Blow-Up Phenomena and Decay for the Periodic Degasperis-Procesi Equation with Weak Dissipation

Wu, Shuyin; Yin, Zhaoyang

doi:10.2991/jnmp.2008.15.s2.3

Cited by 21 publications

(7 citation statements)

References 55 publications

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“…Ott and Sudan [69] [74] and on the line [75] and the weakly dissipative Degasperis-Procesi equation (Equ. (1.1) with m = 0, b = 3, λ ∈ R) on the circle [76] and on the line [77,78]. Niu and Zhang [67] considered the blow-up phenomena and global existence for the nonuniform weakly dissipative b-equation (Equ.…”

Section: Shouming Zhou Chunlai Mu and Liangchen Wangmentioning

confidence: 99%

Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation

Zhou¹,

Mu²,

Wang³

2013

DCDS-A

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This paper deals with the Cauchy problem for a weakly dissipative shallow water equation with high-order nonlinearities yt + u m+1 yx + bu m uxy + λy = 0, where λ, b are constants and m ∈ N, the notation y :which includes the famous b-equation and Novikov equations as special cases. The local well-posedness of solutions for the Cauchy problem in Besov space B s p,r with 1 ≤ p, r ≤ +∞ and s > max{1 + 1 p , 3 2 } is obtained. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and conditions that lead to the development of singularities in finite time for the solutions are acquired, moreover, the propagation behaviors of compactly supported solutions are also established. Finally, the weak solution and analytic solution for the equation are considered. 2010 Mathematics Subject Classification. Primary: 35G25, 35L05; Secondary: 35Q50.

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Section: Shouming Zhou Chunlai Mu and Liangchen Wangmentioning

confidence: 99%

Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation

Zhou¹,

Mu²,

Wang³

2013

DCDS-A

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“…The system (2.1) reduces to (1.2) when b = 2 and σ(t, x) ≡ 0. When b = 3 and σ(t, x) ≡ 0, it reduces to the weakly dissipative Degasperis-Procesi (DP) equation [8,10,14,32,36]. If the second component σ(t, x) is allowed to be nonzero, the choices b = 2 and b = 3 in (2.1) give rise to the weakly dissipative two-component Camassa-Holm and Degasperis-Procesi equations, cf.…”

Section: Statement Of Resultsmentioning

confidence: 99%

On the weakly dissipative Camassa–Holm, Degasperis–Procesi, and Novikov equations

Lenells

Wunsch

2013

Journal of Differential Equations

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We show that the weakly dissipative Camassa-Holm, Degasperis-Procesi, Hunter-Saxton, and Novikov equations can be reduced to their non-dissipative versions by means of an exponentially time-dependent scaling. Hence, up to a simple change of variables, the non-dissipative and dissipative versions of these equations are equivalent. Similar results hold also for the equations in the so-called b-family of equations as well as for the two-component and µ-versions of the above equations.2010 Mathematics Subject Classification. 35Q35,35C05,35C99.

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“…They also studied the weakly dissipative Degasperis-Procesi equation y t + uy x + 3u x y + λy = 0, y = u − u xx , (1.4) on the line [18,19] and on the circle [20], where λ > 0 denotes the weak viscosity. In [21], Yan, Li and Zhang considered the weakly dissipative Novikov equation y t + u 2 y x + 3uu x y + λy = 0, y = u − u xx , (1.5) and obtained the global existence and blow-up phenomenon for this equation.…”

Section: Introductionmentioning

confidence: 98%

On the optimal control problem for the Novikov equation with strong viscosity

Zhou

Yang

Chen

et al. 2016

Journal of Mathematical Analysis and Applications

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Please cite this article in press as: J. Zhou et al., On the optimal control problem for the Novikov equation with strong viscosity, Highlights• The existence of a unique weak solution to the viscous Novikov equation is obtained.• The existence of an optimal solution to the optimal control problem is shown. • The first-order necessary optimality condition is deduced.• Two second-order sufficient optimality conditions are established. AbstractIn this paper, we consider an optimal control problem for the Novikov equation with strong viscosity. Using the Faedo-Galerkin method we derive the existence of a unique weak solution to this equation. Applying Lions' theory, we obtain the existence of an optimal solution to the control problem for this equation. We also deduce the first-order necessary optimality condition. Moreover we establish two second-order sufficient optimality conditions, which require coercivity of the augmented Lagrangian functional on the whole space or on a suitable subspace.

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Blow-Up Phenomena and Decay for the Periodic Degasperis-Procesi Equation with Weak Dissipation

Cited by 21 publications

References 55 publications

Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation

Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation

On the weakly dissipative Camassa–Holm, Degasperis–Procesi, and Novikov equations

On the optimal control problem for the Novikov equation with strong viscosity

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