Proof of Theorem 3.2

3.2 Well-posedness of the System

3.2.1 Proof of Theorem 3.2

The proof of Theorem 3.2 will be given in steps. Firstly, we observe that in (3.15)-(3.17), the initial smoothness of solutions are due to the fact that the analytic semigroup (1.3) is also aC⁰−semigroup, and hence [67, 68] yield the assertions using (3.7).

We also have, using the second integral formula in (3.7), and the estimate in (1.6) for

γ =α if σ = 1, or for 0≤γ−α < _σ¹0 if 1< σ≤ ∞, that kv(t)k_γ ≤ ke^(∆−λ)tv₀k_γ+a

Z t 0

ke^{(∆−λ)(t−s)}u(s)k_{γ ds}

≤ M t^−(γ−α)kv₀k_α+M a Z t

0

(t−s)^−(γ−α)ku(s)k_α ds

≤ M

"

t^−(γ−α)kv₀k_α+a Z t

0

(t−s)^{−σ0(γ−α)}ds

_σ¹0 Z t 0

ku(s)k^σ_αds _σ¹#

≤ M

"

t^−(γ−α)kv₀k_α+a

1 1−σ⁰(γ−α)

_σ¹0

t^{σ10−(γ−α)}ku(t)k_Lσ(0,T ,E_q^α)

# ,

where the second inequality above is obtained from (1.6) and the third one from Hölder's inequality. This means that v(t) is bounded on nite intervals away from t = 0, and v(t)∈E_q^γ fort >0.

In particular,

|kvk|_γ,γ−α = sup

t∈(0,T]

t^γ−αkv(t)k_γ

≤ sup

t∈(0,T]

t^γ−αMh

t^−(γ−α)kv₀k_α+t^{σ10−(γ−α)}ku(t)k_Lσ(0,T ,E_q^α)

i

= sup

t∈(0,T]

Mh

kv₀k_α+t^σ10ku(t)k_Lσ(0,T ,E^α_q)

i

≤ C

kv₀k_α+kuk_Lσ((0,T),E^α_q)

,

for some C >0, which proves that v∈ L^∞_γ−α( ˙I, E^γq).

To prove continuity, x t >0 (or event= 0 if v₀ ∈Eq^γ),h >0. Then, from the second formula in (3.7) we compute

v(t+h)−v(t) =e^(∆−λ)hv(t)−v(t) +a Z t+h

t

e(∆−λ)(t+h−s)u(s)ds,

so that, when we take the norm, (1.6) gives

kv(t+h)−v(t)k_γ ≤ ke^(∆−λ)hv(t)−v(t)k_γ +aM

Z t+h t

(t+h−s)^−(γ−α)ku(s)k_αds.

(3.22)

Now, as h → 0, e^(∆−λ)hv(t) → v(t), and thus the rst term goes to zero. By the Hölder's inequality, the second term is bounded by

aM

Z t+h t

(t+h−s)^{−σ0(γ−α)}ds

1

σ0 Z t+h t

ku(s)k^σ_αds

1 σ

≤M_1−σ⁰_(γ−α)kuk_σ,αh^{σ10−(γ−α)}, which also goes to zero ash→0. Hence, continuity of thev−solution component follows.

Next, for any α, γ ∈Rsuch thatα≤γ ≤α+_σ¹0, if we let

cα,γ(t) =e^(∆−λ)t∈L¹(0,∞),

then it is not bounded att= 0, unlessα=γ. Also, if σ= 1, we let

φ(t) = Z _t

0

e^{(∆−λ)(t−s)}u(s)ds,

then, since

e^{(∆−λ)(t−s)}v0 ∈L¹(0,∞;E_q^γ),

provided thatv₀ ∈Eq^γ, we only need to prove that φ(t)∈L¹(0,∞;Eq^γ). Thus, ifs=tρ for ρ∈[0,1] xed, then we get

kφ(t)k_1,γ ≤ Z t

0

kφ(t)k_1,γ dρ= Z t

0

Z ∞ 0

e(∆−λ)t(1−ρ)u(tρ) γ dt

≤ Z _t

0

Z ∞ 0

r

ρ²c_α,γ(r(1−ρ)ρ⁻¹)ku(r)k_α drdρ

≤

Z ∞ 0

c_α,γ(s)ds

Z ∞ 0

ku(r)k_αdr

,

following other changes to the time variables r = tρ, s = r_1−ρ

ρ

, then integrating with respect to ρ. It then follows that

kv(t)k_1,γ ≤ kc_γ,γ(t)k₁kv₀k_γ+kc_α,γ(s)k₁ku(r)k_1,α. Note that forσ =∞ the second term in (3.22) is bounded by

aM

ku(t)k_L∞([0,T],E_q^α)

Z t+h t

(t+h−s)^−(γ−α)ds≤M_1−σ⁰_(γ−α)kuk_∞,αh^{1−(γ−α)}. If0≤γ−α≤1, then we have continuity. The rest follows by using interpolation, thus, the second from last result in (3.16) is valid.

