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

Authors: Ahmed Hamrouni; Abdelbaki Choucha; Asma Alharbi; Sahar Ahmed Idris
Journal: Journal of Mathematics
 (2021)

---

## 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.


---

## Body

## 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]:(1)Ψt−DΨxx=μΨK−Ψin0,T×0,K,where x,t denotes the position and the time, respectively, as Ψx,t is the population density, D is the propagation constant, and K is the maximum density, with the homogeneous boundary Dirichlet conditions:(2)Ψt,0=Ψt,K=0,t∈0,T,(3)Ψ0,x=Ψ0x,x∈0,K.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(4)Ψt−ΔΨ=μΨ1−Ψin0,T×Ω.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:(5)Ψt−ΔΨ=μΨ1−Ψin0,T×ΩΨ=0onΓ=0,T×∂ΩΨ0,x=Ψ0xinΩ.Our paper is divided into several sections. In Section2, the existence of local solution is proved. In Section 3, the maximum principle under suitable condition on Ψ0 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∈W0,T and verify(6)Ψ′t,Φ+∇Ψt,∇Φ=μΨt1−Ψt,Φ∀Φ∈H01Ω,t∈0,T,(7)Ψ0,x=Ψ0x.


## 2. Local Existence

In this section, we state and prove the local existence result of our problem.Theorem 1.
Suppose thatb0,t0∈L2Ω×0,T. Then, ∃t1=t1t0,b0∈t0,T, and there exists a weak solution Ψ∈Wt0,t1 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:
SinceH01Ω is separable, ∃hii=0∞ is a basis for H01Ω. For all m, the approximate solution Ψm of (6) given by(8)Ψm=∑i=1mfimhi,which satisfies(9)Ψm′t,hi+∇Ψmt,∇hi=μΨmt1−Ψmt,hit∈t0,t1i=1,2,…,m,(10)Ψmt0=b0m,where b0m=∑j=1mξjmhj is a L2Ω projection of b0 onto the span of h1,h2,…,hm.
Properties of projection operators imply(11)b0mL2Ω≤b0L2Ω.
FromH01Ω which is dense in L2Ω and hi which is a basis for H01Ω, we obtain(12)b0m⟶b0inL2Ωasm⟶∞.
Systems (9) and (10) write(13)∑j=1mhj,hiddtfjmt+∑j=1m∇hj,∇hifjmt=μ∑j=1mfjmthj−∑j,k=1mfjmtfjmthjhk,hi,i=1,…,m,(14)∑j=1mhj,hifjmt0=b0,hi,i=1,…,m.
Since the functionshii=1m are linearly independent, this means that the matrix with entries hj,hi is nonsingular, to use the inverse of this matrix to reduce (13) and (14) to the following system:(15)ddtfimt−μfimt+∑j=1mαijfjmt+∑j,k=1mβijkfjmtfkmt=0,i=1,…,m,(16)fimt0=ξi,i=1,…,m,for i=1,2,…,m, where αij, βijk, ξi∈ℝ, and they depend on hii=1m. Systems (15) and (16) have a solution defined on a maximal right interval t0,τm. Or equivalently, systems (9) and (10) have a solution Ψmt defined on a maximal interval (see, e.g., [22]).Step 2.
a priori estimates forΨmt.
Fort∈t0,τm, multiplying (9) by fimt,i=1,…,m, and adding these equations up, we obtain(17)Ψm′t,Ψmt+∇ΨmtL2Ω2=μΨmt1−Ψmt,Ψmt.
Hence, by using Young’s inequality, we obtain(18)ddtΨmtL2Ω2+∇ΨmtL2Ω2≤μΨmtL2Ω2+μΨmtL2Ω3.
Furthermore, fromH1/2Ω⟶L3Ω, using the interpolation between L2Ω and H01Ω, Young’s and Poincare’s inequalities, we obtain(19)μΨmtL3Ω3≤μCΨmtH1/2Ω23≤μC˜ΨmtL2Ω1/2ΨmtH01Ω1/23,≤μC˜˜ΨmtL2Ω3/2∇ΨmtL2Ω3/2,≤C2ΨmtL2Ω6+ϵ∇ΨmtL2Ω2,≤C2ΨmtL2Ω6+∇ΨmtL2Ω2.
