Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 04, pp. 1–17.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

SEMIGROUP THEORY AND ASYMPTOTIC PROFILES OF

SOLUTIONS FOR A HIGHER-ORDER FISHER-KPP PROBLEM

IN RN

JOSÉ LUIS DÍAZ PALENCIA

Abstract. We study a reaction-diffusion problem formulated with a higher-

order operator, a non-linear advection, and a Fisher-KPP reaction term de-
pending on the spatial variable. The higher-order operator induces solutions

to oscillate in the proximity of an equilibrium condition. Given this oscillatory
character, solutions are studied in a set of bounded domains. We introduce

a new extension operator, that allows us to study the solutions in the open

domain RN , but departing from a sequence of bounded domains. The anal-
ysis about regularity of solutions is built based on semigroup theory. In this

approach, the solutions are interpreted as an abstract evolution given by a

bounded continuous operator. Afterward, asymptotic profiles of solutions are
studied based on a Hamilton-Jacobi equation that is obtained with a single

point exponential scaling. Finally, a numerical assessment, with the function

bvp4c in Matlab, is introduced to discuss on the validity of the hypothesis.

1. Description of the problem and main results

Reaction-diffusion models have been a source of research under different scopes.
From a physical perspective, an intuition of diffusion is introduced by the con-
cept of Random Walk (see [28] and references listed there) that permits to model
complex scenarios related with spatially distributed motion including heterogeneous
media. Other physical approach to diffusion has been pursuit based on the Landau-
Ginzburg free energy concept [12, 13]. The free energy permits to consider a gen-
eralization beyond the classical Fick law. In particular, the authors in [12] derived
a mathematical expression for the free energy in a heterogeneous media leading to
the formation of spatial patterns of solutions. It is particularly relevant to briefly
discuss the mathematical arguments used by the authors in [12]: Firstly, they con-
sidered that the free energy shall be dependant of the gradient of a concentration,
v, i.e. 1

2k(∇v)2. Considering this general form and making use of the chemical po-
tential, the authors obtained a parabolic partial differential equation with an order
four spatial operator, generally of the form vt = −vxxxx.

Let us return now to the origin: Nonlinear reaction-diffusion models were for-
mally and systematically introduced by Kolmogorov, Petrovskii and Piskunov in
[24] and by Fisher in [18] to study the behaviour of flames in combustion theory

2020 Mathematics Subject Classification. 35K92, 35K91, 35K55.
Key words and phrases. Higher order diffusion; semigroup theory; Fisher-KPP equation;

Hamilton-Jacobi equation.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted June 15, 2022. Published January 16, 2023.

1



2 J. L. DÍAZ P. EJDE-2023/04

and the interaction of genes respectively. The approaches followed by the authors
were supported by the classical Fick law that ends in a gaussian order two operator.
The authors employed a paramount technique, that opened new areas for exploring
analytical expressions for the solutions. These were named as Travelling Waves and
were particularly interesting to predict the dynamics of diffusion acting in a wave
front. The Fisher-KPP model is ubiquitous and is given in different applied scopes
(refer to [1, 2, 3] as representative examples). Recently, the Fisher-KPP models
have been studied with different kind of parabolic operators: higher order opera-
tors [20], fractional operators [8] and with a p-Laplacian Porous Medium Equation
[4].

In addition to the mention ideas, the higher order operators can be regarded as
perturbation terms that supplement the regular order two diffusion operators (see
[32, 14, 30, 6] for some extensions of the Fisher-Kolmogorov equation).

Some other notable analyses can be mentioned in relation with heterogeneous
diffusion and applications. In [31], the authors studied a biological dynamics with
advection, that precluded a non-linear diffusion. Further, the authors in [22] studied
the haptotaxis cancer invasion given by a degenerate diffusivity and by spectral
stability methods.

The proposed equation, under discussion in this analysis, is formed of a higher-
order operator (that make solutions to oscillate and proceed in a non-homogeneous
evolution), a non-linear advection and a Fisher-KPP term:

vt = −∆2v + c · ∇vp + v(g(x)− v),

v0(x), g(x) ∈Wm,p
0 (RN ) ∩ C0(RN ), N > 1,

(1.1)

where c ∈ RN is the advection vector, m = 4 for our purposes and typically p = 2.
Given the oscillatory character of solutions, induced by the higher order opera-

tor (see [20]), the analysis presented in this paper is provided in a set of bounded
domains. Based on the proposed study, we think that the oscillating character of
solutions can be assessed, in a more comprehensive view, for a free domain as RN .
This idea is in fact the main motivation of the presented analysis and requires to
include some evidences to support the extension from the sequence of bounded do-
mains to the whole space RN . The analyses of regularity, existence and uniqueness
of the solutions are given by making use of the semigroup theory. The opera-
tor −∆2 is briefly shown to be an infinitesimal generator of a strongly continuous
semigroup. Afterward, an exponential scaling, along with some hypothesis under
an asymptotic approach, are introduced to explore forms of analytical profiles of
solutions. Eventually, a numerical assessment, for the equation (1.1), is provided
in order to compare it with the asymptotic solutions and state about their validity.

2. Preliminary results

To start we consider the set Γr = {x ∈ B(0, r) ⊂ RN} with Ca-boundary (a ≥ 1).

