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Khramtsov, O. V.

doi:10.1023/a:1014471307972

“…In [5], problem (c)-(e) with the function F = u|u| p−1 − µ|∇u| q , µ > 0, was treated as a model of population dynamics describing the evolution of the population density of a species. The present paper is a continuation of [6]. It is known (e.g., see [7, p. 37]) that, owing to the degeneration of Eq.…”

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confidence: 91%

Relative stabilization of solutions of a degenerating parabolic equation with nonlinear lower-order terms

Khramtsov

¹

2004

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For a nonlinear degenerate parabolic equation of a perturbed process, we use a nonlinear feedback control to solve the stabilization problem for solutions with initial functions in some class to be constructed.Consider the Cauchy problemwhere the output u ∈ R 1 is the state of the perturbed process, the initial perturbed state f ∈ R 1 is a continuous function (according to the theory in [1, pp. 180-185], it is meaningful to consider nonnegative functions u and f ), the input v ∈ R 1 is a control, and the parameters a, c, α, and λ are real numbers such that ac = 0, α > 1, and, and produces only nonnegative solutions of problem (1), (2). Conditions (a) and (b) imply that the output v and the input u of Eq. (1) are comparable in magnitude: condition (a) implies the inequality |v|/u ≤ 1 for u ≥ 1, and condition (b) gives |u − |v|| ≤ 1 for u ∈ [0, 1].Parabolic equations with gradient nonlinearities have been comprehensively studied [2] from various viewpoints. The first boundary value problemin domains D bounded and unbounded in R n was considered in [2], and for a wide class of such problems, it was shown that global solutions are absent for sufficiently large initial data. Simplest sufficient conditions for the existence and absence of a global solution were given in [3] for the first boundary value problem (c)-(e) with the function F = λu p − |∇u| q , where D is a bounded domain in R n , n ≥ 1, with smooth boundary, λ > 0, p > 1, and q > 1. The lifespan (finite or infinite) of a solution of problem (c)-(e) was found in [4] in the case of the function F = u p − µ|∇u| q , µ > 0, and an unbounded domain D for various ranges of the parameters p, q, and µ. In [5], problem (c)-(e) with the function F = u|u| p−1 − µ|∇u| q , µ > 0, was treated as a model of population dynamics describing the evolution of the population density of a species. The present paper is a continuation of [6]. It is known (e.g., see [7, p. 37]) that, owing to the degeneration of Eq. (1) for u = 0, problem (1), (2) cannot have a classical solution even for smooth initial data. It is reasonable [8] to define a solution of the equation as follows.Definition 1. A nonnegative continuous function u(x, t) inS is called a solution of Eq. (1) for a given control v(x, t) if u(x, t) has a continuous derivative u x (x, t) and satisfies the integral identity t2 t1 R uh t + u α h xx + a |u x | λ h + cvh dx dt − R uh t2 t1 dx = 0 ( * )

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“…

For a nonlinear degenerate parabolic equation of a perturbed process, we use a nonlinear feedback control to solve the stabilization problem for solutions with initial functions in some class to be constructed.

Consider the Cauchy problem

where the output u ∈ R 1 is the state of the perturbed process, the initial perturbed state f ∈ R 1 is a continuous function (according to the theory in [1, pp. The present paper is a continuation of [6]. Conditions (a) and (b) imply that the output v and the input u of Eq.

