Mild and Strong Periodic Solutions to
Semilinear Evolution Equations
Matthias Hieber
TU Darmstadt
Bedlewo
April 25, 2017
joint work with
M. Geissert, H. Nguyen, R. Takada, P. Galdi, T. Kashiwabara,
N. Kashiwara, K. Kress, P. Tolksdorf
Periodic solutions to Linear Evolution Equations
For a Banach space X consider
u ′ (t) + Au(t) = f (t),
u(0) = u0 ,
−A generator of C0 -semigroup on X
f : R+ → X periodic with period T > 0, i.e. f (t + T ) = f (t), t > 0
Mild solution
u(t) = e −tA u0 +
Z
t
e −(t−s)A) f (s)ds
0
Then : (ACP) admits a (unique) T -periodic mild solution ⇔
there exists (unique) u0 ∈ X with
(Id − e −TA )u0 =
Z
0
T
e −(T −s)A) f (s)ds
Proof (easy)
Observe that due to periodicity of f
Z T
Z t+T
u(t) = u(t + T ) = e −tA e −TA u0 +
e −(t+T −s)A f (s)ds +
e −(t+T −s)A f (s)ds
0
T
Z T
Z t
e −(T −s)A f (s)ds] +
e −(t−s)A f (s)ds, t ≥ 0
= e −tA [e −TA u0 +
0
0
Hence :
there exists a (unique) T -periodic solution ⇔
there exist (unique) u0 with
Z T
(Id − e −TA )u0 =
e −(T −s)A) f (s)ds
0
Assume that spectral mapping theorem holds for e −tA
−TA ) is invertible and
∈
̺(A)
for
all
n
∈
Z,
then
(Id
−
e
If 2πin
T
u0 = (Id − e −TA )−1
Z
T
0
e −(T −s)A) f (s)ds
Generalization to Nonautonomous Situation
For a Banach space X consider
u ′ (t) + A(t)u(t) = f (t),
u(0) = u0 ,
Assume
A(t + T ) = A(t) for t ≥ 0, i.e. A(·) is periodic
there exists an evolution operator U(t, s) associated to (A(t))t≥0
f : R+ → X periodic with period T > 0, i.e. f (t + T ) = f (t), t > 0,
Define period map S via S(s) := U(s + T , s)
Then
If 1 ∈ ̺(S(0)), then (ACP) has unique T -periodic solution given by
Z t
u(t) = U(t, 0)x +
U(t, s)f (s)ds, where
0
x := (Id −
S(0))−1
RT
0
U(T , s)f (s)ds
The case 0 ∈ σ(A)
Consider now the situation where :
0 ∈ σ(A) : situation is more delicate
examples in mind : : elliptic operators, Ornstein-Uhlenbeck operators
on Rn or exterior domains, Stokes operator on exterior domains,
bidomain operator, ...
semilinear situation : again 0 ∈ σ(A) :
examples in mind : flow past rotating obstacles, primitive equations,
Allen-Cahn, FitzHugh-Nagumo, ...
Above examples give rise to following questions :
(unique) existence of mild or strong periodic solutions for small or
large forces f ?
Three approaches being discussed :
approach to mild/strong solutions via evolution equations,
interpolation theory
smoothing by Lp − Lq -estimates or dispersive estimates
PDE-approach via weak solutions and weak-strong uniqueness
Further : extensions to almost periodic functions
Abstract Setting for Mild Solutions
Consider the situation
X Banach space, (Y1 , Y2 ) interpolation couple of Banach spaces Y1
and Y2
(Y1 , Y2 )θ,∞ real interpolation space between Y1 and Y2 for 0 < θ < 1
u ′ (t) + Au(t) = Bf (t),
u(0) = u0 ∈ (Y1 , Y2 )θ,∞ ,
