Starting with the ideas delineated by the pioneering work of Perron (see [9]) and developed later in remarkable works by Coppel (see [10]), Daleckii and Krein (see [11]), Massera and Schäffer (see [12]) one of the most operational tool in the study of the asymptotic behavior of an evolution equation is represented by the input-output conditions.
Even if this assumption is not justified, the qualitative behavior of all equations remains unchanged.
We consider existence and asymptotic behavior of solutions for an equation of the formε2 Δu−V x) u+f u)=0, u>0, u∈H10whereΔis a smooth domain in RN, not necessarily bounded.
As an application, we use the result to the study of an asymptotic behavior of that equation.
Then, the method of moments is used to approximate the dominant behavior of a population balance equation (PBE) describing the evolution of the crystal volume distribution through the three leading moments.
It is not a specific place (you can't go to infinity), but rather the behavior of a number or equation if it is done forever.
In studying the behavior of the solutions of an equation, we often need to consider that its solutions are stable and attractive.
Qi and Huang [26] studied the oscillation behavior of the equation of the form ( a ( t ) [ r ( t ) D − α x ( t ) ] ′ ) ′ + p ( t ) [ r ( t ) D − α x ( t ) ] ′ − q ( t ) ∫ t ∞ ( ξ − t ) − α x d ξ = 0, t ∈ [ t 0, ∞ ), where D − α x ( t ) also denotes the Liouville right-sided fractional derivative and some sufficient conditions for the oscillation of the equation have been given.
It is well known that inequality technique is an important tool for investigating dynamical behavior of differential equation.
To evaluate the long term behavior of the equations, a simulation model was constructed by linking all of the component equations and using them to project each plot from its initial to final measurement.