Definition 2.1. The extension operator is defined as

E : Wm,p(Γr)→Wm,p(RN ), (2.1)

where (Ev)(x) = v(x) a.e. in Γr.

Proposition 2.2. The extension operator E satisfies

‖Ev‖Wm,p(RN ) ≤ C‖v‖Wm,p
0 (Γr), (2.2)



EJDE-2023/04 HIGHER-ORDER FISHER-KPP PROBLEM 3

where C is a constant obtained as a relation of Borel measures in partitions of
unity.

Proof. With no loss of generality, first we assume r → ∞ and p = 2. Then v ∈
Hm(Γr→∞) if

(1 + |ξ|2)m/2v̂(ξ) ∈ L2(Γr→∞), (2.3)

where v̂ refers to the Fourier transformation. In addition, for k < m − N
2 and a

suitable constant C1, it is easily checked that

‖(1 + |ξ|2)
k
2 v̂(ξ)‖L2(Γr→∞) ≤ C1‖Ev‖Wm,p(Γr→∞). (2.4)

Then, it follows to state that each ∂βv is bounded and continuous (in the sense
of Holder) for |β| < m− N

2 . Based on the denseness property for Wm,p, given any

v ∈Wm,p(Γr), define a sequence

vr = {|max v(∂Γr)|, r = 1, 2, 3, dots}. (2.5)

Based on the ordered property, it is possible to find regions of ∂Γr where the
following holds:

|max v(RN )| = lim
r→∞

|max v(∂Γr)| <∞. (2.6)

As |max v(RN )| is finite and Γr is bounded with smooth Ca-boundary (a ≥ 1),
it is possible to approximate any function in Wm,p(Γr) by functions in C∞(RN )
restricted to Γr. In addition, there exists a partition of unity for each r represented
as {ρrj}j∈J , with J being a set of indexes. Then, if vr ∈ Wm,p(Γr), then ρrj vr ∈
Wm,p(Γr). Further, it holds that spt(ρrj vr) ⊂ Γr and is compact.

Consider now the standard mollifier φ (See [17]). Given a small εj > 0, we define
function

hrj := (ρrj vr) ∗ φεj ∈ C∞0 (Γr). (2.7)

In the limit for r →∞ , a unity partition is given as {ρj} = limr→∞{ρrj}, such that
the following function is defined:

hj := (ρj v) ∗ φεj ∈ C∞0 (RN ). (2.8)

Given the bound of each ∂βv, |β| < m − N
2 , as previously shown, considering a

Borel measure µ in each {ρrj} and by a spatial translation to make hj below hrj , the
following holds

(µ(ρj)v) ∗ φεj ≤ (µ(ρrj)vr) ∗ φεj , (2.9)

uniformly in Γr and Γr→∞.
Considering the involved norms in Wm,p

0 and the denseness properties of Wm,p
0

(see [23]) in Γr with ∪∞r=1Γr = RN ,

‖Ev‖Wm,p(RN ) ≤ ‖v‖Wm,p(RN ) ≤
µ(ρrj)

µ(ρj)
‖v‖Wm,p

0 (Γr), (2.10)

as intended. �

Given the basic equation vt = −∆2v, the function S : R+ → R+ with S(t) =

e−∆2 t defines a family {S(t)}t∈R+ of bounded linear operators in L2. The bound-
edness of each mapping follows from the Plancherel´s theorem and after application
of the standard norm in L2. The family {S(t)}t∈R+ satisfies the basic condition of
the semigroup theory: S(0) = I, S(t+ s) = S(t)S(s), S(−t) = (S(t))−1.



4 J. L. DÍAZ P. EJDE-2023/04

Consequently, the operator −∆2 can be regarded as the infinitesimal generator
of the semigroup family {S(t)}t∈R+ . Indeed, it complies with:

−∆2 = lim
t→0+

S(t)− I
t

, (2.11)

where I = limt→0+ S(t) = I. In addition, the following proposition holds.

Proposition 2.3. For the bounded linear operator −∆2 in L2, the family {S(t)}t∈R+

where

e−∆2t :=

∞∑
j=1

(−∆2)j tj

j!
, (2.12)

with (−∆2)0 = I, forms a uniformly continuous semigroup in L2 where −∆2 is the
infinitesimal generator.

Proof. The proof follows from standard ideas of the semigroup theory (see [29]).
Nonetheless, some basic principles are introduced, and in particular for the operator
−∆2,

‖S(t)‖2 = ‖e−∆2t‖2 ≤
∞∑
j=1

(‖ −∆2‖2 t)j

j!
= e‖−∆2‖2 t, t > 0. (2.13)

Consequently, the family {S(t)}t∈R+ is well defined and satisfies S(0) = I, being I,
the identity map in L2. To show the uniform continuity,

‖S(t)− I
t

− (−∆2)‖2 ≤
1

t

∞∑
j=2

(‖ −∆2‖2 t)j

j!
=

1

t
(e‖−∆2‖2t−I− t‖−∆2‖2), (2.14)

that tends to zero whenever t→ 0+. Therefore, −∆2 is the infinitesimal generator
of the uniformly continuous semigroup family {S(t)}t∈R+ with domain D(−∆2) =
L2. �

The next objective is to show that the family {S(t)}t∈R+ is strongly continuous
(also referred as C0-semigroup).

