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

Abstract

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

Nonlinear stability of strong traveling waves for a chemotaxis model with logarithmic sensitivity and periodic perturbations

“…Then a series of works has been established for the transformed system (); we can summarize the results as follows: •Existence and stability of traveling wave solutions for $$$ \varepsilon =0 $$$ [11–20] and for $$$ \varepsilon >0 $$$ [10, 21, 22]. •Global dynamics of solutions in one‐dimensional space [23–28] and in higher spaces [29–38]. •Boundary layer problem [39–42].…”

“…• Global dynamics of solutions in one-dimensional space [23][24][25][26][27][28] and in higher spaces [29][30][31][32][33][34][35][36][37][38].…”

Global weak solutions and asymptotics of a singular PDE-ODE chemotaxis system with discontinuous data

“…There has been a considerable amount of interesting works carried out for the transformed system (1.3). The one-dimensional problem has been first studied extensively from various aspects such as the traveling wave solutions [22,32,34,35,39,41,44], global dynamics of solutions in the whole space [12,30,37,49] or in the bounded interval [19,31,33,48]. For the multidimensional whole space R 2 , the nonlinear stability of planar traveling wave was recently established in [3].…”

Asymptotic behaviors and existence of traveling wave solutions to the Keller-Segel model with logarithmic sensitivity

“…D describes the diffusivity of cell, χ and φ(V ) are chemotactic sensitivity coefficient and chemotactic sensitivity function, respectively, ≥ 0 denotes the chemical diffusion coefficient and the function g(U, V ) describes the chemical kinetics. Since then, this model has obtained a large amount of attention and came up with many related models [22,10,26,9,6,15,23,2,3,21]. Especially, when φ(V ) = ln V , g(U, V ) = −U V m , the model (1) becomes the logarithmic sensitively model as follows…”