scholarly journals Asymptotic behavior of solutions to a tumor angiogenesis model with chemotaxis–haptotaxis

2019 ◽  
Vol 29 (07) ◽  
pp. 1387-1412 ◽  
Author(s):  
Peter Y. H. Pang ◽  
Yifu Wang
Keyword(s):  
Tumor Angiogenesis ◽  
Asymptotic Behavior Of Solutions ◽  
Time Behavior ◽  
System Of Differential Equations ◽  
One Dimensional ◽  
Long Time ◽  
Bounded Smooth ◽  
The One ◽  
Global Boundedness ◽  
Angiogenesis Model

This paper studies the following system of differential equations modeling tumor angiogenesis in a bounded smooth domain [Formula: see text] ([Formula: see text]): [Formula: see text] where [Formula: see text] and [Formula: see text] are positive parameters. For any reasonably regular initial data [Formula: see text], we prove the global boundedness ([Formula: see text]-norm) of [Formula: see text] via an iterative method. Furthermore, we investigate the long-time behavior of solutions to the above system under an additional mild condition, and improve previously known results. In particular, in the one-dimensional case, we show that the solution [Formula: see text] converges to [Formula: see text] with an explicit exponential rate as time tends to infinity.

FROM BOUNDARY VALUE PROBLEMS TO DIFFERENCE EQUATIONS: A METHOD OF INVESTIGATION OF CHAOTIC VIBRATIONS

1999 ◽  
Vol 09 (07) ◽  
pp. 1285-1306 ◽  
Author(s):  
E. YU. ROMANENKO ◽  
A. N. SHARKOVSKY
Keyword(s):  
Difference Equation ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Boundary Value ◽  
Dynamical Behavior ◽  
Time Behavior ◽  
One Dimensional ◽  
Long Time ◽  
The Difference ◽  
The One

Among evolutionary boundary value problems for partial differential equations, there is a wide class of problems reducible to difference, differential-difference and other relevant equations. Of especial promise for investigation are problems that reduce to difference equations with continuous argument. Such problems, even in their simplest form, may exhibit solutions with extremely complicated long-time behavior to the extent of possessing evolutions that are indistinguishable from random ones when time is large. It is owing to the reduction to a difference equation followed by the employment of the properties of the one-dimensional map associated with the difference equation, that, it is in many cases possible to establish mathematical mechanisms for one or other type of dynamical behavior of solutions. The paper presents the overall picture in the study of boundary value problems reducible to difference equations (on which the authors have a direct bearing over the last ten years) and demonstrates with several simplest examples the potentialities that such a reduction opens up.

scholarly journals Asymptotic Behavior of a Tumor Angiogenesis Model with Haptotaxis

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 664
Author(s):  
Chi Xu ◽  
Yifu Wang
Keyword(s):  
Asymptotic Behavior ◽  
Tumor Angiogenesis ◽  
Smooth Domain ◽  
Asymptotic Behavior Of Solutions ◽  
Exponential Rate ◽  
Bounded Smooth Domain ◽  
System P ◽  
Bounded Smooth ◽  
Angiogenesis Model

This paper considers the existence and asymptotic behavior of solutions to the angiogenesis system p t = Δ p − ρ ∇ · ( p ∇ w ) + λ p ( 1 − p ) , w t = − γ p w β in a bounded smooth domain Ω ⊂ R N ( N = 1 , 2 ) , where ρ , λ , γ > 0 and β ≥ 1 . More precisely, it is shown that the corresponding solution ( p , w ) converges to ( 1 , 0 ) with an explicit exponential rate if β = 1 , and polynomial rate if β > 1 as t → ∞ , respectively, in L ∞ -norm.

scholarly journals Limit of 𝑝-Laplacian obstacle problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Raffaela Capitanelli ◽  
Maria Agostina Vivaldi
Keyword(s):  
Asymptotic Behavior ◽  
Uniform Convergence ◽  
Explicit Expression ◽  
Positive Measure ◽  
Sufficient Conditions ◽  
Obstacle Problems ◽  
Dimensional Case ◽  
Asymptotic Behavior Of Solutions ◽  
One Dimensional ◽  
The One

AbstractIn this paper, we study asymptotic behavior of solutions to obstacle problems for p-Laplacians as {p\to\infty}. For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case, we provide sufficient conditions to assure the uniform convergence of the whole family of the solutions of obstacle problems either for data f that change sign in Ω or for data f (that do not change sign in Ω) possibly vanishing in a set of positive measure.

Long-time behavior for the full one-dimensional Frémond model for shape memory alloys

2000 ◽  
Vol 12 (6) ◽  
pp. 423-433 ◽  
Author(s):  
Pierluigi Colli ◽  
Philippe Laurençot ◽  
Ulisse Stefanelli
Keyword(s):  
Shape Memory Alloys ◽  
Shape Memory ◽  
Time Behavior ◽  
Long Time Behavior ◽  
One Dimensional ◽  
Long Time
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