Proposition 2.4. Assume that S(t) is a C0-semigroup. Then, there exist two
constants, m and w, such that for t ≥ 0,

‖S(t)‖2 ≤ mewt. (2.15)

Proof. Previously, we showed that S(t) is uniformly continuous, then for m ≥ 1
and τ > 0 with 0 ≤ t ≤ τ :

‖S(t)‖2 ≤ m. (2.16)

Assume that this last inequality does not hold, this is, there exists a sequence {tn}
converging to zero such that ‖S(tn)‖2 ≥ n as n → ∞. Based on the semigroup
property S(tn)v → v as n → ∞, we state that {S(tn)v} is bounded ∀v ∈ L2.
Considering the Banach-Steinhause theorem, we conclude on the boundedness of
{S(tn)}, which is a contradiction to the initial assumption. Then, the initial in-
equality (2.15) holds as ewt ≥ 1, ∀t ≥ 0. �

Based on the above Proposition, the strongly continuous condition for S(t) fol-
lows easily: For any τ > 0, it holds that:

‖S(t+ τ)v − S(t)v‖2 ≤ ‖S(t)‖2 ‖S(τ)v − v‖2 ≤ mewt‖S(τ)v − v‖2. (2.17)

Then, for τ → 0+, it follows that ‖S(t+ τ)v − S(t)v‖2 → 0.



EJDE-2023/04 HIGHER-ORDER FISHER-KPP PROBLEM 5

Definition 2.5. The following norm is defined to support the analysis of existence
and uniqueness of solutions:

‖v‖2Υ =

∫
Γr

Υ(ξ)

4∑
k=0

|Dkv(ξ)|2dξ, ξ ∈ Γr ⊂ RN (2.18)

where Dk = ∂|k|

∂ξ
k1
1 ∂ξ

k2
2 ...∂ξ

kN
N

, with |k| =
∑N
i=1 ki, and k = (k1, k2, . . . , kN ) belongs

to (N ∪ {0})N. Further, v ∈ H4
Υ(Γr) ⊂ L2

Υ(Γr) ⊂ L2(Γr) and the weight Υ is given
as per the following expression (see [20] together with [27]):

Υ(ξ) = exp
(
a0|ξ|4/3 −

1

|ξ|q
1

tγ

∫ t

0

(‖c · ∇v(ξ, s)‖p2 + 1)ds
)
, (2.19)

such that |ξ| =
∑N
i=1 |ξi|, a0 > 0 is sufficiently small and γ > p+ 1.

2.1. Inequalities and relations in functional spaces. Let L = (−∆2 +pvp−1c ·
∇) be the spatial operator and consider the basic equation

vt = Lv. (2.20)

Although the initial data was requested to belong to Wm,p
0 (RN )∩C0(RN ), further

conditions will be specified on v0 in this section to support the embedding analysis
to come.

Lemma 2.6. Given v0 ∈ L2(RN ), we have

‖v‖L2 ≤ ‖v0‖L2 . (2.21)

For r ∈ R+, and considering v0 ∈ Hr(RN ) ∩ L2(RN ), now we have

‖v‖Hr ≤ ‖v0‖Hr , (2.22)

‖v‖2Hr ≤ e
r2

8t ‖v0‖2L2 . (2.23)

In addition,

‖v‖Υ ≤ κ‖v‖Hr ≤ κ‖v0‖Hr , κ2 = 5 sup
|ξ|∈R
{|v|2, |D1v|2, |D2v|2, |D3v|2, |D4v|2}.

Proof. A fundamental solution to the basic equation in (2.20) is given by

v(x, t) = etLv0(x), (2.24)

and considering the Fourier transformation in the domain (ξ),

v̂(ξ, t) = et(−|ξ|
4+pv̂p−1c·ξi)v̂0(ξ). (2.25)

Considering Plancherel‘s theorem or the isometric Fourier condition in L2:

‖v‖2L2 =

∫ ∞
−∞
|et2(−|ξ|4+pv̂p−1c·ξi)||v̂0(ξ)|2dξ

=

∫ ∞
−∞

e−2|ξ|4t|v̂0(ξ)|2dξ

≤ sup
|ξ|∈R

(e−2|ξ|4t)

∫ ∞
−∞
|v̂0(ξ)|2dξ = ‖v0‖2L2 .

(2.26)

Then ‖v‖L2 ≤ ‖v0‖L2 .



6 J. L. DÍAZ P. EJDE-2023/04

Now, consider a mollifying norm for r ∈ R+ and 0 ≤ t ≤ τ < ∞ that complies
with the Ap-condition (see [21]) for p = 1,

‖v‖2Hr =

∫ ∞
−∞

er|ξ|
2

|v̂(ξ, t)|2dξ. (2.27)

Then

‖v‖2Hr =

∫ ∞
−∞

er|ξ|
2

|v̂(ξ, t)|2dξ

=

∫ ∞
−∞

er|ξ|
2

|et2(−|ξ|4+pv̂p−1c·ξi)||v̂0(ξ)|2dξ

≤ sup
|ξ|∈R

(e−2|ξ|4t)

∫ ∞
−∞

er|ξ|
2

|v̂0(ξ)|2dξ = ‖v0‖2Hr .