By adding up (19) and then applying the resulting estimate, we find(20)ddtΨmtL2Ω2+∇ΨmtL2Ω2≤μΨmtL2Ω2,≤C1ΨmtL2Ω2+C2ΨmtL2Ω6+∇ΨmtL2Ω2,(21)⇒ddtΨmtL2Ω2≤C1ΨmtL2Ω2+C2ΨmtL2Ω6.
Integrating (21) over t0,t, where t∈t0,τm, we obtain(22)ΨmtL2Ω2−Ψmt0L2Ω2≤∫t0t2C1ΨmsL2Ω2+2C2ΨmsL2Ω6ds.
Settingzmt=ΨmtL2Ω2, we obtain(23)zmt≤zmt0+∫t0tfσ,zmσdσ,where ft,z=C1z+C2z3, and from it, f∈C0,T×ℝ;ℝ and f is increasing in z, ∀t.
Setzt as a maximal solution of the following equation:(24)dydt=2C1z+2C2z3,with(25)z0t=zt0L2Ω2.
There existJ=t0,t1 with t1∈t0,T (we have (24) and (25) and t1 are independent of m).
Inequality (11) implies zm0≤z¯t0. Then, by setting a=τm−t0, we obtain(26)ΨmtL2Ω2=zmt≤z¯t≤maxt∈t0,τz¯t=Ct1∀t∈t0,τm.
Sincet0,τm is the maximal interval of existence for (23) and (26), we deduce that τm=t1, and the existence interval is t0,t1. Furthermore, by (26), we find(27)supt∈t0,τΨmtL2Ω2≤Ct1∀t∈t0,t1m=1,2,…
Hence, we get∃t1∈t0,T such that(28)Ψmm=1∞belongs to a bounded set ofL∞t0,t1;L2Ω.
By (22), (11), and (28), we obtain(29)Ψmm=1∞belongs to a bounded set ofL∞t0,t1;H01Ω.Step 3.
passage to limits.
A priori estimates (28) and (29) allow us to draw a subsequence of Ψm such that(30)Ψm⇀Ψweakly star inL∞t0,t1;L2Ω(31)Ψm⇀Ψweakly inL2t0,t1;H01Ω,for some ΨinL∞t0,t1;L2Ω∩L2t0,t1;H01Ω. Hence, we obtain(32)Ψm⇀Ψstrongly inL2t0,t1;L2Ω.
Letχt∈C1t0,t1;ℝ with χt1=0.
Multiplying (9) by χt and integrating by parts, we obtain(33)−b0m,hiχt0−∫t0t1Ψmt,χ′thidt+∫t0t1∇Ψmt,χt∇hidt,=∫t0t1μΨmt1−Ψmt,χthidt.
According to (30)–(32), as m⟶∞, we obtain(34)∫t0t1Ψmt,χ′thidt⟶∫t0t1Ψt,χ′thidt,(35)∫t0t1∇Ψmt,χt∇hidt⟶∫t0t1∇Ψt,χt∇hidt,(36)∫t0t1Ψmt,χthidt⟶∫t0t1Ψt,χthidt,(37)b0m,hiχt0⟶b0,hiχt0.
By using Hölder’s inequality andH01Ω⟶L4Ω, we obtain(38)∫t0t1Ψm2t−Ψ2t,χthidt,≤∫t0t1χt∫ΩΨmt,x−Ψt,x.Ψmt,x+Ψt,x.hixdxdt,≤∫t0t1χtΨmt,x−Ψt,xL2ΩΨmt,x+Ψt,xL4Ω.hiL4Ωdt,≤CχtL∞t0,t1hiH01Ω∫t0t1Ψmt−ΨtL2Ω.Ψmt,x+Ψt,xH01Ω.dt,≤CχtL∞t0,t1hiH01ΩΨm−ΨL2t0,t1,L2Ω.Ψm+ΨL2t0,t1,H01Ω.
According (29), (32), and (38), we get, as m⟶∞,(39)∫t0t1Ψm2t,χthidt⟶∫t0t1Ψ2t,χthidt.
Relations (34), (35), and (37) allow us to pass to the limits in (9) to find(40)−b0,hiχt0−∫t0t1Ψt,χ′thidt+∫t0t1∇Ψt,χt∇hidt,=∫t0t1μΨt1−Ψt,χthidt,∀i=1,2,…. We use the linearity in hi of (40) and hi as total in H01Ω, we obtain(41)−b0,ψχt0−∫t0t1Ψt,χ′tψdt+∫t0t1∇Ψt,χt∇ψdt,=∫t0t1μΨt1−Ψt,χtψdt∀ψ∈H01Ω.
Since, (41) holds for any ψ∈D0,T such that Ψ satisfies(42)Ψ′t,ψ+∇Ψt,∇ψ=μΨt1−Ψt,ψdt∀ψ∈H01.
Using [23], we get Ψ′∈L20,T,H−1Ω in the sense of distributions (in time).
Finally, it rests to show thatΨ verifies the initial condition Ψt0=b0.
Multiplying (42) by χt and integrating by parts, we find(43)−Ψt0,ψχt0−∫t0t1Ψt,χ′tψdt+∫t0t1∇Ψt,χt∇ψdt,=∫t0t1μΨt1−Ψt,χtψdt∀ψ∈H01Ω.
A comparison of (41) and (43) yields b0−Ψt0,ψχt0=0∀ψ∈H01Ω, and we pick χ with χt0=1.
b0−Ψt0,ψ=0∀ψ∈H01Ω, i.e., b0=Ψt0 a.e. Ω. This is end of the proof of Theorem 1.


