共查询到20条相似文献,搜索用时 15 毫秒
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In this paper, we give a probabilistic interpretation for a coupled system of Hamilton–Jacobi–Bellman equations using the
value function of a stochastic control problem. First we introduce this stochastic control problem. Then we prove that the
value function of this problem is deterministic and satisfies a (strong) dynamic programming principle. And finally, the value
function is shown to be the unique viscosity solution of the coupled system of Hamilton–Jacobi–Bellman equations. 相似文献
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Thomas Strömberg 《Journal of Evolution Equations》2007,7(4):669-700
Let H (t, x, p) be a Hamiltonian function that is convex in p. Let the associated Lagrangian satisfy the nonstandard minorization condition
where m > 0, ω > 0, and C ≥ 0 are constants. Under some additional conditions, we prove that the associated value function is the unique viscosity
solution of S
t
+ H(t, x, ∇S) = 0 in , without any conditions at infinity on the solution. Here ωT < π/2. To the Hamilton–Jacobi equation corresponding to the classical action integrand in mechanics, we adjoin the continuity
equation and establish the existence and uniqueness of a viscosity–measure solution (S, ρ) of
This system arises in the WKB method. The measure solution is defined by means of the Filippov flow of ∇S.
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《Nonlinear Analysis: Theory, Methods & Applications》2004,57(3):341-348
In this article, we consider (component-wise) positive radial solutions of a weakly coupled system of elliptic equations in a ball with homogeneous nonlinearities. The existence is well-known in general: We give a result for the remaining cases. The uniqueness is less studied: We complement the known results. 相似文献
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This paper is concerned with a parabolic–elliptic–parabolic system arising from ion transport networks. It shows that for any properly regular initial data, the corresponding initial–boundary value problem associated with Neumann–Dirichlet boundary conditions possesses a global classical solution in one-dimensional setting, which is uniformly bounded and converges to a trivial steady state, either in infinite time with a time-decay rate or in finite time. Moreover, by taking the zero-diffusion limit of the third equation of the problem, the global weak solution of its partially diffusive counterpart is established and the explicit convergence rate of the solution of the fully diffusive problem toward the solution of the partially diffusive counterpart, as the diffusivity tends to zero, is obtained. 相似文献
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In the present article we consider several issues concerning the doubly parabolic Keller–Segel system and in the plane, when the initial data belong to critical scaling-invariant Lebesgue spaces. More specifically, we analyze the global existence of integral solutions, their optimal time decay, uniqueness and positivity, together with the uniqueness of self-similar solutions. In particular, we prove that there exist integral solutions of any mass, provided that ε>0 is sufficiently large. With those results at hand, we are then able to study the large time behavior of global solutions and prove that in the absence of the degradation term (α=0) the solutions behave like self-similar solutions, while in the presence of the degradation term (α>0) the global solutions behave like the heat kernel. 相似文献
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In this paper, we shall study the problem of optimal control of the parabolic–elliptic system
and ut+(f(t,x,u))x+g(t,x,u)+Px−(a(t,x)ux)x=f0+B∗ν
−Pxx+P=h(t,x,u,ux)+k(t,x,u)
u|t=0=u0.
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We obtain existence and multiplicity results for the solutions of a class of coupled semilinear bi-harmonic Schrödinger equations. Actually, using the classical Mountain Pass Theorem and minimization techniques, we prove the existence of critical points of the associated functional constrained on the Nehari manifold.Furthermore, we show that using the so-called fibering method and the Lusternik–Schnirel’man theory there exist infinitely many solutions, actually a countable family of critical points, for such a semilinear bi-harmonic Schrödinger system under study in this work. 相似文献
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《Journal of Computational and Applied Mathematics》2012,236(2):253-264
A nonlinear iteration method named the Picard–Newton iteration is studied for a two-dimensional nonlinear coupled parabolic–hyperbolic system. It serves as an efficient method to solve a nonlinear discrete scheme with second spatial and temporal accuracy. The nonlinear iteration scheme is constructed with a linearization–discretization approach through discretizing the linearized systems of the original nonlinear partial differential equations. It can be viewed as an improved Picard iteration, and can accelerate convergence over the standard Picard iteration. Moreover, the discretization with second-order accuracy in both spatial and temporal variants is introduced to get the Picard–Newton iteration scheme. By using the energy estimate and inductive hypothesis reasoning, the difficulties arising from the nonlinearity and the coupling of different equation types are overcome. It follows that the rigorous theoretical analysis on the approximation of the solution of the Picard–Newton iteration scheme to the solution of the original continuous problem is obtained, which is different from the traditional error estimate that usually estimates the error between the solution of the nonlinear discrete scheme and the solution of the original problem. Moreover, such approximation is independent of the iteration number. Numerical experiments verify the theoretical result, and show that the Picard–Newton iteration scheme with second-order spatial and temporal accuracy is more accurate and efficient than that of first-order temporal accuracy. 相似文献
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Jia-Yong Wu 《Journal of Mathematical Analysis and Applications》2012,396(1):363-370
We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li–Yau differential Harnack inequality for a nonlinear parabolic equation to the constrained trace Chow–Hamilton Harnack inequality for this nonlinear equation with respect to evolving metrics related to the Ricci flow on a 2-dimensional closed manifold. This result can be regarded as a nonlinear version of the previous work of Y. Zheng and the author [J.-Y. Wu, Y. Zheng, Interpolating between constrained Li–Yau and Chow–Hamilton Harnack inequalities on a surface, Arch. Math., 94 (2010) 591–600]. 相似文献
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《中国科学 数学(英文版)》2015,(8)
Consider the following system of double coupled Schr¨odinger equations arising from Bose-Einstein condensates etc.,-△u+u=μ1u3+βuv2-κv,-△v+v=μ2v3+βu2v-κu,u≠0,v≠0 and u,v∈H1(RN),whereμ1,μ2are positive and fixed;κandβare linear and nonlinear coupling parameters respectively.We first use critical point theory and Liouville type theorem to prove some existence and nonexistence results on the positive solutions of this system.Then using the positive and non-degenerate solution to the scalar equation-△ω+ω=ω3,ω∈H1r(RN),we construct a synchronized solution branch to prove that forβin certain range and fixed,there exist a series of bifurcations in product space R×H1r(RN)×H1r(RN)with parameter κ. 相似文献
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This paper analyzes the surface/subsurface flow coupled with transport. The flow is modeled by the coupling of Navier–Stokes and Darcy equations. The transport of a species is modeled by a convection-dominated parabolic equation. The two-way coupling between flow and transport is nonlinear and it is done via the velocity field and the viscosity. This problem arises from a variety of natural phenomena such as the contamination of the groundwater through rivers. The main result is existence and stability bounds of a weak solution. 相似文献