scholarly journals Global well-posedness of advective Lotka–Volterra competition systems with nonlinear diffusion

2019 ◽  
Vol 150 (5) ◽  
pp. 2322-2348
Author(s):  
Qi Wang ◽  
Jingyue Yang ◽  
Feng Yu
Keyword(s):  
Global Existence ◽  
Elliptic System ◽  
Nonlinear Diffusion ◽  
Uniform Boundedness ◽  
Parabolic Systems ◽  
Growth Conditions ◽  
Diffusion Models ◽  
Well Posedness ◽  
Sensitivity Functions ◽  
Advection Diffusion

AbstractThis paper investigates the global well-posedness of a class of reaction–advection–diffusion models with nonlinear diffusion and Lotka–Volterra dynamics. We prove the existence and uniform boundedness of the global-in-time solutions to the fully parabolic systems under certain growth conditions on the diffusion and sensitivity functions. Global existence and uniform boundedness of the corresponding parabolic–elliptic system are also obtained. Our results suggest that attraction (positive taxis) inhibits blowups in Lotka–Volterra competition systems.

scholarly journals Global well-posedness and uniform boundedness of a higher-dimensional crime model with exponential decay

2021 ◽  
Author(s):  
YANI FENG ◽  
Deqi Wang
Keyword(s):  
Global Solution ◽  
Exponential Decay ◽  
Uniform Boundedness ◽  
Diffusion System ◽  
Well Posedness ◽  
Mild Conditions ◽  
Modeling Approaches ◽  
Advection Diffusion ◽  
Higher Dimensional

We study a class of reaction–advection–diffusion system which is in fact motivated by recent modeling approaches in criminology. While there are results regarding the existence of global solution of the original crime model, the restriction of the parameters is existed for n >=2 . Under mild conditions, our result expands the parameters to be arbitrary.

scholarly journals Global and non global solutions for a class of coupled parabolic systems

2020 ◽  
Vol 9 (1) ◽  
pp. 1383-1401 ◽  
Author(s):  
T. Saanouni
Keyword(s):  
Global Existence ◽  
Exponential Decay ◽  
Existence Of Solutions ◽  
Global Solutions ◽  
Parabolic Systems ◽  
Heat Equations ◽  
Well Posedness ◽  
Potential Well ◽  
Global Existence Of Solutions ◽  
Coupled Parabolic Systems

Abstract In the present paper, we investigate the global well-posedness and exponential decay for some coupled non-linear heat equations. Moreover, we discuss the global and non global existence of solutions using the potential well method.

scholarly journals The Muskat problem with surface tension and equal viscosities in subcritical $$L_p$$-Sobolev spaces

2021 ◽  
Author(s):  
Anca-Voichita Matioc ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Mathematical Model ◽  
Surface Tension ◽  
Global Existence ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Well Posedness ◽  
Global Existence Of Solutions ◽  
Muskat Problem ◽  
The Mathematical Model ◽  
Quasilinear Parabolic

AbstractIn this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $$W^s_p(\mathbb {R})$$ W p s ( R ) , where $${p\in (1,2]}$$ p ∈ ( 1 , 2 ] and $${s\in (1+1/p,2)}$$ s ∈ ( 1 + 1 / p , 2 ) . This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in $$W^{\overline{s}-2}_p(\mathbb {R})$$ W p s ¯ - 2 ( R ) , where $${\overline{s}\in (1+1/p,s)}$$ s ¯ ∈ ( 1 + 1 / p , s ) . Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.

Attractors for multi-valued lattice dynamical systems with nonlinear diffusion terms

2021 ◽  
Author(s):  
Panpan Zhang ◽  
Anhui Gu
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Diffusion ◽  
Upper Semicontinuity ◽  
Growth Conditions ◽  
Pullback Attractor ◽  
Random Lattice ◽  
Lipschitz Continuous ◽  
Lattice Dynamical Systems ◽  
Long Term Behavior

This paper is devoted to the long-term behavior of nonautonomous random lattice dynamical systems with nonlinear diffusion terms. The nonlinear drift and diffusion terms are not expected to be Lipschitz continuous but satisfy the continuity and growth conditions. We first prove the existence of solutions, and establish the existence of a multi-valued nonautonomous cocycle. We then show the existence and uniqueness of pullback attractors parameterized by sample parameters. Finally, we establish the measurability of this pullback attractor by the method based on the weak upper semicontinuity of the solutions.

scholarly journals Global existence and nonexistence for a class of finitely degenerate coupled parabolic systems with high initial energy level

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yuxuan Chen ◽  
Jiangbo Han
Keyword(s):  
Energy Level ◽  
Global Existence ◽  
Blow Up ◽  
Parabolic Systems ◽  
Initial Energy ◽  
Blowup Time ◽  
Positive Initial Energy ◽  
Existence And Nonexistence ◽  
Nonexistence Of Solutions ◽  
Coupled Parabolic Systems

<p style='text-indent:20px;'>In this paper, we consider a class of finitely degenerate coupled parabolic systems. At high initial energy level <inline-formula><tex-math id="M1">\begin{document}$ J(u_{0})&gt;d $\end{document}</tex-math></inline-formula>, we present a new sufficient condition to describe the global existence and nonexistence of solutions for problem (1)-(4) respectively. Moreover, by applying the Levine's concavity method, we give some affirmative answers to finite time blow up of solutions at arbitrary positive initial energy <inline-formula><tex-math id="M2">\begin{document}$ J(u_{0})&gt;0 $\end{document}</tex-math></inline-formula>, including the estimate of upper bound of blowup time.</p>

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