
. 相似文献
– | • ifp≤(N+2)/(N+1), thenI ∞=∞ for allu 0; |
– | • if (N+2)/(N+1)<p<2, then bothI ∞=∞ andI ∞<∞ occur; |
– | • ifp≥2, thenI ∞<∞ for allu 0. |
12.
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
13.
I. Capuzzo Dolcetta 《Applied Mathematics and Optimization》1983,10(1):367-377
An approximation of the Hamilton-Jacobi-Bellman equation connected with the infinite horizon optimal control problem with discount is proposed. The approximate solutions are shown to converge uniformly to the viscosity solution, in the sense of Crandall-Lions, of the original problem. Moreover, the approximate solutions are interpreted as value functions of some discrete time control problem. This allows to construct by dynamic programming a minimizing sequence of piecewise constant controls. 相似文献
14.
Necessary and sufficient conditions for the optimality of a pair subject to are given. Here is a selfadjoint operator with closed range on a Hilbert space and . The case - unbounded is also discussed, which leads to some open problems. This general functional scheme includes most of the previous results on the optimal control of the -periodic wave equation for all in a dense subset of . It also includes optimal control problems for some elliptic equations.
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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
18.
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
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