(2.28)

Now we consider v0 ∈ L2(RN ). Then

‖v‖2Hr =

∫ ∞
−∞

er|ξ|
2

|v̂(ξ, t)|2dξ ≤ sup
|ξ|∈R

(er|ξ|
2

e−2|ξ|4t)

∫ ∞
−∞
|v̂0(ξ)|2dξ. (2.29)

Making standard operations and rearranging terms,

‖v‖2Hr ≤ e
r2

8t ‖v0‖2L2 , (2.30)

as initially stated. Eventually,

‖v‖2Υ =

∫
Γr

Υ(ξ)

4∑
k=0

|Dkv(ξ)|2dξ

≤
∫

Γr

er|ξ|
2

4∑
k=0

|Dkv(ξ)|2dξ

≤ κ2

∫
Γr

er|ξ|
2

|v(ξ)|2dξ

≤ κ2‖v‖2Hr ,

(2.31)

where κ2 = 5 sup|ξ|∈R{|v|2, |D1v|2, |D2v|2, |D3v|2, |D4v|2}.
The scaling variable κ is defined in accordance with the results about continuous

inclusions in Sobolev spaces ([23], p. 79). Indeed, assume that the function v is
regularly differentiable up to the third order. The fourth order derivative in κ can
be considered as a controlling variable. If such fourth order derivative is regular,
then the mollifying norm (2.27) bounds the norm ‖ · ‖Γ �

As previously shown, −∆2 is the infinitesimal generator of a strongly continuous
semigroup. As a consequence, the following representation holds based on the
Duhamel‘s principle,

v(t) = e−∆2tv0 +

∫ t

0

[
c ·∇(e−∆2(t−s)pvp(s))+e−∆2(t−s)v(s)(g(x)−v(s))

]
ds. (2.32)

Consider now the basic problem vt = −∆2v with v(x, 0) = δ(x), a solution is given
by the Fourier transformation

v̂(t) = e−|ξ|
4tv̂0(ξ). (2.33)



EJDE-2023/04 HIGHER-ORDER FISHER-KPP PROBLEM 7

As a consequence, the kernel can be obtained as

K(x, t) = F−1(e−|ξ|
4t) =

1

(2π)N/2

∫
RN

e−|ξ|
4t−iξ·xdξ

=
1

(2π)N/2

∫
RN

e−|ξ|
4tcos(ξ · x)dξ.

(2.34)

Note that the left-hand integral is bounded for ξ in RN . Hence, the abstract
evolution in (2.19) can be rewritten in terms of such existing kernel. But firstly,
the following operator in H4

Υ(Γr) is defined,

Tv0,t : H4
Υ(Γr)→ H4

Υ(Γr), (2.35)

which is given as

Tv0,t(v) = K(x, t) ∗ v0(x) +

∫ t

0

[
c · ∇K(x, t− s) ∗ vp(s)

+K(x, t− s) ∗ v(x, s)(g(x)− v(x, s))
]
ds,

(2.36)

where ‘ ∗ ‘ refers to the spatial convolution and t is a single parameter. Note that
in the previous expression, the following assessment has been implicitly conducted
in the advection term

K(x, t) ∗ c · ∇vp(x, s) =

∫ ∞
−∞

K(x− β, t)c · ∇vp(β, s) dβ

= −
∫ ∞
−∞

vp(β, s)c · ∇K(x− β, t) dβ

= −
∫ ∞
−∞

vp(β, s)c · ∇(x−β)K(x− β, t)∂(x− β)

∂β
dβ

=

∫ ∞
−∞

vp(β, s)c · ∇(x−β)K(x− β, t)

= c · ∇K(x, t) ∗ vp(x, t).

The next step is to show the boundedness properties of the single parametric
operator Tv0,t. Then, the following lemma holds.

Lemma 2.7 (Operator bound). The single parametric operator Tv0,t is bounded in
H4

Υ(Γr) with the norm (2.18). In addition, the extended operator given by ETv0,t
(see Definition 2.1) is bounded in H4(RN ).

Proof. Firstly, we shoe inequality

k0‖v0‖Υ ≤ ‖v‖Υ. (2.37)



8 J. L. DÍAZ P. EJDE-2023/04

Indeed,

‖v‖2Υ =

∫
Γr

Υ(ξ)

4∑
k=0

|Dkv̂(ξ)|2dξ

=

∫
Γr

Υ(ξ)

4∑
k=0

|Dk
[
et(−|ξ|

4+pv̂p−1c·ξi)v̂0

]
|2dξ

≥
∫

Γr

Υ(ξ)

4∑
k=0

∣∣Dk
[
et(−|ξ|

4+pv̂p−1c·ξi)]∣∣2 4∑
k=0

∣∣Dkv̂0

∣∣2dξ
≥ k2

0

∫
Γr

Υ(ξ)

4∑
k=0

|Dkv̂0|2dξ = k2
0‖v0‖2Γ,

(2.38)

such that

k2
0 = inf

ξ∈Br
{

4∑
k=0

∣∣Dk
[
et(−|ξ|

4+pŵp−1c·ξi)]∣∣2} > 0, (2.39)

in Br = {ξ : |ξ| < r}, for any r > 0.
Now, we return to the operator Tv0,t.

