Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions

In this paper, we consider a particular type of nonlinear McKend-rick-von Foerster equation with a diffusion term and Robin boundary condition. We prove the existence of a global solution to this equation. The steady state solutions to the equations that we consider have a very important role to play in the study of long time behavior of the solution. Therefore we address the issues pertaining to the existence of solution to the corresponding state equation. Furthermore, we establish that the solution of McKendrick-von Foerster equation with diffusion converges pointwise to the solution of its steady state equations as time tends to infinity.

In [5,6,7], Gladkov et al have studied equation (1) with a particular type of nonlinear nonlocal boundary condition. Furthermore, the authors of those papers have proved existence, uniqueness and blow-up phenomena of the solutions in finite time. In [3], Day has considered a linear thermoelasticity model similar to (1) and established its dynamic theories. Pao has developed the method of subsolution and supersolution to prove an existence and uniqueness result for a reaction diffusion equation with nonlocal boundary conditions. Moreover, the asymptotic behaviour and dynamical properties for some specific type of nonlinear differential equations are also studied using this method (see [12]- [15]). In [10], Iannelli et al have considered a structured population model that is hyperbolic in nature and they have used an appropriate approximating regularized parabolic problem whose solution converges to the solution of original problem. The authors of [10] have used the semigroup theory techniques to study the regularized parabolic problem.
In [1,8,9], the authors have considered the renewal equation with diffusion with a † = +∞ and studied the existence, uniqueness of weak solution. Moreover, the long time behavior has been studied in some particular cases.
The paper is organized as follows. An existence result for equation (1) has been discussed in Section 2. We demonstrate that there exists a unique solution to (1) which is bounded above by a supersolution and bounded below by a subsolution. We define the steady state equations corresponding to (1) and show the existence of its solution in Section 3. In Section 4, we study the long time behavior of the solution of (1). In particular, we prove that the solution of (1) converges pointwise to the solution of the corresponding steady state equations. In Section 5, we present some numerical results to justify the convergence results that are discussed in the article.
2. Existence of solution. In this section, we prove an existence result to equation (1) using the method of subsolution and supersolution. Strong maximum principle for the parabolic equations plays a crucial rule in order to prove the existence of solution of (1). To this end, we first define supersolution and subsolution.
Similarly, one can define a subsolution of (1) as a function which satisfies all the reversed inequalities present in (2).
For M > max{M 0 , u 0 L ∞ }, thanks to the nonnegativity of g, the constant functionũ ≡ M is a supersolution andû ≡ 0 is a subsolution of (1). This ensures the existence of subsolutions and supersolutions of (1).
We consider the following sequence of problems for n ≥ 1, If u (0) =ũ then we denote the sequence of solutions to the problems in (3) where u andū are the pointwise limits of the sequences {u (n) } and {ū (n) }, respectively.
Again, from the maximum principle we getū (1) ≥ u (1) . So far, we have proved Now, using the principle of induction and the arguments used in the proof so far, one can easily prove the required inequalitŷ Therefore the pointwise limits of {u (n) } and {ū (n) }, say u, andū, respectively, exist and satisfy the relation (4). This completes a proof of the promised result.
In the next theorem we prove that the limits u andū which are obtained in Theorem 2.1 are indeed solutions to (1). The main ingredient in the proof is the representation of classical solution to the linear parabolic equations given in the books of Friedman and Pao (see [4,11]).

Theorem 2.2. (Existence) Consider the notation that is introduced in Theorem 2.1. Then both u andū are solutions to equation (1).
Proof. First, observe that the solution u (n) to equation (3) is given by (see [4] for more details) Here Γ(t, x; τ, ξ) is the fundamental solution to (3) and the corresponding density function φ (n) , for x = 0 and a † , is given by Define φ(t, x), the density function for x = 0 and a † by where We show that u has the integral representation given by Denote Since u (n) → u pointwise, φ (n) → φ pointwise. From this, it is straight forward to conclude that u satisfies (6)- (8). Furthermore, it is known that a solution of (1) is given implicitly by relations (6)-(8) (see [4,11]). Therefore we conclude that u is a solution of (1). By repeating the same argument with u replaced byū we can deduce thatū is also a solution.
For, consider (û, u * ) as a pair of subsolution and supersolution. From Theorem 2.1, we getû ≤ u ≤ u * . On the other hand, considering (u * ,ũ) as a pair of subsolution and supersolution we readily obtain u * ≤ū ≤ũ. This completes the proof of Remark 2.2.

