Global Existence of Solution for the Fisher Equation via Faedo–Galerkin’s Method

Abstract

In this study, we consider the Fisher equation in bounded domains. By Faedo–Galerkin’s method and with a homogeneous Dirichlet conditions, the existence of a global solution is proved.

1. Introduction and Preliminaries

The Fisher equation arises in abundance in many fields, including chemistry, biology, and the environment [1–15]. It also has a common name, the Fisher–Kolmogorov–Petrovsky–Piskunov equation (KPP), where it describes the following equation by the process of population progress in space (F.KPP equation) [16]:where denotes the position and the time, respectively, as is the population density, is the propagation constant, and is the maximum density, with the homogeneous boundary Dirichlet conditions:

Also, this equation is closely related to biology, applied mathematics, parasites, bacteria, and genes. For more detail, we refer the reader to the following research papers, see, for example, [17–21].

The simplest version of the FK equation is

Based on the previous work, we will shed light on problem (4), which is a multidimensional model of Fisher’s equation, under a Dirichlet boundary condition:

Our paper is divided into several sections. In Section 2, the existence of local solution is proved. In Section 3, the maximum principle under suitable condition on is established. In Section 4, the existence and uniqueness of solution are proved. Finally, we give some concluding remarks in Section 5.

Firstly, we define the solution of (5) as a solution of the following weak formulation: and verify

2. Local Existence

In this section, we state and prove the local existence result of our problem.

Theorem 1. Suppose that . Then, , and there exists a weak solution of problem (5), satisfying (6) and (7).

Proof. To reach our goal, we shall use the so-called Faedo–Galerkin method.

Step 1. solution of the approximate problem:
Since is separable, is a basis for . For all , the approximate solution of (6) given bywhich satisfieswhere is a projection of onto the span of .
Properties of projection operators implyFrom which is dense in and which is a basis for , we obtainSystems (9) and (10) writeSince the functions are linearly independent, this means that the matrix with entries is nonsingular, to use the inverse of this matrix to reduce (13) and (14) to the following system:for , where , , , and they depend on . Systems (15) and (16) have a solution defined on a maximal right interval . Or equivalently, systems (9) and (10) have a solution defined on a maximal interval (see, e.g., [22]).

Step 2. a priori estimates for .
For , multiplying (9) by , and adding these equations up, we obtainHence, by using Young’s inequality, we obtainFurthermore, from , using the interpolation between and , Young’s and Poincare’s inequalities, we obtainBy adding up (19) and then applying the resulting estimate, we findIntegrating (21) over , where , we obtainSetting , we obtainwhere , and from it, and is increasing in , .
Set as a maximal solution of the following equation:withThere exist with (we have (24) and (25) and are independent of ).
Inequality (11) implies . Then, by setting , we obtainSince is the maximal interval of existence for (23) and (26), we deduce that , and the existence interval is . Furthermore, by (26), we findHence, we get such thatBy (22), (11), and (28), we obtain

Step 3. passage to limits.
A priori estimates (28) and (29) allow us to draw a subsequence of such thatfor some . Hence, we obtainLet with .
Multiplying (9) by and integrating by parts, we obtainAccording to (30)–(32), as , we obtainBy using Hölder’s inequality and , we obtainAccording (29), (32), and (38), we get, as ,Relations (34), (35), and (37) allow us to pass to the limits in (9) to find. We use the linearity in of (40) and as total in , we obtainSince, (41) holds for any such that satisfiesUsing [23], we get in the sense of distributions (in time).
Finally, it rests to show that verifies the initial condition .
Multiplying (42) by and integrating by parts, we findA comparison of (41) and (43) yields , and we pick with .
, i.e., a.e. . This is end of the proof of Theorem 1.

3. Maximum Principles

Consider a solution of (5) satisfying (6) and (7), , in Section 5. Under suitable hypothesis on , we prove that the solution verifies . This result proves the global existence in Section 4.

Theorem 2. Suppose that and a.e in . If is a solution of problem (5) satisfying (6) and (7), where and , then a.e in .

Proof. For a.e , we let in (6); we haveBy (21), we haveIntegration over to giveswhere . We haveand deduce that .
Setting , , and , from (46), we findFurthermore, by Gronwall’s inequality, we get , i.e., and a.e in .

Theorem 3. Suppose that the suppositions of Theorem 2 and a.e in hold. Then, is the local solution of (6) and (7) which satisfies a.e in .

Proof. For a.e. , we let in (6) to findEquation (49) can be written asThen, a.e in .

4. Global Existence

In this section, we will show the global existence and uniqueness.

By the result (Theorems 1–3), under suitable hypothesis on , we deduce that there exists a solution for our problem (5), satisfying (6) and (7) on some interval and a.e in . Hence, we expect a global solution to exist on any interval .

On the contrary, the first global existence theorem below is a consequence of the local existence theorem and maximum principles; for the theorem of the second global existence, it requires further work.

For this purpose, we give now the following result.

Theorem 4. Let . Suppose that a.e. . Then, such that is a solution of (6) and (7) and a.e. in .

Proof. Using Theorem 1, such that as a solution of (6) and (7) on . Applying Theorems 2 and 3, we obtainWe let . Hence, must equal . Furthermore, Theorem 1 would allow us to continue the solution beyond , and this would contradict the maximality supposition of (if and for some ; then, .
To show the solution is a unique, we assume and are two solutions of (6) and (7) and set . Hence, satisfiesBy letting in (52), we obtainso thatHence, by Gronwall’s inequality and (53), we findWe obtain . The proof is complete.

Now, we present a new a priori estimate:

Lemma 1. Suppose that a.e. . From the solution of (6) and (7),

Proof. Using Theorems 2 and 3, we get a.e in . Hence, for a.e., upon letting in (6), we obtainwhere is constant. Integrating over , we obtainFrom (6), we findfor all and a.e. .
We pick the supremum over , and we obtainfor a.e. . According (59) and (61), we obtain (57).

Theorem 5. Let . a.e. . Then, such that is a solution of (5) satisfying (6) and (7), , a.e. in , and

Proof. For any , Theorem 4 deduces the existence of so thatFurthermore, we have that a.e. . By Lemma 1, we obtainInequality (65) allows us to extract a subsequence of so thatSimilar to Step 3 in Theorem 1 and passing to the limits in (63), we obtainWe get (62) from (65). Theorems 2 and 3 imply a.e. in , and by repeating the proof of Theorem 4, the uniqueness of solution is obtained.
This is end of the proof.

5. Conclusion

The objective of this work is the study of the Fisher equation in bounded domains. By Faedo–Galerkin’s method and with a homogeneous Dirichlet conditions, we establish the existence of a global solution. This type of problem is frequently found in many fields, including chemistry, biology, and the environment.

In the next work, we will try to using the same method with same problem but by adding other conditions and damping.

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this manuscript.

Acknowledgments

The fourth author extends appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups’ program, under Grant RGP2/53/42.