Blow-Up Solution of Modified-Logistic-Diffusion Equation

Modified-Logistic-Diffusion Equation ut = Duxx + u|1 − u| with Neumann boundary condition has a global solution, if the given initial condition ψ satisfies ψ(x) ≤ 1, for all x ∈ [0, 1]. Other initial conditions can lead to another type of solutions; i.e., an initial condition that satifies ∫1 0 ψ(x)dx > 1 will cause the solution to blow up in a finite time. Another initial condition will result in another kind of solution, which depends on the diffusion coefficient D. In this paper, we obtained the lower bound of D, so that the solution of Modified-Logistic-Diffusion Equation with a given initial condition will have a global solution.


Introduction
Logistic-Difussion Equation was first introduced by Fisher at 1937.It describes the growth of mutant gene on long habitat  [1].The equation is given by with (, ) being the number of mutant genes at time  > 0, at location  ∈  with initial condition  ∈  2 ().Lots of researches had been done on this type of partial differential equation (PDE), including how to approximate the solution with various methods [2][3][4][5].This PDE has been applied on different disciplines of knowledge, such as in chemical reaction and economic growth [6][7][8][9].This research will focus on the behavior of solution if the nonlinear factor in (1) is modified with absolute function; i.e., the rate of growth is always positive, PDE (2) explains that the rate of growth is slowing down around  = 1 and is always positive.The state  = 1 for all  is the steady state of (1) and (2).In (1),  = 1 is a stable equilibrium, while in (2) it is a semi-stable equilibrium.This research is conducted on a bounded domain [0, 1] ⊂ R with Neumann boundary condition.
(3) Equation ( 3) has a global solution if 0 ≤  0 ≤ 1. Completely, the solution of (3) for initial condition  0 > 0 is given by The behavior of the solution for the three types of initial conditions can be seen in Figure 1.Equation (4) shows that the interval of the solution of (3) for  0 > 1 is [0,  * ) with On the contrary, (3) has a global solution for  0 < 1.
According to the solution (4), we can see that, for given initial condition  with 0 ≤ () < 1 or 1 < () for all  ∈ , the diffusion factor does not play many roles in the behavior of solution. 2 International Journal of Differential Equations Now, we will discuss the behavior of solution of (2) for other initial conditions , that is, The effect of the diffusion coefficient  will be observed to guarantee the existence of global solution.
Definition 1.Let  be the initial condition for (2). is called -initial condition if  satisfies (6).
The function () = |1 − | ≥ 0 for all  ≥ 0. Thus, in the absence of diffusion,   ≥ 0 for all  ≥ 0. This means that the value of (, ) will grow for every  ∈ [0, 1].If there exist  ∈ [0, 1] such that () > 1, then the solutions interval of (2) is [0,  * ] with On the value of initial condition  with ∫ 1 0 () > 1, the diffusion coefficient is not affected much.It is because the diffusion coefficient is just distributing the concentration from high to low without addition or substraction of the total of concentration on interval [0, 1].As a result, there is any  ∈ [0, 1] such that (, ) > 1 for all  > 0. Furthermore the solution of (2) will blow up at finite time.
Notice the following usual diffusion equation: If  1 and  2 are the solutions of ( 2) and ( 9), respectively, with the same initial condition , then  1 (, ) ≥  2 (, ) for every (, ) ∈ [0, 1] × [0, ∞).Hence, the diffusion time of ( 9) with respect to  is last or equal to diffusion time of (2) with respect to .Let   be the diffusion time of ( 9) with respect to ; then   is a lower bound for diffusion time of ( 2) with respect to .In [11], the solution of ( 9) for initial condition  is with   = ∫ 1 0 () cos().For certain initial condition, the   value can be obtained easily, while for other initial conditions, we could only determine the lower bound of   .
The diffusion time for this class of function is given by this theorem.

Reaction Time
The function () = |1 − | ≥ 0 for all  ≥ 0. It means the reaction factor is causing the increase of concentration for all time.If  could contribute to the increasing of the concentration such that (, ) > 1 for time  < ∞, the solution will blow up.
For initial condition that has minimum value at [0, 1], the reaction time is given by Theorem 7 below.

Theorem 7.
Let  be the -initial condition of ( 2) and have minimum value on [0, 1] with  = min  (); then is an upper bound for   ().
Let min  () = ; then Furthermore, let If () = , then  is an upper bound of   ().The time  that satisfies this condition is Since () > 1 for every  ∈ Ω 2 , then This shows that  > 0 for some -initial condition .
The upper bound of reaction time for initial condition of linear and sinusoidal function class is obtained as follows.
Theorem 8. Let the -initial condition be given by () =  + ,  > 0 with  + /2 < 1,  ̸ =  and  ̸ Proof.For -initial condition () =  + ,  > 0, that is, 0 <  < 1 and 1 −  <  < 2 − 2, we obtain Therefore, by Theorem 7 for -initial condition () =  + ,  > 0 we obtain the upper bound of reaction time   () as In the same way, we obtain the upper bound of reaction time for -initial condition () =  −  as From the first result in Theorem 8 we obtain the upper bound of the reaction time for some family of linear functions with positive gradient, while the second result is for the negative gradient.
Proof.Let () =  −  cos() satisfying condition (6); that is, We have  ( This result will be used to show the relationship between diffusion coefficient  and the behavior of the solution of (2) for a given -initial condition .

The Lower Bound of 𝐷 for Global Solution
The solution of (2) with -initial condition  will blow up for  * < ∞ if   () <   ().This is because the reaction term is growing much faster to supply the volume than the suppressing of diffusion term.From Theorems 4, 5, 8, and 9, we obtained the lower bound for diffusion coefficient  such that the global solution for linear and sinusoidal function family of initial condition exists.

Discussion
The diffusion-reaction equation with modified logistic function of reaction term and Neumann's boundary condition at [0, 1] can have a global or blow-up solution.If the initial conditions given are -initial conditions, then the diffusion term plays an important role in determining whether the system will have a global solution or a blow-up solution.
In this study, we obtain the lower bound of the diffusion coefficient  such that the system has a global solution.The objective for further investigation is determining the limiting value of the diffusion coefficient  such that, for -initial condition, the solution has a global soluton or a blow-up solution.