‖Tv0,t(v)‖Υ
≤ ‖Tv0,t‖Υ ‖v‖Υ ≤ ‖K‖Υ‖v0‖Υ

+

∫ t

0

[
‖c · ∇K‖Υ‖vp‖Υ + ‖K‖Υ ‖v‖Υ‖g(x)− v‖Υ

]
ds

≤
[
‖K‖Υ

1

k0 t
+

∫ t

0

[
‖c · ∇K‖Υ ‖vp−1

0 ‖Hr + ‖K‖Υ|‖g(x)‖Υ − k0‖v0‖Υ|
]
ds
]
t‖v‖Γ.

Note that inequalities (2.28) and (2.31) have been employed for the term ‖vp−1‖Υ,
indeed,

‖vp−1‖Υ ≤ ‖vp−1‖Hr ≤ ‖vp−1
0 ‖Hr . (2.40)

Then

‖Tv0,t‖Υ ≤
[
‖K‖Υ

1

k0 t
+

∫ t

0

[
‖c · ∇K‖Υ‖vp−1

0 ‖Hr

+ ‖K‖Υ|‖g(x)‖Υ − k0‖v0‖Υ|
]
ds
]
t <∞,

(2.41)

for 0 < t < ∞. Further, note that the right-hand side is locally bounded for any
single parameter t value. The operator action can be extended to RN by a simple
application of the extension operator given in Proposition 2.2 and the relation
‖Tv0,t‖H4

0 (Γr) ≤ ‖Tv0,t‖Υ, a.e. in Γr. Then

‖ETv0,t‖H4(RN ) ≤ C‖Tv0,t‖H4
0 (Γr) ≤ C‖Tv0,t‖Υ. �

Now, we obtain existence results for any solution to (1.1) in H4
0 (Γr). To this

end, the following theorem holds.

Theorem 2.8. Assume that locally v(t) = Tv0,t(v(t)) for 0 < t ≤ τ . Then there
exists a weak solutions to (1.1) and it satisfies v1 ∈ L2(0, τ ;H4

0 (Γr)).

Proof. Given the sequence subset Γr and the space H4
0 (Γr), it is well known that

there exist eigenfunctions of a reproducing kernel that form an orthonormal basis



EJDE-2023/04 HIGHER-ORDER FISHER-KPP PROBLEM 9

(refer to [9] for polynomial basis as a representative example). Consequently, as-
sume {ϕj(x)}, j = 1, 2, . . . is an orthogonal basis of H4

0 (Γr). Then, the following
sequence of solutions {vn} is defined for problem (1.1),

vn(x, t) =

n∑
j=1

gn,j(t)ϕj(x), n = 1, 2, . . . (2.42)

Note that each element of the sequence vn satisfies an evolution ODE

(vn,t, ϕj)− (∇(∇ · ∇vn),∇ϕj) + (cvpn,∇ϕj)− (vn(g(x)− vn), ϕj) = 0, (2.43)

where (·, ·) denotes the inner product in H4.
The initial condition for each element of the sequence is expressed as

vn(x, 0) =

n∑
j=1

gn,j(0)ϕj(x), n = 1, 2, . . . (2.44)

where gn,j(0) = (vn(x, 0), ϕj(x)) are constants.
Considering standard results of ODE theory, in particular the Peano‘s theorem,

there exist solutions for

gn,j(t) ∈ C1([0, τ ];H4
0 (Γr)), (2.45)

so that, vn ∈ C1([0, τ ];H4
0 (Γr)).

The intention now is to obtain estimates for each element vn(t). Firstly, we
consider equation (2.43) and multiply by gn,j(t). After the sum from j = 1 to
j = n:

1

2

d

dt
‖vn‖22 + ‖∇ · ∇vn‖22 + ‖cvpn‖2‖∇vn‖2 ≤ ‖vn‖22‖g(x)‖∞. (2.46)

Given the Sobolev space Wm,p(Γr), we define α = int{m− N
p }, then the following

Holder-continuous inclusion holds (refer to [23, page 79]),

Wm,p(Γr) ↪→ Cα(Γr). (2.47)

In the present analysis, any solution shall be generally differentiable up to order
fourth, hencem = 4, and typically p = 2, then α = int{4−N2 }. Given the continuity

embedding in (2.47) and the compact support for each function ϕj(x) ∈ H4
0 (Γr), it

is possible to conclude that the terms ‖∇ · ∇vn‖22 and ‖cvpn‖2‖∇vn‖2 are bounded
in Γr. In addition and since g(x) ∈ Wm,p

0 (RN ) ∩ C0(RN ), it is possible to define
M = ‖g(x)‖∞ <∞ in Γr. As a consequence, inequality (2.46) is reduced to

1

2

d

dt
‖vn‖22 ≤M‖vn‖22, (2.48)

such that the following estimate is obtained for each element of the sequence
‖vn‖22 ≤ e2Mt, 0 < t ≤ τ .