Theorem 2.3. Consider the notation that is introduced in Theorem
If u * is a solution to (1) such thatû ≤ u * ≤ũ, then u = u * =ū.
Proof. From Remark 2.2, we immediately get that u ≤ u * ≤ū. In order to prove the required result it is enough to proveū = u. To this end, we introduce w(t, Moreover, thanks to Theorem 2.2, w(t, x) satisfies the following equation, for t ∈ [0, T ], If w attains its positive maximum, then from the maximum principle, it is attained on the boundary, say at (t 0 , 0). Since w(0, x) = 0, for all x ∈ D, therefore we have t 0 > 0. From the mean value theorem there exists ξ ≥ 0 such that Thus, we have On the other hand, we know that from the strong maximum principle w x (t 0 , 0) < 0. This contradicts assumption (9). Similarly, we can get a contradiction if we assume that the maximum is attained at (t 1 , a † ) for some t 1 > 0. Therefore we have w ≡ 0. This shows that u ≡ū. 3. Steady state equation. In this section, we define the steady state equations corresponding to (1). Furthermore, we prove an existence result for these steady state equations. In order to do that, we use the method of sub and supersolutions which has been discussed in detail in Section 2. We say that U is a steady state corresponding to (1) if it solves the following second order ordinary differential equation, Furthermore, we call (10) as the equation of steady state corresponding to (1).
is said to be a supersolution of (10) if it satisfies the following inequalities, Similarly, we can define a subsolution of (10) as a function which satisfies the reversed inequalities in (11). As in Section 2, we construct the following sequence of ordinary differential equations, In what follows, unless specified otherwise, we always useŨ andÛ to denote a nonnegative supersolution and nonnegative subsolution to equation (10), respectively. If U (0) =Ũ then denote the sequence of solutions to the problems in (12) by {Ū (n) }. If U (0) =Û then the sequence of solutions to the problems in (12) is denoted by {U (n) }. With this notation we have the following result similar to Theorems 2.1 and 2.3. (10), respectively, such thatÛ ≤Ũ . Then the sequences {Ū (n) } and {U (n) } possess the following monotonicity property:

Proposition 3.1. (Existence) LetŨ andÛ be a nonnegative supersolution and a nonnegative subsolution to
where U andŪ are the pointwise limits of the sequences {U (n) } and {Ū (n) }, respectively. Moreover U andŪ are solutions to (10).
The inequalities in (13) can be obtained from the same arguments given in the proof of Theorem 2.1. Using the arguments given in [11,16] one can show that U andŪ are solutions to (10). Therefore we omit the proof. Now, we have the following remark similar to Remark 2.2.
We conclude this section by stating a uniqueness result for the steady state equation (10), which says that there is a unique solution for (10) between any pair of subsolutionŪ and subsolutionÛ withÛ ≤Ū . Details are given in the following result.
One can prove this result by following similar lines of the proof of Theorem 2.3.