where −A is generator of C0 -semigroup e −tA on Y1 and Y2
f (t) ∈ X for all t ≥ 0 and T -periodic
B ’connection’ operator between X and Y := (Y1 , Y2 )θ,∞ such that
e −tA B ∈ L(X , Yi ) for i = 1, 2 and t ≥ 0
Given T > 0, u ∈ C ([0, T ]; (Y1 , Y2 )θ,∞ ) is called mild solution of
(ACPB) if u satisfies
Z t
u(t) = e −tA u(0) +
e −(t−τ )A Bf (τ )dτ, t ∈ [0, T ].
0
Assumptions
Assumptions (A) :
Let −A be the generator of a C0 -semigroup e −tA on Y1 and Y2
Assume Yi has separable pre-dual Zi for i = 1, 2 such that Z1 ∩ Z2 is
dense in Zi for i = 1, 2
Key Assumption : there exist α1 , α2 ∈ R with 0 < α2 < 1 < α1 and
L > 0 such that
ke −tA1 Bv kY1 ≤Lt −α1 kv kX ,
t > 0,
ke −tA2 Bv kY2 ≤Lt −α2 kv kX ,
t > 0.
Motivation for choice of Y are stationary solutions of semilinear
problem, i.e. by solutions to
Au(t) = div G (u)(t),
t > 0,
u(0) = u0
(1)
stationary solutions exist often in Lnweak (Ω)
existence of 2−m -periodic solutions for all m ∈ N yields existence of
stationary solutions
Boundedness of solution in interpolation spaces
Lemma (boundedness of solution) : Let θ ∈ (0, 1) with
1 = (1 − θ)α1 + θα2 . Then (ACPB) admits mild solution u satisfying
ku(t)kY ≤ Me ωt ku0 kY + M̃kf kL∞ (R+ ;X ) ,
t>0
Sketch of Proof :
duality :
′ −tA′
ϕkX ′ ≤Lt −α1 kϕkY1′ ,
t > 0,
′ −tA′
ϕkX ′ ≤Lt −α2 kϕkY2′ ,
t > 0.
kB e
kB e
Since (Z1 , Z2 )′θ,1 = (Y1 , Y2 )θ,∞ ϕ ∈ (Z1 , Z2 )θ,1
Z t
′
kf (τ )kX kB ′ e −(t−τ )A ϕkX ′ dτ
| < u(t), ϕ > | ≤ ke −tA u0 k(Y1 ,Y2 )θ,∞ kϕk(Z1 ,Z2 )θ,1 +
0
aim :
R∞
0
′
kB ′ e −ξA ϕkX ′ dξ ≤ M̃kϕk(Z1 ,Z2 )θ,1
Sketch of Proof
Sublinear operator : T : Z1 + Z2 ∋ ϕ 7→ v (t) :=
Assumption (A) : v (t) ≤ Ct −αj kϕkZj ,
T is of weak type
1
sj
L1
′
′
−tA
ϕkX ′ .
kB e
j = 1, 2
:= αj in Lorentz spaces, i.e. kv ksj ,w ≤ Cj kϕkZj
Key observation :
can be viewed as real interpolation space of
Lorentz spaces of order (θ, 1)
(Lsw1 (0, ∞), Lsw2 (0, ∞))θ,1 = L1 (0, ∞)
interpolation of pre-duals and operators : kv kL1 ≤ M̃kϕk(Z1 ,Z2 )θ,1
finally : global bound for u(t) in (Y1 , Y2 )θ,∞ since
| < u(t), ϕ > | ≤ Me ωt ku0 k(Y1 ,Y2 )θ,∞ kϕk(Z1 ,Z2 )θ,1 +M̃kf k∞,X kϕk(Z1 ,Z2 )θ,1 , t > 0
claim follows since (Y1 , Y2 )θ,∞ = (Z1 , Z2 )′θ,1
Main Result for Linear Situation
Theorem (linear)
Assume Assumption (A). Let θ ∈ (0, 1) with 1 = (1 − θ)α1 + θα2 . Let
f ∈ L∞ (R+ ; X ) be T -periodic for some T > 0. Then linear Cauchy
problem with B, (ACPB), admits a T -periodic mild solution û satisfying
kûkL∞ (R+ ;Y ) ≤ C (T )kf kL∞ (R+ ;X ) ,
Furthermore, û is unique provided
2nπi
T
6∈ σp (−A) for all n ∈ Z
Sketch of Proof
suffices to show : there exists x ∈ (Y1 , Y2 )θ,∞ with
Z T
(Id − e −TA )x =
e −(T −s)A Bf (s)ds
0 R
(k−1)T −(kT −s)A
e
Bf (s)ds
Define sequence (ak )k∈N by ak := 0
Pk−1 −jTA R T −(T −s)A
Bf (s)ds
then ak = j=1 e
o e