Now, integrating (2.48) on (0, τ ],

1

2
‖vn‖22 −

1

2
‖vn(0)‖22 ≤M

∫ τ

0

‖vn‖22 ≤
1

2
e2Mτ , (2.49)

which yields a global bound of each element of the sequence for 0 < t ≤ τ .
Considering the obtained bounds for 0 < t ≤ τ , there exists a function v ∈

L2(0, τ ;H4
0 (Γr)) together with a sub-sequence {vn}, n = 1, 2, . . . such that given

any t ∈ (0, τ ] and for n→∞, it holds that

vn → v, (2.50)



10 J. L. DÍAZ P. EJDE-2023/04

in a weak sense in L2(0, τ ;H4
0 (Γr)).

Now, consider the global bound in (2.49) for 0 < t ≤ τ , the weak converge of
{vn} as described and the Aubin-Lions-Dubinskii compactness theorem (see [11]):
Then, we state that vn → v strongly in C(0, τ ;L2(Γr)).

We recall that the Aubin-Lions-Dubinskii theorem requires ∂vn
∂t to be bounded

in a Banach space. This can be found easily from inequality (2.48) in 0 < t ≤ τ .
Once the convergence of the sequence {vn} has been shown, the following for-

mulation holds for the solution v,

(vt, ϕj)− (∇(∇ · ∇v),∇ϕj) + (c vp,∇ϕj)− (v(g(x)− v), ϕj) = 0, ∀j. (2.51)

For any arbitrary function Ψ obtained as a linear combination of {ϕj}, it holds that

(vt,Ψ)− (∇(∇ · ∇v),∇Ψ) + (cvp,∇Ψ)− (v(g(x)− v),Ψ) = 0. (2.52)

This last expression, along with the condition v0(x) ∈Wm,p
0 (RN )∩C0(RN ) (m =

4, p = 2) and the convergence in these dense spaces for any initial data distribution,
permit to conclude that the problem (1.1) admits weak solutions in 0 < t < τ where
τ ≤ ∞. �

Note that in Lemma 2.7, we showed the global bound of solutions by the bound-
ing properties of the single parametric operator Tv0,t. In addition, in Theorem 2.8,
the solutions have been shown to exist locally in time based on arguments related
with convergence. Such convergence can be extended to any τ � 1 while keeping
the global bound and the local convergence shown. As a consequence, the existence
of solutions is shown as a global basis for any 0 ≤ t ≤ τ < ∞. Further, the re-
sults can be made applicable to RN upon application of the extended operator in
Definition 2.1 together with ‖ETv0,t‖H4(RN ) ≤ C‖Tv0,t‖H4

0 (Γr).
Based on the observations made, we conclude on the existence of global solutions

v(x, t) for (x, t) ∈ RN × (0, τ ].

2.2. Uniqueness of the solution. The uniqueness analysis is based on a fix
point argument. The map Tv0,t in (2.35) shall comply with v(x, t) = Tv0,t(v(x, t)),
(x, t) ∈ Γr × (0, τ ]. For this, consider that there exist two solutions v1(x, t) and
v2(x, t) satisfying (2.36) and with the same initial data v0(x), such that making the
difference and for any t ∈ (0, τ ],

‖Tv0,t(v1)− Tv0,t(v2)‖Υ

≤
∫ t

0

‖c · ∇K(x, t− s) ∗ (vp1 − v
p
2)

+K(x, t− s) ∗ [v1(g − v1)− v2(g − v2)]‖Υds

=

∫ t

0

‖
∫ s

t

{c · ∇K(x, t− s− r)(vp1 − v
p
2)

+K(x, t− s− r)[v1(g − v1)− v2(g − v2]}dr‖Υds

≤
∫ t

0

∫ s

t

{‖c · ∇K(x, t− s− r)(vp1 − v
p
2)‖Υ

+ ‖K(x, t− s− r)[v1(g − v1)− v2(g − v2)]‖Υ}dr ds

=

∫ t

0

∫ s

t

{‖c · ∇K(x, t− s− r)‖Υ‖vp1 − v
p
2‖Υ

+ ‖K(x, t− s− r)‖Υ‖v1(g − v1)− v2(g − v2)‖Υ}dr ds



EJDE-2023/04 HIGHER-ORDER FISHER-KPP PROBLEM 11

≤M1

∫ t

0

∫ s

t

{‖vp1 − v
p
2‖Υ + ‖v1(g − v1)− v2(g − v2)‖Υ}dr ds, (2.53)

Note that K and ∇K are bounded as per the expression (2.34), hence

M1 = sup{‖K(x, t−s−r)‖Υ, ‖c ·∇K(x, t−s−r)‖Υ; ∀t ∈ (0, τ ], x ∈ Γr}, (2.54)

and for any s > 0 and r > 0.
To assess the integrals involved in (2.53), consider the function

A(ε, s) =

{v1(ε,s)p−v2(ε,s)p

v1(ε,s)−v2(ε,s) for v1 6≡ v2

pvp−1
1 otherwise.

(2.55)

For a fixed value in ε and with s = τ , the last expression is bounded and satisfies
0 ≤ A(ε, s) ≤ C0(p, ‖v0‖∞, τ). Then

‖vp1 − v
p
2‖Υ ≤ C∗0‖v1 − v2‖Υ, (2.56)

where C∗0 = ‖C0‖Γ.
The remaining term involving the reaction-absorption in (2.53) is assessed based

on the definition of a norm given in (2.18).

