scholarly journals Global solutions and asymptotic behavior for a parabolic degenerate coupled system arising from biology

2010 ◽  
Vol 72 (1) ◽  
pp. 77-98 ◽  
Author(s):  
Gabriela Liţcanu ◽  
Cristian Morales-Rodrigo
Keyword(s):  
Asymptotic Behavior ◽  
Global Solutions ◽  
Coupled System

scholarly journals ASYMPTOTIC BEHAVIOR OF GLOBAL SOLUTIONS TO A MODEL OF CELL INVASION

2010 ◽  
Vol 20 (09) ◽  
pp. 1721-1758 ◽  
Author(s):  
GABRIELA LIŢCANU ◽  
CRISTIAN MORALES-RODRIGO
Keyword(s):  
Asymptotic Behavior ◽  
Cell Invasion ◽  
A Priori Estimates ◽  
A Priori ◽  
Global Solutions ◽  
Coupled System ◽  
Regularity Properties ◽  
Priori Estimates ◽  
Degradative Enzymes ◽  
Key Events

In this paper, we analyze a mathematical model focusing on key events of the cell invasion process. The three equations of the corresponding coupled system describe the behavior of the invasive cells, the extracellular matrix and the degradative enzymes. We employ a fix-point method and a priori estimates to prove local and global existence, uniqueness and regularity properties of the solutions. Our approach enable us to find estimates that are uniform in time. This is essential in proving, in the last part of the paper, new results that establish the asymptotic behavior of the solutions.

CONVERGENCE TO EQUILIBRIUM FOR A PARABOLIC–HYPERBOLIC PHASE-FIELD SYSTEM WITH NEUMANN BOUNDARY CONDITIONS

2007 ◽  
Vol 17 (01) ◽  
pp. 125-153 ◽  
Author(s):  
HAO WU ◽  
MAURIZIO GRASSELLI ◽  
SONGMU ZHENG
Keyword(s):  
Boundary Conditions ◽  
Type Inequality ◽  
Global Solutions ◽  
Neumann Boundary ◽  
Coupled System ◽  
Neumann Boundary Conditions ◽  
Convergence To Equilibrium ◽  
Relative Temperature ◽  
Real Analytic Functions ◽  
Phase Field System

This paper is concerned with the asymptotic behavior of global solutions to a parabolic–hyperbolic coupled system which describes the evolution of the relative temperature θ and the order parameter χ in a material subject to phase transitions. For the system with homogeneous Neumann boundary conditions for both ¸ and χ, under the assumption that the nonlinearities λ and ϕ are real analytic functions, we prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable Łojasiewicz–Simon type inequality.

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.

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