assumption α1 > 1 implies limk→∞ ak = a in Y1
above Lemma : (ak ) bounded in (Y1 , Y2 )θ,∞ .
w ∗ − limk→∞ ak = a in (Y1 , Y2 )θ,∞ as well.
R T −(T −s)A
Bf (s)ds. Then x satisfies desired equality.
set x := a + 0 e
Remarks
Solution operator ST ∈ L(L∞ (R+ ; X ), Cb (R+ ; (Y1 , Y2 )θ,∞ )) maps
T -periodic f to a T -periodic mild solution û. ST is given by
Z t
ST f (t) := e−At x +
e−A(t−s) Bf (s)ds, t > 0,
where Z
x = a+
0
T
−A(T −s)
e
∗
Bf (s)ds and a = w −lim
Z
k→∞ 0
0
(k−1)T
e −(kT −s)A Bf (s)ds.
solution may not be unique
uniqueness ⇔ 2nπi
T 6∈ σp (−A) for all n ∈ Z
Example : Stokes operator A on Lqσ (Ω) with Ω ⊂ Rn exterior domain
Then 0 ∈ σ(A) but 0 ∈
/ σp (A)
A different uniqueness condition : there exists X1 such that
lim ke −tA xkX1 = 0 for all x ∈ (Y1 , Y2 )θ,∞ .
t→∞
Let f be T /2 periodic. Then ST /2 f = ST f
The Semilinear Situation
Consider
u ′ (t) + Au = BG (u)(t),
u(0) = u0 ∈ (Y1 , Y2 )θ,∞ ,
A and B satisfy Assumption (A)
G maps Cb (R+ , (Y1 , Y2 )θ,∞ ) into L∞ (R+ , X )
mild solution to (SLCP) : u ∈ C ([0, T ); Y ) satisfying
Z t
u(t) = e −tA u(0) +
e −(t−τ )A BG (u)(τ )dτ.
0
For T -periodic function up ∈ Cb (R+ ; Y ) we set
BR (up ) := {w ∈ Cb (R+ ; (Y1 , Y2 )θ,∞ ) : w is T-periodic, kw −up kCb (R+ ;Y ) ≤ R}
Assumption (G).
G : Cb (R+ ; Y ) → L∞ (R+ ; X ) maps T -periodic functions to
T -periodic functions, G (u) ∈ L∞ (R+ , X ) and
kG (u) − G (v )kL∞ (R+ ;X ) ≤ Lku − v kCb (R+ ;Y ) ,
for suitable L, R > 0.
u, v ∈ BR (up ),
Main result for Semilinear Situation
Theorem : (semilinear)
Assume Assumptions (A) and (G). Let up ∈ Cb (R+ ; Y ) be a T -periodic
function. Suppose 2nπi
/ σp (−A) or the second uniqueness condition.
T ∈
There exists a unique T -periodic solution û of (SLCP) in BR (up ).
If G (u) and up are independent of t, then there exists a stationary
mild solution of (SLCP)
Sketch of Proof : Define
ΦT : Cb (R+ ; (Y1 , Y2 )θ,∞ ) → Cb (R+ ; (Y1 , Y2 )θ,∞) ,
where ST is the above solution operator of
v 7→ ST G (v ),
u ′ (t) + Au(t) = BG (v )(t), t > 0,
show : ΦT maps BR (up ) into itself and is a contraction on BR (up )
If G (u) and up t-independent, then G (u)(t) and up (t) T -periodic for
any T > 0
Consider (S2−n ). By above Remark S2−n = S2−m for any n, m ∈ N
By above proof : there is fixed point v of Φ2−n (v ) = v which is
2−n -periodic for all n ∈ N. Hence, v is independent of t.
Almost Periodic Functions
Above approach generalizes to situation of almost periodic functions.
A bounded function f : R+ → X is call almost periodic if for each
ε > 0 there exists Lǫ > 0 such that, for every a ∈ R+ there is
T ∈ [a, a + Lε ] such that
sup kf (t + T ) − f (t)kX < ε
t∈R+
existence of almost periodic solutins similar as above
uniqueness question : delicate
Application : Navier-Stokes Flow around Rotating Obstacle

