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

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.


Introduction and Preliminaries
e Fisher equation arises in abundance in many fields, including chemistry, biology, and the environment [1][2][3][4][5][6][7][8][9][10][11][12][13][14][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 (x, t) denotes the position and the time, respectively, as Ψ(x, t) is the population density, D is the propagation constant, and K 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][18][19][20][21]. e 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 Ψ 0 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: Ψ ∈ W(0, T) and verify

Local Existence
In this section, we state and prove the local existence result of our problem. (5), satisfying (6) and (7).
Proof. To reach our goal, we shall use the so-called Faedo-Galerkin method.

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Step 1. solution of the approximate problem: Since . For all m, the approximate solution Ψ m of (6) given by which satisfies Properties of projection operators imply From Systems (9) and (10) write Since the functions h i m i�1 are linearly independent, this means that the matrix with entries (h j , h i ) is nonsingular, to use the inverse of this matrix to reduce (13) and (14) to the following system: for i � 1, 2, . . . , m, where α ij , β ijk , ξ i ∈ R, and they depend on h i m i�1 . Systems (15) and (16) have a solution defined on a maximal right interval [t 0 , τ (m) ]. Or equivalently, systems (9) and (10) have a solution Ψ m (t) defined on a maximal interval (see, e.g., [22]).
Step 2. a priori estimates for Ψ m (t).
For t ∈ [t 0 , τ (m) ], multiplying (9) by f (m) i (t), i � 1, . . . , m, and adding these equations up, we obtain Hence, by using Young's inequality, we obtain 2 Journal of Mathematics , using the interpolation between L 2 (Ω) and H 1 0 (Ω), Young's and Poincare's inequalities, we obtain By adding up (19) and then applying the resulting estimate, we find Setting where Set z(t) as a maximal solution of the following equation: with (24) and (25) and t 1 are independent of m ).
Finally, it rests to show that Ψ verifies the initial condition Ψ(t 0 ) � b 0 .

Global Existence
In this section, we will show the global existence and uniqueness. By the result ( eorems 1-3), under suitable hypothesis on Ψ 0 (x), we deduce that there exists a solution Ψ for our problem (5), satisfying (6) and (7)  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.
is is end of the proof.

Conclusion
e 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.
is 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
e authors declare that there are no conflicts of interest regarding the publication of this manuscript.