Global existence and asymptotic behavior of smooth solutions to a coupled hyperbolic-parabolic system

Zhang, Yinghui; Tan, Zhong; Sun, Mingbao

doi:10.1016/j.nonrwa.2012.07.009

Cited by 26 publications

(22 citation statements)

References 30 publications

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“…• either d = 1 and α = 2 [26,17], • or d = 2, 3, α = 2 and the initial data verifies some smallness condition on strong Sobolev spaces H s , s ≥ 2 [24,27]. Our results removed some of the previous conditions.…”

Section: 4mentioning

confidence: 53%

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Global solutions for a hyperbolic–parabolic system of chemotaxis

Granero-Belinchón

2017

Journal of Mathematical Analysis and Applications

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Section: 4mentioning

confidence: 53%

“…In particular, Fan [23]). The case with general f (u) was studied by Zhang, Tan & Sun [26] and Li & Wang [17].…”

Section: Introductionmentioning

confidence: 99%

Global solutions for a hyperbolic–parabolic system of chemotaxis

Granero-Belinchón

2017

Journal of Mathematical Analysis and Applications

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“…Since the initial value (u 0 , v 0 ) considered in the current work has only L p (R)-regularity, the energy estimate framework in previous works (cf. [17,22,25,26,40]) relying on the higher regularity of initial values no longer applies. We have employed a few new approaches, such as mollifying technique, time-weight function and effective viscous flux, to obtain desired results (see details in Remark 2.2).…”

Section: Introductionmentioning

confidence: 99%

On a parabolic-hyperbolic chemotaxis system with discontinuous data: Well-posedness, stability and regularity

Peng

Wang

2020

Journal of Differential Equations

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The global dynamics and regularity of parabolic-hyperbolic systems is an interesting topic in PDEs due to the coupling of competing dissipation and hyperbolic effects. This paper is concerned with the Cauchy problem of a parabolic-hyperbolic system derived from a chemotaxis model describing the dynamics of the initiation of tumor angiogenesis. It is shown that, as time tends to infinity, the Cauchy problem with large-amplitude discontinuous data admit global weak solutions which converge to a constant state (resp. a viscous shock wave) if the asymptotic states of initial values at far field are equal (resp. unequal). Our results improve the previous results where initial value was required to be continuous and have small amplitude. Numerical simulations are performed to verify our analytical results, illustrate the possible regularity of solutions and speculate the minimal regularity of initial data required to obtain the smooth (classical) solutions of the concerned parabolic-hyperbolic system. MSC 2000: 35A01, 35B40, 35B44, 35K57, 35Q92, 92C17 KEYWORDS: Parabolic-hyperbolic system, discontinuous initial data, weak solutions, effective viscous flux, regularity with the initial valueand the far-field behavior (i.e., asymptotic state at ±∞):where u ± ≥ 0, χ and D are positive constants. The system (1.1) is transformed from the following PDE-ODE singular chemotaxis model proposed in [19] (see [18,28] for mathematical derivation) to describe

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“…The one dimensional case with α = 2 and a general f (u) was studied by Zhang, Tan & Sun [65] and Li & Wang [46].…”

Section: Introductionmentioning

confidence: 99%

On the fractional Fisher information with applications to a hyperbolic–parabolic system of chemotaxis

Granero-Belinchón

2017

Journal of Differential Equations

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Global existence and asymptotic behavior of smooth solutions to a coupled hyperbolic-parabolic system

Cited by 26 publications

References 30 publications

Global solutions for a hyperbolic–parabolic system of chemotaxis

Global solutions for a hyperbolic–parabolic system of chemotaxis

On a parabolic-hyperbolic chemotaxis system with discontinuous data: Well-posedness, stability and regularity

On the fractional Fisher information with applications to a hyperbolic–parabolic system of chemotaxis

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