ut + (u · ∇)u − ∆u + ∇p = (ω × x) · ∇u − ω × u + divF
in Ω × (0, ∞),



∇·u =0
in Ω × (0, ∞),
u =ω×x
on ∂Ω × (0, ∞



u(0) = u0 .
ω = (0, 0, a) angular velocity of obstacle D ⊂ R3 , Ω := R3 \D.
Set bω := − 12 ∇ × (ϕ(x)|x|ω) and
Lu:= −P [∆u + (ω × x) · ∇u − ω × u],
D(L) := {u ∈ Lpσ (Ω) ∩ W 2,p (Ω) : u|∂Ω = 0 and (ω × x) · ∇u ∈ Lp (Ω)},
rewrite original equation as equation for v := u − bω
v ′ (t) + Lv (t) = P div G (v )(t), t > 0, v (0) = v0 ,
G (v ) = F + Fω + (ω × x)∇bω + ∇bω + bω ⊗ v + v ⊗ v + bω ⊗ v + bω ⊗ bω
Geissert, Heck, H. :−L generates a bounded C0 -semigroup e −tL on
Lpσ (Ω) for p ∈ (1, ∞), which is, however, not analytic ;
Hishida, Shibata : global Lp − Lq and gradient smoothing estimates
use interpolation to transfer these estimates to (solenoidal) Lorentz
spaces
Result on Periodic solutions for flow past rotating obstacle
Theorem :
2
∞
3/2,∞
d
Suppose that F ∈ L (R+ ; L
(Ω) ) is T -periodic.
a) Assume that kG (0)kL∞ (R ;L3/2,∞ (Ω)) is small enough. Then there exists
+ σ
a mild T -periodic solution v̂ of our problem, which is unique in BR (0) =
≤ R} for suitable R
{u ∈ Cb (R+ ; L3,∞
3,∞
σ (Ω)) : kukC (R+ ;Lσ
(Ω))
b
b) If F is independent of t, then v̂ is a stationary solution
Semilinear Ornstein-Uhlenbeck equations
For exterior domain Ω ⊂ Rd consider the Ornstein-Uhlenbeck equation
ut − ∆u − Mx · ∇u = G (u)(t),
u = 0,
t > 0, x ∈ Ω,
t > 0, x ∈ ∂Ω,
u(0) = u0 ,
x ∈ Ω,
G (u) = |u|r −1 u + F for some r ∈ N
M ∈ Rd×d and T -periodic function F
For Lu(x) := ∆u(x) + Mx · ∇u(x) define
Ornstein-Uhlenbeck operators L on Lp (Ω) by
Lu := Lu,
D(L) := {u ∈ W 2,p (Ω) ∩ W01,p (Ω) : Mx · ∇u ∈ Lp (Ω)}
Then
u ′ (t) − Lu(t) = G (u)(t),
u(0) = u0
t > 0,
Periodic solutions to Ornstein-Uhlenbeck
Assumption (A) and (G) are satisfied for
B = Id
d(r−1)
X = L 2r (Ω)
Y1 := L
2d(r−1)
,∞
5−r
(Ω), Y2 := L
2d(r−1)
,∞
r+3
(Ω), Y = L
d(r−1)
,∞
2
(Ω)
Theorem B
Let 1 < r < 5 and d > 2. Assume M ∈ Rd×d satisfies tr M = 0 and
F ∈ L∞ (R+ ; X ) T -periodic. Then
the linear equation
u ′ (t) − Lu(t) = F (t),
t > 0,
admits a unique T -periodic mild solution u
If kF kL∞ (R+ ;X ) is small enough, then there exists a mild T -periodic
solution u to semilinear equation
uniqueness in BR (0) := {u ∈ Cb (R+ ; Y ) : kukCb (R+ ;Y ) ≤ R}, R small
If F independent of t, then u is a stationary solution of semilinear
equation.
Proof, Variations and Generalizations
Proof
Use Lp − Lq estimates for Ornstein-Uhlenbeck operators,
since B = Id , the result follows provided (2πni )/T are not
eigenvalues of L
Metafune’s result : 2πin ∈ σ(L)
maximum priciple implies : (2πin)/T ∈ σ(L) are not eigenvalues
Approach extends to more general settings :
ok, whenever e tL on L2 (Ω) satifies heat kernel bound, i.e. rough
diffusion, L = b∆, b ∈ L∞ , b > δ > 0, (Duong,Ouhabaz)
ok, if e tL is analytic on L2 (G ) for certain Lie groups G
uniqueness by Liouville type theorems on Lp (G ) due to Lanconelli et
al :
Lu = λu, Reλ ≥ 0 ⇒ u = 0
Then there exists periodic solutions for the linear and semilinear equations
in the framework presented.
Periodic solutions for non-small forces
Viscous incompressible fluids with rotation and stratification