‖[v1(g − v1)− v2(g − v2)]‖2Υ

=

∫
Γr

Υ(ξ)

4∑
k=0

|Dk[v1(g − v1)− v2(g − v2)]|2dξ

=

∫
Γr

Υ(ξ)
{
|v1(g − v1)− v2(g − v2)|2

+

4∑
k=1

|Dk[v1(g − v1)− v2(g − v2)]|2
}
dξ

=

∫
Γr

Υ(ξ)
{
|(v1 − v2)(g − (v1 − v2))|2

+

4∑
k=1

k∑
i=1

∣∣(k
i

)
(v1 − v2)(i)(a− (v1 − v2)(k−i))

∣∣2}dξ
≤ 25B2

∫
Γr

Υ(ξ)
{
|(v1 − v2)|2 +

4∑
k=1

k∑
i=1

∣∣(k
i

)
(v1 − v2)(i)

∣∣2}dξ
= 25B2

∫
Γr

Υ(ξ)

4∑
k=0

|Dk[v1 − v2]|2dξ

= 25B2‖v1 − v2‖2Υ,

(2.57)

where B2 = sup{|g − (v1 − v2)|2, |g − (v1 − v2)]k−i|2}. Eventually, it holds

‖Tv0,t(v1)− Tv0,t(v2)‖Υ ≤M1(5B + C∗0 )

∫ t

0

∫ s

t

‖v1 − v2‖Υdsdr

= M1(5B + C∗0 )t(t− s)‖v1 − v2‖Υ.
(2.58)

Given a local time interval with 0 < t < s ≤ τ , the uniqueness is shown for
v1 ↙ v2 leading to a contractive mapping Tv0,t, that complies with the convergence
condition Tv0,t(v1)↙ v1 in the space of functions H4

Υ.



12 J. L. DÍAZ P. EJDE-2023/04

The uniqueness result can be made applicable to solutions in RN . For this,
consider the extended operator in Definition 2.1 with the bound ‖ETv0,t‖H4(RN ) ≤
C‖Tv0,t‖H4

0 (Γr).

3. Profiles of solutions

Profiles of solutions are obtained under the non-linear point transformation:

v = ew. (3.1)

For particular discussions in relation to the proposed exponential scaling, the reader
can consult [7] together with the formal introduction in [26, 25, 5].

Generally, the function w shall be understood as a complex mapping, and this
can be foreseen given the oscillatory nature of the solutions profiles (see [20]):
w : X × [0, τ ]→ C.

Now, following some previous ideas discussed in [10], the function w satisfies the
Hamilton-Jacobi type of equation

wt = H4(w,∇w) + P4

(
w,

∂|k|w

∂xk11 ∂x
k2
2 . . . ∂xkNN

)
,

|k| =
N∑
i=1

ki, k = (k1, k2, . . . , kN ) ∈ (N ∪ {0})N.
(3.2)

where

H4(w) = −(∇w)2(∇w)2 + cp∇we(p−1)w + g(x)− ew. (3.3)

Furthermore,

P4(w) = −∆2w −∆(∇w · ∇w)− 2∇w · ∇∆w − 2(∇w · ∇w)∆w

− 2∇w · ∇(∇w · ∇w)− (∆w)2.
(3.4)

The operator P4 is of algebraic order 3 while the Hamilton-Jacobi operator H4 is
of algebraic order 4. This can be easily shown by considering a smooth function
Σ ∈ C∞0 , such that

|P4(λΣ)| = O(λ3)� H4(λΣ) = O(λ4), (3.5)

for λ� 1. Balancing the leading terms, the equation (1.1) can be rewritten as

wt = −(∇w)2 (∇w)2 + c p∇w e(p−1)w + g(x)− ew. (3.6)

Now, the idea is to explore standing wave solutions to (3.6). Assume that such
solutions are given in the form of separated variables (refer to [10] for further
details),

w(x, t) = (σ + t)−1/3Θ(x), (3.7)

where σ < t ≤ τ .
Upon substitution in (3.6) and in the asymptotic approach with σ � 1,

− 1

3
Θ = (∇Θ)4 + c p(σ + t)∇Θ + (g(x)− 1)(σ + t)4/3. (3.8)

Recall that our intention is to introduce an asymptotic solution. Consequently, in
the last expression, we have considered that

ew = e(σ+t)−1/3Θ(x) → 1, e(p−1)w → 1, (3.9)



EJDE-2023/04 HIGHER-ORDER FISHER-KPP PROBLEM 13

with t→∞. We consider now the leading terms in (3.8):

− 1

3
Θ = cp(σ + t)∇Θ + g∗(·, t), (3.10)

where σ, t are parameters and g∗(·, t) = |(g(·)−1)(σ+t)4/3|. Note that the function
g∗ is expressed as a function of t to stress the asymptotic evaluation of g for t→∞
and the point ‘ · ‘ indicates that the function is assessed in each x. Hence

Θ(x) = 3
(
e−

x
3|c|p (σ+t) − g∗(·, t)

)
, (3.11)

where |c| =
∑N
i=1 |ci|.