## 3. Maximum Principles

ConsiderΨ a solution of (5) satisfying (6) and (7), ∃t0,t1, in Section 5. Under suitable hypothesis on Ψ0, we prove that the solution Ψ verifies 0≤Ψ≤1. This result proves the global existence in Section 4.Theorem 2.
Suppose thatb0∈L2Ω and b0x≥0 a.e in Ω. If Ψt,x is a solution of problem (5) satisfying (6) and (7), where t0∈0,T and t1∈t0,T, then Ψ≥0 a.e in t0,t1∈Ω.Proof.
For a.et∈t0,t1, we let ψx=Ψ−t,x=max−Ψt,x,0 in (6); we have(44)ddtΨ−tL2Ω2+∇Ψ−tL2Ω2=∫ΩΨ−t21−Ψ−tdx.≤μΨ−tL2Ω2+μΨmtL3Ω3.
By (21), we have(45)ddtΨ−tL2Ω2≤2μΨ−tL2Ω2+2CμΨmtL2Ω6.
Integration overt0 to t gives(46)Ψ−tL2Ω2≤Ψ−t0L2Ω2+∫t0t2μ1+Ψ−sL2Ω4ΨmsL2Ω2ds,where t∈t0,t1. We have(47)Ψ−=max−Ψ,0=Ψ−Ψ2,and Ψ∈Ct0,t1;L2Ω deduce that Ψ−∈Ct0,t1;L2Ω.
SettingK=Ψ−t0L2Ω2=0, Y=Ψ−sL2Ω2, and G=2μ1+Ψ−sL2Ω4, from (46), we find(48)Yt≤K+∫atYsGsds∀t∈t0,t1.
Furthermore, by Gronwall’s inequality, we getYt≡0,ont0,t1, i.e., Ψ−tL2Ω2≡0 and Ψ−t,x=0 a.e in t0,t1×Ω.Theorem 3.
Suppose that the suppositions of Theorem2 and b0x≤1 a.e in Ω hold. Then, Ψt,x is the local solution of (6) and (7) which satisfies Ψ≤1 a.e in t0,t1×Ω.Proof.
For a.e.t∈t0,t1, we let ψx=Ψ−1+t,x=maxΨt,x−1,0 in (6) to find(49)Ψ′t,Ψ−1++∫Ω∇Ψ∇Ψ−1+dx=∫ΩΨ1−ΨΨ−1+dx.
Equation (49) can be written as(50)12ddtΨ−1+L2Ω2+∇Ψ−1+L2Ω2=−∫ΩΨΨ−1+2dx≤0.
Then,Ψ−1+t,x=0 a.e in t0,t1×Ω.