3,

∂
v
+
(v
·
∇)v
=
∆v
−
Ωe
×
v
−
∇p
+
θe
+
g
t
∈
R,
x
∈
R
t
3
3

t ∈ R, x ∈ R3 ,
∂t θ + (v · ∇)θ = ∆θ − N 2 v3 + h


∇·v = 0
t ∈ R, x ∈ R3 ,
rewrite as evolution equation with µ = Ω/N
e · (u ⊗ u) = P∇
e · F, ∇
e · u = 0,
∂t u − ∆u + NPJµ Pu + P∇


0 −µ 0 0
where Jµ is defined by
 µ 0 0 0 

Jµ := 
 0 0 0 −1  .
0 0 1 0
n
o
|ξµ |
σ[−P(ξ)Jµ P(ξ)] = ± i
, 0, 0 , ξµ = (ξ1 , ξ2 , µξ3 )
|ξ|
The semigroup e −tLN generated by LN := −∆ + NPJµ P is given by
e
−tLN
ϕ=e
t∆ iNtOp(
e
|ξµ |
)
|ξ|
P+ ϕ + e
where PF = P+ F + P− F + P0 F .
t∆ −iNtOp(
e
|ξµ |
)
|ξ|
P− ϕ + e t∆ P0 ϕ,
Dispersive estimates
For 1 < p 6 q ′ 6 2 6 q < ∞, there exists C > 0 such that
t∆ iNtOp( |ξ|ξ|µ | ) f
e e
Lq
− 21
6 C (1 + Nt)
3 1
1
2
1− q − 2 p − q
t
kf kLp
for all t > 0, N > 0 and all f ∈ Lp (R3 ).
above approach yields the following result.
Theorem : (periodic solutions for large f if rotation is large) :
If p is large enough, there exist δ, K > 0 such that for every N > 0
and every T -periodic force F ∈ BC (R+ , Lp,∞ (R3 ) ∩ L3/2,∞ (R3 ))
satisfying
3 2
1
(
−
)
sup kPF (t)kLp,∞ ≤ δN 2 3 p
t>0
there exists a unique T -periodic solution to our problem in a ball of
size K .
Idea of how to obtain the dispersive estimate
suffices to consider e itNR1
Key Lemma : if q ∈ [2, ∞] and r ∈ [2, ∞), then
ke
itNR1
f kLqt Lrx
C
≤ 1/q kf kḢ 3/2−3/r
N
provided 1/q + 1/r ≤ 1/2.
loss of derivative s = 3/2 − 3/r is sharp. Further, no smoothing in
space variable
This follows from abstract result by Keel-Tao :
Let (U(t))t∈R be a family of operators with
kU(s)U(t)∗ f k∞ ≤ C (1 + |t − s|)−σ kf k1
kU(s)U(t)∗ f k2 ≤ C kf k2 .
Then kU(t)f kLqt Lrx ≤ C kf k2 for all q, r ∈ [2, ∞] with
(q, r , σ) 6= (2, ∞, 1) satisying 1/q + σ/r ≤ σ/2.
C
suffices thus to prove ke itNR1 f k∞ ≤ 1+Nt
kf k1
Strong Periodic Solutions for Arbitrary Large Forces by
Weak-Strong Uniqueness
Consider primitive equations, model for atmospheric dynamics
∂t v + u · ∇v − ∆v + ∇H π = f
in Ω × (0, T ),
∂z π = 0
in Ω × (0, T ),
div u = 0
in Ω × (0, T ),
v (0) = a.
Ω = G × (−h, 0), where G = (0, 1)2 and h > 0
u = (v , w ) with v = (v1 , v2 ), v and w denote horizontal and vertical
components of u