Now, in the asymptotic approximation t → τ � 1, there exists a moving wave
front as

|x| = 3|c|p ln(g∗(·, t))t. (3.12)

Note that the time exponent in the function g∗ and the Log function recall the
results in [20], where a Log-Front shift is analyzed for another higher order problem.
This is a confirmation of the ubiquity of the Log shift in the moving front.

Now, making the balance of the nabla distribution O(∇Θ) < O(σ+t) and making
the asymptotic condition with |x| � 1 and σ � 1,

− 1

3
Θ = (∇θ)4 + g∗(·, t), (3.13)

for which a general solution is

Θ(x) = 3
(1

4
J(i)|x|

)4/3

− 3g∗(·, t), (3.14)

being J(i) = (−1)
1
4 and i the imaginary unit. Based on all exposed, a profile of

asymptotic solution to w is

w(x, t) = 3t−1/3
((1

4
J(i)|x|

)4/3

− g∗(·, t)
)
. (3.15)

In return to the original scaling (3.1), a profile of asymptotic solution to (1.1), for
t→ τ � 1, is

v(x, t) = e−3g∗(·,t)t−1/3

exp
(

3t−1/3
(1

4
J(i)|x|

)4/3)
. (3.16)

Note on the existence of oscillations in the obtained profile. This is affirmative
given the complex number J(i). The oscillation behaviour of solutions in higher-
order operators has been widely discussed in [20] and will be further discussed in
the numerical approach introduced in the next section.

4. Numerical validation of the asymptotic approach

Once an analytical asymptotic solution has been shown to hold in the expression
(3.16), and under certain hypothesis, it is the intention to compare the asymptotic
approach with a numerical solution of (1.1). The numerical assessment is based
on the function bvp4c in Matlab. The initial condition and the reaction function
shall satisfy v0(x), g(x) ∈ H4

0 (R3) ∩ C0(R3). For the sake of simplicity, and with
no loss of generality, both functions have been considered as the same: v0(x) =
g(x) = 1 − cs |x|4 where cs is a suitable constant to be determined based on the
numerical trials and has the intention of providing fully readable representations.
Note that the bvp4c function set is based on a Runge-Kutta implicit algorithm



14 J. L. DÍAZ P. EJDE-2023/04

Figure 1. Solutions for t = 1 (left) and t = 10 (right). In both
cases, for |x| > xm = 5.352, the global distance between the nu-
merical solutions and the asymptotic profile (3.16) is ≤ 10−3.

Figure 2. Solutions for t = 100 as a global picture (left) and a
zoom (right). For |x| > xm = 11.781, the global distance between
the numerical solutions and the asymptotic profile (3.16) is ≤ 10−3.
Note that in this case the oscillatory behaviour of solutions is made
apparent.

with interpolant extensions [16]. The pseudo-boundary condition, required by the
collocation methods in the function bvp4c, has been considered as v(|x| � 1) = 0.

The domain of integration is considered sufficiently large so as to minimize any
influence of the collocation method at the boundaries. In this case, the integration
domain is given by |x| ∈ (−500, 500). The number of nodes to execute the Runge-
Kutta is 100000 with absolute error of 10−5.

Given the difficulty to make a full representation in several dimensions, the
results are given in different time levels and for a single spatial variable of the
form |x| =

∑3
i=1 |xi|. As it can be observed, the asymptotic profile in (3.16) fits

with the numerical solution of (1.1) for |x| � 1 as it was required to obtained the
transport equation (3.13). To properly characterize the adequacy of the solutions, it
is considered that both solutions are sufficiently close whenever the global distance



EJDE-2023/04 HIGHER-ORDER FISHER-KPP PROBLEM 15

Figure 3. Solutions for t = 500 globally (left) and with a zoom
(right). For |x| > xm = 22.612, the global distance between the
numerical solutions and the asymptotic profile (3.16) is ≤ 10−3.
Note that the oscillatory behaviour increases when increasing the
time levels.

(or error) between the numerical solution and the asymptotic profile is |vnum −
vasym| ≤ 10−3 for |x| > xm (see each figure footprint for specific values of xm).

5. Conclusions

The Fisher-KPP problem in RN , (N > 1) with a nonlinear advection and a
higher order diffusion was studied with analytical and numerical approaches. The
operator −∆2 was shown to be the infinitesimal generator of a strongly continu-
ous semigroup by application of standard techniques. Consequently, the analyses
about regularity and uniqueness of the solutions were supported by the semigroup
theory. Afterward, the asymptotic profiles of solutions were obtained with a single
point exponential scaling, that led to a Hamilton-Jacobi equation. Eventually, a
numerical assessment, with the function bvp4c in Matlab, permitted to validate the
hypothesis made during the asymptotic analysis.

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EJDE-2023/04 HIGHER-ORDER FISHER-KPP PROBLEM 17

José Luis D́ıaz Palencia

Department of Mathematics and Education, Universidad a Distancia de Madrid, 28400,

Madrid, Spain
Email address: joseluis.diaz.p@udima.es


	1. Description of the problem and main results
	2. Preliminary results
	2.1. Inequalities and relations in functional spaces
	2.2. Uniqueness of the solution

	3. Profiles of solutions
	4. Numerical validation of the asymptotic approach
	5. Conclusions
	References