Long time behavior.
In this section, we prove that the solution of (1) converges to the solution of the steady state problem (10) with time. First we consider the case when g(0) = 0. In this case, it is easy to observe that U ≡ 0 is the solution to (10). Furthermore, we prove that the solution of (1) converges to the trivial steady state with an exponential rate. Proposition 4.1. Assume that inequality (9) holds and g(0) = 0. Then ∃C > 0 such that where α is given in (H3).
Proof. Set C := u 0 L ∞ . It is easy to verify that the functionũ := Ce −αt is a supersolution and the constant functionû = 0 is a subsolution to (1). From Theorems 2.1-2.3, there exists a unique solution to (1), say u, that lies betweenû andũ. Therefore, 0 ≤ u(t, x) ≤ Ce −αt , for all x ∈ D, t > 0. This completes the proof. Now, we turn our attention to the case when g(0) = 0. In this case, the steady state cannot be trivial. We show that the solution to (1) converges pointwise to the nontrivial steady state.
Suppose not, then w(t, x) > 0 for some (t, x) ∈ (0, ∞) × D and w attains its positive maximum on the boundary, say at (t 0 , 0). Since w(0, x) ≤ 0, we must have t 0 > 0. Now, there exists ξ > 0 such that This readily implies By the same argument that is given in the proof of Theorem 2.3, we conclude that w ≤ 0. Thus, we have Similarly, we can get the other inequality Step 2. We begin with the observation thatŨ andÛ are supersolution and subsolution to (1), respectively. Hence, Theorems 2.1 and 2.2 ensure thatρ and ρ satisfy the relationÛ (18) On the other hand, for δt > 0, setw(t, x) =ρ(t + δt, x) −ρ(t, x). Then we get Using the arguments that are given in Step 1, we obtainw(t, x) ≤ 0. Since δt is arbitrary,ρ decreases with time. Similarly, one can show that ρ increases with time. Hence the pointwise limits By regularity arguments (see [11]), one can show that U * ,Ū * are the solutions to equation (10). From the uniqueness result in Proposition 3.  (1). We broadly divide these examples into two categories. In the first category, we provide some examples that satisfy the main hypothesis of Section 4 i.e., inequality (9). In the next category, we give another set of examples which do not satisfy (9). We use the finite difference method to discretize the the problem. Let ∆x and ∆t denote the discretization step sizes for space and time, respectively. In all these examples we have taken a † = 20, B(a † , x) = 0, ∆x = 0.05 and ∆t = 0.0012 in our numerical computations.
Example 5.1. Here we assume that d, B(0, x), g and u 0 are given by It is easy to verify that the functions B and g satisfy (9). In Figure 1 (left), the graphs of the solution u to equation (1) at t = 12 and solution U to (10) are presented. From this figure, we readily observe at t = 12, the solution to (1) is in well agreement with the corresponding steady state. The graph of the absolute difference between u at t = 12 and U , i.e., |u(12, x) − U (x)| is plotted in Figure  1 (right). From both the graphs we find that u indeed converges to the nontrivial steady state U with time.
Example 5.2. In this example, we consider the case when We notice that these functions also satisfy (9). We have given the graphs of the solution of (1) at t = 12 and solution of (10) in Figure 2(left). In Figure 2(right), we have depicted the absolute value of the difference between u(12, .) and U (.). We notice that u converges to U as time increases.
Example 5.3. In this example, we assume The asymptotic behavior of the solution u to (1) when g(0) = 0 is discussed in Proposition 4.1, but the functions g and B in this example do not satisfy assumption (9) which is vital in it's proof. In this example, we want to investigate whether u converges pointwise (if not uniformly) to a steady state. To this end, first we have found a nonzero numerical solution U to (10). In Figure 3 (left), we have depicted the graphs of solution u to (1) at t = 12 and a nontrivial solution U to (10). In Figure 3 (right), we have shown the corresponding absolute error. We have computed u(t, x) at x = 2, 3 and 4 to see the asymptotic behavior of u at these points. Graphs of u(t, x) at x = 2, 3 and 4 are presented in Figure 4 (left). From this figure, we readily observe that u(., x) eventually tends to a constant which  depends only on x. In Figure 4 (right) we have plotted |u(12, x) − U (x)| for x = 2, 3 and 4 to confirm that u indeed converges to the nonzero steady state U pointwise.
In this example also we notice that inequality (9) does not hold for the functions B and g. As in the previous example, we have computed a nontrivial steady state U (a numerical solution to (10)). In Figure 5 (left), we have presented the solution to (1) at t = 12 and a nontrivial solution to (10). In Figure 5 (right) we have shown the absolute difference between u(12, .) and U . In Figure 6 (left), graphs of u(t, 1), u(t, 2) and u(t, 4) are depicted. In this example also we observe that for every fixed value of x, the solution is eventually constant. Furthermore, we have computed |u(12, x) − U (x)| for x = 1, 2 and 4 to establish that u(t, x) converges to U (x) at x = 1, 2 and 4, see Figure 6 (right). These numerical experiments suggest that the u converges pointwise to a corresponding nonzero steady state with time.    6. Conclusions. We have proved the existence and uniqueness of a bounded classical solution to the nonlocal McKendric-von Foerster equation with diffusion. Moreover, an existence and uniqueness result has been established for the corresponding steady state equation also. If the nonlinear function g in the boundary condition is such that g(0) = 0, then the solution to (1) converges uniformly to the trivial steady state with an exponential rate in time. On the other hand, we have established that if the initial data is trapped in between a pair of sub and supersolutions of the steady state problem then the solution to (1) converges pointwise to the nontrivial solution to (10) for large time. We have also performed numerical simulations to show that the solution u converges to the nontrivial solution U to the steady state equation for large time. We have provided some examples where the crucial assumption in Section 4, i.e., inequality (9), does not hold but still the numerical experiments show that u converges to U as time increases. In view of these empirical results,  we believe that the assumption (9) can be weakened in order to get the pointwise convergence of u(t, .) to a nontrivial steady state U (.) as t → ∞.