model goes back to Lions, Temam and Wang
breakthrough result by Cao and Titi in 2007 : unique global strong
solution for arbitrary large data a ∈ H 1
Periodic Solutions for Large Forces
Aims :
show existence of strong time-periodic solutions for arbitrary
(time-periodic) f ∈ L2 (0, T , L2 (Ω)), without assuming any smallness
condition on f
Consequence : analogous result for steady-state solutions
Our approach is based on three steps :
construct a suitable weak time-periodic solution v by combining
Galerkin’s method with Brouwer’s fixed point theorem.
show existence of a unique, strong solution u to the initial-value
problem for arbitrary f ∈ L2 (0; T ; L2 (Ω))
look at v as a weak solution to the initial-value problem, employ
weak-strong uniqueness argument : This yields v ≡ u
Weak and Strong Periodic Solutions
v is a weak T -periodic solution provided
v ∈ C (J; L2 (Ω)) ∩ L2 (J; H 1 (Ω)) is a weak solution
v satisfies strong energy inequality
Z t
Z t
kv (t)k22 + 2
k∇v (τ )k22 dτ ≤ kv (s)k22 + 2
(f (τ ), v (τ ))dτ
s
s
v (t + T ) = v (T ) for all t ≥ 0
A weak T -periodic solution v is strong if in addition
v ∈ C (J; H 1 (Ω)) ∩ L2 (J; H 2 (Ω))
Proposition : Let f ∈ L2 (J; L2 (Ω)) be T -periodic. Then there exists at
least one weak T -periodic solution v
Proof : Galerkin procedure and Brouwer’s fixed point theorem
Strong Periodic Solutions via Weak-Strong Uniqueness
Proposition 1 : Let f ∈ L2 (J; L2 (Ω)) be T -periodic. Then there exists
unique global strong solution u for arbitrary large a ∈ H 1 (Ω)
Proposition 2 : weak-strong uniqueness theorem : u = v
Idea of Proof :
◮ weak theory : there is t0 > 0 with v (t0 ) ∈ H 1
◮ take v (t0 ) as initial data for strong solution u
◮ take u as test function
◮ for w = v − u one has
Z t
Z t
[k∇H u(s)k42 + k∇H u(s)k22 kD 2 u(s)k22 ]kw (s)k22 ds
kw (t)k22 +
k∇w (s)k22 ds ≤ C
◮
t0
◮
◮
t0
blue term in L1 (t0 , t) due to regularity of strong solutions u
Gronwall : w = 0
Theorem : primitive equations admit a strong, periodic solution for
arbitrary large periodic f ∈ L2 (J, L2 )
Corollary : primitive equations admit a stationary solution for arbitrary
large f ∈ L2 (Ω)
The Bidomain Fitzhugh-Nagumo Model

