scholarly journals A nonlocal parabolic system with application to a thermoelastic problem

1999 ◽  
Vol 22 (4) ◽  
pp. 855-867
Author(s):  
Y. Lin ◽  
R. J. Tait
Keyword(s):  
Representation Theory ◽  
Parabolic System ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
System Modeling ◽  
Parabolic Systems ◽  
Thermoelastic Problem ◽  
Potential Representation

A system modeling the thermoelastic bards contacts is studied. The problem is first transformed into an equivalent nonlocal parabolic systems using a transformation, and then the existence and uniqueness of the solutions are demonstrated via the theoretical potential representation theory of the parabolic equations. Finally some realistic situations in the applications are discussed using the results obtained in this paper.

scholarly journals Global existence and asymptotic behaviour of solutions for a hyperbolic-parabolic model of chemotaxis on network

2021 ◽  
Author(s):  
Yafeng Li ◽  
Chunlai Mu ◽  
Xin Qiao
Keyword(s):  
Global Existence ◽  
Asymptotic Behaviour ◽  
Parabolic System ◽  
Existence And Uniqueness ◽  
System Modeling ◽  
Global Solutions ◽  
Energy Estimates ◽  
Parabolic Model ◽  
Biological Phenomena ◽  
Acyclic Network

In this paper, we discuss a hyperbolic-parabolic system modeling biological phenomena evolving on a network. The global existence of the is obtained by using energy estimates with suitable the transmission conditions at interior. Moreover, for the case of acyclic network, the existence and uniqueness of stationary solution to the system is proposed and it is proved that these ones are asymptotic profiles for a class of global solutions

OPTIMAL CONTROL OF HARVESTING IN A PARABOLIC SYSTEM MODELING TWO SUBPOPULATIONS

2001 ◽  
Vol 11 (07) ◽  
pp. 1129-1141 ◽  
Author(s):  
SUZANNE M. LENHART ◽  
J. A. MONTERO
Keyword(s):  
Optimal Control ◽  
Differential System ◽  
Parabolic System ◽  
Existence And Uniqueness ◽  
System Modeling ◽  
Optimal Harvesting ◽  
Time Interval ◽  
Biological Parameters ◽  
Control Pair ◽  
Partial Differential System

An optimal harvesting problem for a parabolic partial differential system modeling two subpopulations of the same species is investigated. The two subpopulations are competing for resources. Under conditions on the smallness of the time interval and certain biological parameters, existence and uniqueness of an optimal control pair are established.

Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

2002 ◽  
Vol 7 (1) ◽  
pp. 93-104 ◽  
Author(s):  
Mifodijus Sapagovas
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Numerical Algorithms ◽  
Uniqueness Of The Solution

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

Stability in 𝐿𝑞-norm for inverse source parabolic problems

2020 ◽  
Vol 28 (6) ◽  
pp. 797-814
Author(s):  
Elena-Alexandra Melnig
Keyword(s):  
Parabolic Equations ◽  
Main Tool ◽  
Parabolic Systems ◽  
Carleman Estimates ◽  
Parabolic Problems ◽  
Lebesgue Spaces ◽  
First Order ◽  
Lipschitz Estimates ◽  
Systems Of Parabolic Equations

AbstractWe consider systems of parabolic equations coupled in zero and first order terms. We establish Lipschitz estimates in {L^{q}}-norms, {2\leq q\leq\infty}, for the source in terms of the solution in a subdomain. The main tool is a family of appropriate Carleman estimates with general weights, in Lebesgue spaces, for nonhomogeneous parabolic systems.

scholarly journals Lorentz spaces in action on pressureless systems arising from models of collective behavior

2021 ◽  
Author(s):  
Raphaël Danchin ◽  
Piotr Bogusław Mucha ◽  
Patrick Tolksdorf
Keyword(s):  
Initial Data ◽  
Collective Behavior ◽  
Existence And Uniqueness ◽  
Maximal Regularity ◽  
Parabolic Systems ◽  
Lorentz Spaces ◽  
Regularity Estimate ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Time Existence

AbstractWe are concerned with global-in-time existence and uniqueness results for models of pressureless gases that come up in the description of phenomena in astrophysics or collective behavior. The initial data are rough: in particular, the density is only bounded. Our results are based on interpolation and parabolic maximal regularity, where Lorentz spaces play a key role. We establish a novel maximal regularity estimate for parabolic systems in $$L_{q,r}(0,T;L_p(\Omega ))$$ L q , r ( 0 , T ; L p ( Ω ) ) spaces.

scholarly journals EXISTENCE, UNIQUENESS AND APPROXIMATION OF A DOUBLY-DEGENERATE NONLINEAR PARABOLIC SYSTEM MODELLING BACTERIAL EVOLUTION

2007 ◽  
Vol 17 (07) ◽  
pp. 1095-1127 ◽  
Author(s):  
JOHN W. BARRETT ◽  
KLAUS DECKELNICK
Keyword(s):  
Parabolic System ◽  
Existence And Uniqueness ◽  
Finite Element Approximation ◽  
System Modelling ◽  
Element Approximation ◽  
Nonlinear Parabolic System ◽  
Spatiotemporal Evolution ◽  
Nonlinear Parabolic ◽  
Flux Boundary Conditions ◽  
Bacterium Species

We consider the following nonlinear parabolic system [Formula: see text] subject to no flux boundary conditions, and non-negative initial data u0 and v0 on u and v. Here we assume that c > 0, θ ≥ 0 and that [Formula: see text] is increasing with f(0) = 0. The system is possibly doubly-degenerate in that [Formula: see text] is only non-negative, and ψ ∈ C1([0,∞)) ∩ C2((0,∞)) is convex, strictly increasing with ψ(0) = 0 and possibly ψ'(0) = 0. The above models the spatiotemporal evolution of a bacterium species on a thin film of nutrient, where u is the nutrient concentration and v is the bacterial cell density. Under some further mild technical assumptions on b and ψ, we prove the existence and uniqueness of a weak solution to the above system. Moreover, we prove error bounds for a fully practical finite element approximation of this system. All of our results apply to the choices b(r) ≔ rq and ψ(r) ≔ rp with q ≥ 2 and p ≥ 1, for example.

scholarly journals On a nonlinear parabolic system-modeling chemical reactions in rivers

2005 ◽  
Vol 4 (4) ◽  
pp. 889-899 ◽  
Author(s):  
Wenxiong Chen ◽  
◽  
Congming Li ◽  
Eric S. Wright ◽  
◽  
...  
Keyword(s):  
Chemical Reactions ◽  
Parabolic System ◽  
System Modeling ◽  
Nonlinear Parabolic System ◽  
Nonlinear Parabolic
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