…”

mentioning

confidence: 96%

Relative stabilization of solutions of a degenerating parabolic equation with nonlinear lower-order terms

Khramtsov¹

2004

Diff Equat

Self Cite

0

View full text Add to dashboard Cite

For a nonlinear degenerate parabolic equation of a perturbed process, we use a nonlinear feedback control to solve the stabilization problem for solutions with initial functions in some class to be constructed.Consider the Cauchy problemwhere the output u ∈ R 1 is the state of the perturbed process, the initial perturbed state f ∈ R 1 is a continuous function (according to the theory in [1, pp. 180-185], it is meaningful to consider nonnegative functions u and f ), the input v ∈ R 1 is a control, and the parameters a, c, α, and λ are real numbers such that ac = 0, α > 1, and, and produces only nonnegative solutions of problem (1), (2). Conditions (a) and (b) imply that the output v and the input u of Eq. (1) are comparable in magnitude: condition (a) implies the inequality |v|/u ≤ 1 for u ≥ 1, and condition (b) gives |u − |v|| ≤ 1 for u ∈ [0, 1].Parabolic equations with gradient nonlinearities have been comprehensively studied [2] from various viewpoints. The first boundary value problemin domains D bounded and unbounded in R n was considered in [2], and for a wide class of such problems, it was shown that global solutions are absent for sufficiently large initial data. Simplest sufficient conditions for the existence and absence of a global solution were given in [3] for the first boundary value problem (c)-(e) with the function F = λu p − |∇u| q , where D is a bounded domain in R n , n ≥ 1, with smooth boundary, λ > 0, p > 1, and q > 1. The lifespan (finite or infinite) of a solution of problem (c)-(e) was found in [4] in the case of the function F = u p − µ|∇u| q , µ > 0, and an unbounded domain D for various ranges of the parameters p, q, and µ. In [5], problem (c)-(e) with the function F = u|u| p−1 − µ|∇u| q , µ > 0, was treated as a model of population dynamics describing the evolution of the population density of a species. The present paper is a continuation of [6]. It is known (e.g., see [7, p. 37]) that, owing to the degeneration of Eq. (1) for u = 0, problem (1), (2) cannot have a classical solution even for smooth initial data. It is reasonable [8] to define a solution of the equation as follows.Definition 1. A nonnegative continuous function u(x, t) inS is called a solution of Eq. (1) for a given control v(x, t) if u(x, t) has a continuous derivative u x (x, t) and satisfies the integral identity t2 t1 R uh t + u α h xx + a |u x | λ h + cvh dx dt − R uh t2 t1 dx = 0 ( * )

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“…The other pressing open question is to find out what is the precise role of the replicanondiagonal saddle points and of the coupling (2.3) in the dual gravity description. We expect that the replica-nondiagonal saddle points in SYK with decoupled replicas, obtained in [32] describe the contributions of nontrivial topologies to the path integral of UV completion of the JT gravity [41], which were discussed in [15]. The work [16,38] has shown that the model with local interaction defined by (5.2) is suitable as a holographic dual to traversable wormhole.…”

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confidence: 97%

Revealing Nonperturbative Effects in the SYK Model

Aref’eva

¹

,

Волович

²

,

Khramtsov

³

2019

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We study the large N saddle points of two SYK chains coupled by an interaction that is nonlocal in Euclidean time. We start from analytic treatment of the free case with q = 2 and perform the numerical study of the interacting case q = 4. We show that in both cases there is a nontrivial phase structure with infinite number of phases. Every phase correspond to a saddle point in the non-interacting two-replica SYK. The nontrivial saddle points have non-zero value of the replica-nondiagonal correlator in the sense of quasi-averaging, when the coupling between replicas is turned off. Thus, the nonlocal interaction between replicas provides a protocol for turning the nonperturbatively subleading effects in SYK into non-equilibrium configurations which dominate at large N . For comparison we also study two SYK chains with local interaction for q = 2 and q = 4. We show that the q = 2 model also has a similar phase structure, whereas in the q = 4 model, dual to the traversable wormhole, the phase structure is different. arXiv:1905.04203v1 [hep-th]

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Cited by 3 publications

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Relative stabilization of solutions of a degenerating parabolic equation with nonlinear lower-order terms

Relative stabilization of solutions of a degenerating parabolic equation with nonlinear lower-order terms

Relative stabilization of solutions of a degenerating parabolic equation with nonlinear lower-order terms

Revealing Nonperturbative Effects in the SYK Model

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