∂t u + g (u, w ) − div (σi ∇ui ) = fi
in





in

 ∂t u + g (u, w ) + div (σe ∇ue ) = −fe
∂t w + h(u, w ) = 0
in



u = ui − ue
in




σi ∇ui · ν = 0, σe ∇ue · ν = 0 on
(0, ∞) × Ω,
(0, ∞) × Ω,
(0, ∞) × Ω,
(0, ∞) × Ω,
(0, ∞) × ∂Ω,
Assumptions : Ω ⊂ Rn bounded domain, ∂Ω smooth
coefficient matrices σ are C 1 (Ω), elliptic and ∇ui ,e · ν = 0 on ∂Ω.
define Ai ,e u := −div (σi ,e ∇u) with domain D(Ai ,e ) :=
u ∈ W 2,q (Ω) ∩ Lqav (Ω) : σi ,e ∇u · ν = 0 a.e. in ∂Ω ⊂ Lqav (Ω)
R
q
q
where Lav (Ω) := {u ∈ L (Ω) : Ω u dx = 0} and let Pav be the
orthogonal projection,
For 1 < q < ∞ define bidomain operator
A = Ai (Ai + Ae )−1 Ae Pav
with D(A) := {u ∈ W 2,q (Ω) : ∇u · ν = 0 a.e. on ∂Ω}
Nonlinear terms and abstract formulation
FitzHugh-Nagumo model :
1 3
g (u, w ) = [u − (a + 1)u 2 + au + w ],
ε
h(u, w ) = −(cu − bw ).
Abstract formulation of the periodic problem :

1 3
2


∂
u
+
εAu
=
f
−
[u
−
(a
+
1)u
+ au + w ]
t


ε


∂t w = cu − bw


u(t) = u(t + T )




w (t) = w (t + T )
in R × Ω,
in R × Ω,
in R × Ω,
in R × Ω.
Giga, Kajiwara ’16 : A is sectorial of angle 0 on Lp (Ω), 1 < p ≤ ∞,
0 ∈ σ(A)
aim : periodic solution in space of maximal regularity
choose real interpolation space DA (θ, p)
The Da Prato-Grisvard Theorem in the periodic setting
Let θ ∈ (0, 1), 1 ≤ p < ∞, 0 < T < ∞ and f : R → DA (θ, p) be periodic
of period T . Consider
(
u ′ (t) + Au(t) = f (t), t ∈ R,
u(t) = u(t + T ),
candidate for solution u is
u(t) :=
Z
t
t ∈ R.
e −(t−s)A f (s)ds.
−∞
Theorem : Let e −tA be a bounded analytic semigroup on a Banach
space X , 0 ∈ ̺(A).
Then there exists C > 0 such that for all periodic functions
f : R → DA (θ, p) with f|(0,T ) ∈ Lp (0, T ; DA (θ, p)),
u ∈ C (R; DA (θ, p)), u is periodic of period T , u(t) ∈ D(A) for
almost every t ∈ R and satisfies
kAukLp (0,T ;DA (θ,p)) ≤ C kf kLp (0,T ;DA (θ,p)) .
Semilinear version
periodic Da Prato Grisvard Theorem remains true in semilinear
setting for small f , i.e.
(
u ′ (t) + Au(t) = F [u](t) + f (t), t ∈ R,
u(t) = u(t + T ),
t ∈ R,
provided F ∈ C 1 (BR , Lp (0, T ); DA (θ, p)), F (0) = 0, DF (0) = 0.
Back to bidomain FitzHugh-Nagumo Equation
three equilibrium points (u1 , w1 ) = (0, 0), (u2 , w2 ), (u3 , w3 )
change of variables
v
u − ui
:=
, i = 1, 2, 3
z
w − wi
this yields rescaled equation
v
εA + ε1 [3ui2 − 2(a + 1)ui + a]
∂t
+
−c
z
3
2
1
v
f − ε [v + (3ui − a − 1)v ]
,
=
0
b
z
1
ε
v (t) = v (t + T ),
z(t) = z(t + T ).
for (u1 , w1 ) = (0, 0) we have 0 ∈ ̺(εA + a/ε)
Then also 0 ∈ ̺(A1 ), where A1 denotes linear part of coupled
equation
1
εA + a/ε ε
A1 :=
−c
b
provided c, b > 0 and we may apply the periodic semilinear Da Prato
Grisvard theorem.
Results
Bidomain FitzHugh-Nagumo Equation admits a strong periodic
solution in E = {u ∈ W 1,p (0, T ; DA (θ, p)) : Au ∈ Lp (0, T ; DA (θ, p))}
for small f and for those parameters a, b, c, which yield stable
equilibrium points.
similar results are true for Allen-Cahn model and related
Aliev-Panfilov and McCulloch models