Morita, Yoshihisa

EPrint Series of Department of Mathematics, Hokkaido University

Entire solutions associated with front waves to
reaction-diffusion equations
Yoshihisa Morita
Department of Applied Mathematics and Informatics
Ryukoku University
Seta Otsu 520-2194 Japan
E-mail:morita@rins.ryukoku.ac.jp
Abstract
We can observe propagation phenomena modeled by reaction-diffusion equa-
tions in various fields of materials science, biology and life science. Correspond-
ing to a wave propagation, the equations allow a traveling wave that has a con-
stant profile and a constant speed. Historically, starting from the pioneering
work by [5, 15], there are enormous number of papers for the study of traveling
wave solutions (for instance see [1], [2], [4], [7], [10], [11], [13], [12], [19], [18],
[17], [20], [21], and references therein).
Here we consider a simple model equation, that is a scalar reaction-diffusion
equation of one-space dimension
ut = uxx + f(u) (x ∈ R) (1)
with the condition
f(0) = f(1) = 0, f ′(0) 6= 0, f ′(1) < 0. (2)
The condition (2) tells that u = 0 and u = 1 are nondegenerate constant
equilibria and that u = 1 is asymptotically stable. In addition to (2), if f
satisfies
f ′(0) > 0, f(u) 6= 0 (u 6= 0, 1),
the equation (1) is called a monostable reaction-diffusion equation while if there
is a number a ∈ (0, 1) such that
f ′(0) < 0, f(a) = 0, f ′(a) > 0, f(u) 6= 0 (u 6= 0, a, 1),
(1) is called a bistable reaction-diffusion equation. Typical examples of those
cases are the Fisher-KPP equation with
f(u) = u(1− u), (3)
and the Nagumo equation with
f(u) = u(1− u)(u− a), 0 < a < 1, (4)
respectively.
Putting u(x, t) = U(x + ct), z = x + ct, a monotone traveling wave (called
front wave) is obtained by solving the equation{
Uzz − cUz + f(U) = 0, U(z) > 0 (z ∈ R),
U(−∞) = 0, U(∞) = 0. (5)
It is known that a monostable equation allows a family of traveling waves with
speeds c ≥ cmin > 0; for instance cmin = 2 if f(u) = u(1 − u). On the other
hand a bistable reaction-diffusion equation has a unique traveling wave up to
translation. For the Nagumo equation the traveling wave is given explicitly as
U01(x+ ct) :=
exp
[
x/
√
2 + (1/2− a)t
]
1 + exp
[
x/
√
2 + (1/2− a)t
]
=
1
2
+
1
2
tanh
(
x+ ct
2
√
2
)
, c =
√
2
(
1
2
− a
)
.
We note that the reflected one U10 := U01(−x + ct) is also a traveling wave
with a monotone decreasing profile.
As for the Nagumo equation there are other exact traveling wave solutions,
which are given as
U0a(x+ c1t) :=
a exp
[
ax/
√
2− (a− a2/2)t
]
1 + exp
[
ax/
√
2− (a− a2/2)t
]
=
a
2
+
a
2
tanh
[
a(x+ c1t)
2
√
2
]
, c1 = −2− a√
2
,
and
Ua1(x+ c2t) :=
a+ exp
[
(1− a)x/√2 + (1− a2)t/2
]
1 + exp
[
(1− a)x/√2 + (1− a2)t/2
]
=
a+ 1
2
+
1− a
2
tanh
[
(1− a)(x+ c2t)
2
√
2
]
, c2 =
1 + a√
2
.
We notice that the former one is a traveling wave connecting u = 0 to u = a
while the latter one connects u = a to u = 1. More interesting thing is that
the Nagumo equation has the following exact solution:
u(x, t) =
exp
[
x/
√
2 + (1/2− a)t
]
+ a exp
[
ax/
√
2− (a− a2/2)t
]
1 + exp
[
x/
√
2 + (1/2− a)t
]
+ exp
[
ax/
√
2− (a− a2/2)t
] (6)
([14]). This solution is not a traveling wave with a constant profile. In fact it
behaves as two traveling waves U0a and Ua1 propagate from the left axis and
right axis respectively until they merge. Then the solution behaves like a single
wave U01.
We call an entire solution to (1) if it is a classical solution defined for all
x and t. Although an equilibrium solution and a traveling wave solution are
examples of entire solutions, the above solution (6) is a different example of an
entire solution from those. The solution (6) suggests us an interesting problem
how we can show the existence of an entire solution which behaves as two
traveling waves for t << 0.
The aim of our talk is to introduce the existence theorem for some entire
solutions [16]. Applying the theorem, we obtain the similar solution to (6) for
a general cubic f . We can also obtain a different type of entire solution than
(6).
Finally we note that there is an entire solution which behaves as two fronts
U01 and U10 propagating from the both sides of x-axis and annihilating even-
tually. This kind of entire solutions are extensively studied in [3], [6], [8], [9],
[22].
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Entire solutions associated with front waves to reaction-diffusion equations

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