Regularity analysis for an abstract system of coupled hyperbolic and parabolic equations

Hao, Jianghao; Liu, Zhuangyi; Yong, Jiongmin

doi:10.1016/j.jde.2015.06.010

Cited by 34 publications

(17 citation statements)

References 14 publications

(24 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then the region of C ∞ smoothness was founded in [16]. A complete picture of the stability and regularity of this model were given in [9,10]. In the case γ > 0, the analysis become more demanding due to the following challenges: How to divide the parameter region E into subregions with specific stability and regularity properties due to the additional parameter γ?…”

Section: Semigroup Ementioning

confidence: 99%

See 1 more Smart Citation

Regularity analysis for an abstract thermoelastic system with inertial term

2021

Self Cite

View full text Add to dashboard Cite

In this paper, we provide a complete regularity analysis for an abstract system of coupled hyperbolic and parabolic equations. The asymptotic stability of this model was investigated in [6]. We are able to decompose the parameter region into three parts where the semigroup associated with the system is analytic, of Gevrey class of specific order, and non-smoothing, respectively. Moreover, by a detailed and creative spectral analysis,, we will show that the order of Gevrey class is sharp, under proper conditions. We also show that the orders of the polynomial stability obtained in [6] is optimal.

show abstract

Section: Semigroup Ementioning

confidence: 99%

“…The following corollary will be useful below. See [10]. for some a ∈ R, then the semigroup e At is not differentiable.…”

Section: Frequency Domain Characterization Of Semigroup's Propertiesmentioning

confidence: 99%

Regularity analysis for an abstract thermoelastic system with inertial term

2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…where (α, β) ∈ [0, 1] × [0, 1], γ 1 ∈ R\{0}, γ 2 ∈ R + , and A denotes a self-adjoint operator on a Hilbert space. The regularity analysis for (1.1) have been developed in [1,2,3,7,10,11,13,14,23,24], where the semigroup associated with the system is analytic, of specific order Gevrey classes, and nonsmoothing have been obtained. Moreover, L p -resolvent estimates and time decay estimates for the solutions in the L q norm with q ∈ [2, ∞] have been derived [7].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic profiles of solutions for regularity-loss-type generalized thermoelastic plate equations and their applications

Liu¹,

Chen²

2020

Z. Angew. Math. Phys.

View full text Add to dashboard Cite

In this paper, we consider generalized thermoelastic plate equations with Fourier's law of heat conduction. By introducing a threshold for decay properties of regularity-loss, we investigate decay estimates of solutions with/without regularity-loss in a framework of weighted L 1 spaces. Furthermore, asymptotic profiles of solutions are obtained by using representations of solutions in the Fourier space, which are derived by employing WKB analysis. Next, we study generalized thermoelastic plate equations with additional structural damping, and analysis the influence of structural damping on decay properties and asymptotic profiles of solutions. We find that the regularity-loss structure is destroyed by structural damping. Finally, we give some applications of our results on thermoelastic plate equations and damped Moore-Gibson-Thompson equation.Mathematics Subject Classification (2010). Primary 35M31; Secondary 35B40, 35Q74, 74F05.

show abstract

“…Therefore the relation between the regularity of solutions and the initial values of parabolic-hyperbolic systems has been an interesting topic and attracted many studies (e.g. see [4,9,30]). This paper is concerned with the Cauchy problem of the following parabolic-hyperbolic system in R:…”

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

View full text Add to dashboard Cite

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

show abstract

Regularity analysis for an abstract system of coupled hyperbolic and parabolic equations

Cited by 34 publications

References 14 publications

Regularity analysis for an abstract thermoelastic system with inertial term

Regularity analysis for an abstract thermoelastic system with inertial term

Asymptotic profiles of solutions for regularity-loss-type generalized thermoelastic plate equations and their applications

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

Contact Info

Product

Resources

About