scholarly journals Flow invariance for perturbed nonlinear evolution equations

1996 ◽  
Vol 1 (4) ◽  
pp. 417-433 ◽  
Author(s):  
Dieter Bothe
Keyword(s):  
Evolution Equations ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Accretive Operator ◽  
Nonlinear Evolution Equations ◽  
Evolution System ◽  
Reaction Diffusion Systems ◽  
Closed Sets

LetXbe a real Banach space,J=[0,a]⊂R,A:D(A)⊂X→2X\ϕanm-accretive operator andf:J×X→Xcontinuous. In this paper we obtain necessary and sufficient conditions for weak positive invariance (also called viability) of closed setsK⊂Xfor the evolution systemu′+Au∍f(t,u)  on  J=[0,a]. More generally, we provide conditions under which this evolution system has mild solutions satisfying time-dependent constraintsu(t)∈K(t)onJ. This result is then applied to obtain global solutions of reaction-diffusion systems with nonlinear diffusion, e.g. of typeut=ΔΦ(u)+g(u)  in  (0,∞)×Ω,   Φ(u(t,⋅))|∂Ω=0,   u(0,⋅)=u0under certain assumptions on the setΩ⊂Rnthe functionΦ(u1,…,um)=(φ1(u1),…,φm(um))andg:R+m→Rm.

INTEGRAL SOLUTIONS TO A CLASS OF NONLOCAL EVOLUTION EQUATIONS

2010 ◽  
Vol 12 (06) ◽  
pp. 1031-1054 ◽  
Author(s):  
JESÚS GARCÍA-FALSET ◽  
SIMEON REICH
Keyword(s):  
Banach Space ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Sufficient Conditions ◽  
Accretive Operator ◽  
Integral Solution ◽  
Nonlinear Evolution Equations ◽  
Nonlocal Evolution Equations ◽  
Integral Solutions

We study the existence of integral solutions to a class of nonlinear evolution equations of the form [Formula: see text] where A : D(A) ⊆ X → 2X is an m-accretive operator on a Banach space X, and f : [0, T] × X → X and [Formula: see text] are given functions. We obtain sufficient conditions for this problem to have a unique integral solution.

scholarly journals Impulsive Evolution Equations with Causal Operators

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 48
Author(s):  
Tahira Jabeen ◽  
Ravi P. Agarwal ◽  
Vasile Lupulescu ◽  
Donal O’Regan
Keyword(s):  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Strongly Continuous Semigroups ◽  
Causal Operators ◽  
Impulsive Evolution Equations ◽  
Impulsive Reaction

In this paper, we establish sufficient conditions for the existence of mild solutions for certain impulsive evolution differential equations with causal operators in separable Banach spaces. We rely on the existence of mild solutions for the strongly continuous semigroups theory, the measure of noncompactness and the Schauder fixed point theorem. We consider the impulsive integro-differential evolutions equation and impulsive reaction diffusion equations (which could include symmetric kernels) as applications to illustrate our main results.

scholarly journals Asymptotic Smoothing and Global Attractors for a Class of Nonlinear Evolution Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Yongqin Xie ◽  
Zhufang He ◽  
Chen Xi ◽  
Zheng Jun
Keyword(s):  
Evolution Equations ◽  
Global Solutions ◽  
Global Attractors ◽  
Nonlinear Evolution Equations ◽  
Time Behavior ◽  
Asymptotic Regularity ◽  
Compact Attractor ◽  
Long Time ◽  
Critical Exponential Growth ◽  
Semilinear Evolution Equations

We prove the asymptotic regularity of global solutions for a class of semilinear evolution equations in H01(Ω)×H01(Ω). Moreover, we study the long-time behavior of the solutions. It is proved that, under the natural assumptions, these equations possess the compact attractor 𝒜 which is bounded in H2(Ω)×H2(Ω), where the nonlinear term f satisfies a critical exponential growth condition.

Large-time dynamics in complex networks of reaction–diffusion systems applied to a panic model

2019 ◽  
Vol 84 (5) ◽  
pp. 974-1000
Author(s):  
Guillaume Cantin ◽  
M A Aziz-Alaoui ◽  
Nathalie Verdière
Keyword(s):  
Large Time ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Energy Estimates ◽  
Invariant Region ◽  
Exponential Attractors ◽  
Reaction Diffusion Systems ◽  
Time Dynamics ◽  
Diffusion Systems ◽  
Theoretical Results

Abstract This paper is devoted to the analysis of the asymptotic behaviour of a complex network of reaction–diffusion systems for a geographical model, which was proposed recently, in order to better understand behavioural reactions of individuals facing a catastrophic event. After stating sufficient conditions for the problem to admit a positively invariant region, we establish energy estimates and prove the existence of a family of exponential attractors. We explore the influence of the size of the network on the nature of those attractors, in correspondence with the geographical background. Numerical simulations illustrate our theoretical results and show the various possible dynamics of the problem.

scholarly journals Blow-Up of Solutions with High Energies of a Coupled System of Hyperbolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Jorge A. Esquivel-Avila
Keyword(s):  
Initial Data ◽  
Blow Up ◽  
Hyperbolic Equations ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Coupled System ◽  
Initial Energy ◽  
Evolution System ◽  
High Energies ◽  
Second Order In Time

We consider an abstract coupled evolution system of second order in time. For any positive value of the initial energy, in particular for high energies, we give sufficient conditions on the initial data to conclude nonexistence of global solutions. We compare our results with those in the literature and show how we improve them.

scholarly journals ON UNIQUENESS OF MILD SOLUTIONS FOR DISSIPATIVE STOCHASTIC EVOLUTION EQUATIONS

2010 ◽  
Vol 13 (03) ◽  
pp. 363-376 ◽  
Author(s):  
CARLO MARINELLI ◽  
MICHAEL RÖCKNER
Keyword(s):  
Evolution Equations ◽  
Broad Class ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Strong Solutions ◽  
Fundamental Issue ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Monotonicity Assumption ◽  
Semigroup Approach

In the semigroup approach to stochastic evolution equations, the fundamental issue of uniqueness of mild solutions is often "reduced" to the much easier problem of proving uniqueness for strong solutions. This reduction is usually carried out in a formal way, without really justifying why and how one can do that. We provide sufficient conditions for uniqueness of mild solutions to a broad class of semilinear stochastic evolution equations with coefficients satisfying a monotonicity assumption.

scholarly journals Solving Some Linear and Nonlinear PDEs Using Laplace Adomian Decomposition Method

2018 ◽  
Vol 1 (2) ◽  
pp. 9-31
Author(s):  
Attaullah
Keyword(s):  
Decomposition Method ◽  
Evolution Equations ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Reaction Diffusion ◽  
Nonlinear Evolution Equations ◽  
Reaction Diffusion Equations ◽  
Nonlinear Pdes ◽  
Adomian Decomposition ◽  
Linear And Nonlinear

In this paper, Laplace Adomian decomposition method (LADM) is applied to solve linear and nonlinear partial differential equations (PDEs). With the help of proposed method, we handle the approximated analytical solutions to some interesting classes of PDEs including nonlinear evolution equations, Cauchy reaction-diffusion equations and the Klien-Gordon equations.

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