## 4. Global Existence

In this section, we will show the global existence and uniqueness.By the result (Theorems1–3), under suitable hypothesis on Ψ0x, we deduce that there exists a solution Ψ for our problem (5), satisfying (6) and (7) on some interval t0,t1 and 0≤Ψ≤1 a.e in t0,t1×Ω. Hence, we expect a global solution to exist on any interval 0,T.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.
LetT∈0,∞. Suppose that 0≤Ψ0x≤1 a.e. x∈Ω. Then, ∃!Ψ∈W0,T such that Ψ is a solution of (6) and (7) and 0≤Ψ≤1 a.e. in 0,T×Ω.Proof.
Using Theorem1, ∃t1>0 such that ∃Ψ∈W0,t1 as a solution of (6) and (7) on 0,t1. Applying Theorems 2 and 3, we obtain(51)0≤Ψt,x≤1,a.e.t,x0,t1×Ω.
We lett¯=supt˜:∃Ψ∈W0,t˜satisfying1.3and1.4hold, fora.e.t∈0,T. Hence, t˜ must equal T. Furthermore, Theorem 1 would allow us to continue the solution beyond t˜, and this would contradict the maximality supposition of t˜ (if Ψ∈W0,t˜ and Ψ∈Wt˜,t˜+δ for some δ>0; then, Ψ∈W0,t˜+δ.
To show the solution is a unique, we assumeΨ1 and Ψ2 are two solutions of (6) and (7) and set Γ=Ψ1−Ψ2. Hence, Γ satisfies(52)Γ′t,ϕ+∇Γt,∇ϕ=μΓt1−Γt,ϕ∀ϕ∈H01Ω,a.e.t∈0,T,(53)Γ0,x=0.
By lettingϕ=Γt,. in (52), we obtain(54)12ddtΓtL2Ω2+∇ΓtL2Ω2=μΓtL2Ω2−μΓ2,Γ,a.e.t∈0,T,so that(55)12ddtΓtL2Ω2+∇ΓtL2Ω2≤μΓtL2Ω2,a.e.t∈0,T,⇒12ddtΓtL2Ω2≤μΓtL2Ω2,a.e.t∈0,T.
Hence, by Gronwall’s inequality and (53), we find(56)ΓtL2Ω2=0,a.e.t∈0,T.
We obtainΓ=0a.e.in0,T×Ω. The proof is complete.Now, we present a new a priori estimate:Lemma 1.
Suppose that0≤b0x≤1 a.e. x∈Ω. From the solution Ψ of (6) and (7), (57)ΨtL2Ω2+ΨL20,T,H01Ω2+Ψ′L20,T,H−1Ω2≤Ψt0L2Ω2+C1T.Proof.
Using Theorems2 and 3, we get 0≤Ψt,x≤1 a.e in 0,T∈Ω. Hence, for a.e.t∈0,T, upon letting ψx=Ψt,x in (6), we obtain(58)12ddtΨtL2Ω2+∇ΨtL2Ω2=∫ΩμΨt,x21−Ψt,xdx,≤∫Ωμdx≤μC1μ,Ω,where C1>0 is constant. Integrating over 0,t, we obtain(59)ΨtL2Ω2+ΨL20,T,H01Ω2≤Ψt0L2Ω2+C1T.
From (6), we find(60)Ψ′t,ψ+∫Ω∇Ψt,x∇ψxdx=∫ΩμΨt,x21−Ψt,xψxdx,≤ΨtH01Ω+ψH01Ω+μC∫Ωψxdx,≤ΨtH01Ω+ψH01Ω+μCμ,Ωψx,for all ψ∈H01Ω and a.e. t∈0,T.
We pick the supremum overψ∈H01Ω, and we obtain(61)Ψ′tH−1Ω2≤ΨH010,T,H01Ω2+Cμ,Ω,for a.e. t∈0,T. According (59) and (61), we obtain (57).Theorem 5.
LetT∈0,∞. 0≤Ψ0x≤1 a.e. x∈Ω. Then, ∃!Ψ∈W0,T such that Ψ is a solution of (5) satisfying (6) and (7), 0≤Ψ≤1, a.e. in 0,T×Ω, and(62)ΨtL2Ω2+ΨL20,T,H01Ω2+Ψ′L20,T,H−1Ω2≤Ψt0L2Ω2+C1T.Proof.
For anym, Theorem 4 deduces the existence of Ψm∈W0,T so that(63)Ψm′t,ψ+∇Ψmt,∇ψ=μΨmt1−Ψmt,ψ∀ψ∈H01Ω,a.e.t∈0,T,(64)Ψ0,x=Ψ0.
Furthermore, we have that0≤Ψnx,t≤1 a.e. t,x∈0,T×Ω. By Lemma 1, we obtain(65)ΨmL2Ω2+Ψm′L20,T,H01Ω2≤Ψt0L2Ω2+C1T.
Inequality (65) allows us to extract a subsequence of Ψm so that(66)Ψm⇀ΨweakyinL20,T;H01Ω,(67)Ψm′⇀Ψ′weakly inL∞0,T;H−1Ω,(68)Ψm⇀ΨstronglyinL20,T;L2Ω.
Similar to Step3 in Theorem 1 and passing to the limits in (63), we obtain(69)Ψ′,ψt+∇Ψt,∇ψ=μΨ1−Ψ,ψ∀ψ∈H01Ω,a.e.t∈0,T.
We get (62) from (65). Theorems 2 and 3 imply 0≤Ψ≤1 a.e. in 0,T×Ω, 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.



---

*Source: 3304917-2021-11-19.xml*