Furthermore, we note that if we apply ∇ to the second equation in (3.7) and take the norm inγ− ¹₂, we get

k∇vk_γ−1 2

≤ k∇

e^(∆−λ)tv0

k_γ−1

2 +a Z t

0

k∇

e^{(∆−λ)(t−s)}u(s)

k_γ−1 2 ds

≤M t^−(γ−α)kv₀k_α+M a Z t

0

(t−s)^−(γ−α)ku(s)k_αds

≤M

"

t^−(γ−α)kv₀k_α+a Z t

0

(t−s)^{−σ0(γ−α)}ds

_σ¹0 Z t 0

ku(s)k^σ_α ds ¹_σ#

≤M

"

t^−(γ−α)kv₀k_α+a

1 1−σ⁰(γ−α)

¹

σ0

t^{σ10−(γ−α)}ku(t)k_Lσ(0,T ,E^α_q)

# ,

(3.23)

since (3.8) is assumed so that if α ≥ _2q^N, then γ − ¹₂ ≥ α. This implies that ∇v ∈ L^∞_γ−α(0,∞;E^γ−

1

q 2).

If we then consider the rst formula in (3.7) and seth(t) =t^γ−αk∇vk_γ−1

2, then we have ku(t)k_γ ≤ M t^−(γ−β)ku₀k_β+

Z _t

0

k∇e^∆(t−s)(uχ∇v)(s)k_γ ds

≤ M t^−(γ−β)ku₀k_β+χM Z t

0

(t−s)^{−12−(γ−α)}k(u∇v)(s)k_αds

≤ M t^−(γ−β)ku₀k_β +χM 2

N eπ

γ+^β₂−¹₂

×

× Z t

0

(t−s)^{−12−(γ−α)}ku(s)k_βk∇v(s)k_γ−1 2 ds

≤ M t^−(γ−β)ku₀k_β +χM 2

N eπ

γ+^β₂−¹

2

×

× Z t

0

(t−s)^{−12−(γ−α)}h(s)s^−(γ−α)ku(s)k_β ds

= M t^−(γ−β)ku₀k_β +χM 2

N eπ

γ+^β₂−¹

2

Φ, whereΦ =Rt

0(t−s)^{−12−(γ−α)}h(s)s^−(γ−α)ku(s)k_β ds. If we then make the changes=ρtin the time variable, then we see that

Φ≤sup

t>0

h(t) Z 1

0

t^{−σ0(12+γ−α)}t^{−σ0(γ−α)} 1

(1−ρ)^{σ0(12+γ−α)}ρ^σ0(γ−α)

! dρ

!¹

σ0

kuk_σ,α

≤t^{−(12+2(γ−α))}sup

t>0

h(t) Z 1

0

1

(1−ρ)^{σ0(12+γ−α)}ρ^σ0(γ−α) dρ

! kuk_σ,α

≤t^{−(12+(γ−α))}sup

t>0

k∇vk_γ−1 2

Z 1 0

1

(1−ρ)^{σ0(12+γ−α)}ρ^σ0(γ−α) dρ

!

kuk_σ,α.

(3.24)

It then follows by a backward substitution into (3.24) that lim sup

t→∞

kuk_γ = 0. If we then allow the exponential decay eect of the semigroup (1.4) in the norm estimates of (3.23) and letσ =∞, we conclude that the last statement in (3.16) is valid. In fact:

lim sup

t→∞

k∇vk_γ−1 2

≤aM Z ∞

0

e^−ωt t^γ−αdt

lim sup

t→∞

kuk_α= 0, from which the result follows.

To complete the proof of (i), we need some extra results on (ii).

Lemma 3.3. Let u ∈ Eq^β be as given in (3.7). Then u ∈ C_loc^ξ (0, T;E_q^α) is of exponent ξ=γ−α∈(0,1). That is, it is Hölder continuous in time.

Proof. Let0< t < t+h < T. Then we have that u(t+h)−u(t) = (e^∆h−I)e^∆tu0+

Z t 0

(e^∆h−I)e^∆(t−s)∇(u(s)χ∇v(s))ds+

+ Z t+h

t

e^∆(t+h−s)∇(u(s)χ∇v(s))ds,

Now, by virtue of Lemma 3.1, taking β =α, and taking the norm of E_q^α on both sides, and estimating, we get that

ku(t+h)−u(t)k_α ≤ k(e^∆h−I)e^∆tu₀k_α+ Z t

0

k(e^∆h−I)e^∆(t−s)∇(u(s)χ∇v(s))k_αds+

+ Z _t+h

t

ke^∆(t+h−s)∇(u(s)χ∇v(s))k_αds

≤Mγ−αh^γ−α e^∆tu0

γ+Mγ−αh^γ−α+ Z t

0

k∇(u(s)χ∇v(s))k_γds+

+ Z t+h

t

ke^∆(t+h−s)∇(u(s)χ∇v(s))k_γds

≤Mγ−αM h^γ−αt^−(γ−α)ku₀k_α+χMγ−αM h^γ−α Z t

0

(t−s)^{−12−(γ−α)}×

× ku(s)∇v(s)k_αds+χM Z t+h

t

(t+h−s)^{−12−(γ−α)}ku(s)∇v(s)k_αds

≤Mγ−αM h^γ−αt^−(γ−α)ku₀k_α+χM 2

N eπ

γ+^α₂−¹₂

Mγ−αh^γ−α×

× Z t

0

(t−s)^{−12−(γ−α)}× ku(s)k_αk∇v(s)k_γ−1 2 ds+

+χM 2

N eπ

γ+^α₂−¹₂ Z t+h t

(t+h−s)^{−12−(γ−α)}ku(s)k_αk∇v(s)k_γ−1 2 ds

≤ Mγ−αM t^−(γ−α)ku₀k_α+χ 2

N eπ

γ+^α₂−¹₂

Mγ−αM_{1−(γ−α)}t^{1−(γ−α)}+ +χ

2 N eπ

γ+^α₂−¹₂

M_{1−(γ−α)} sup

t∈(0,T)

nku(s)k_αk∇v(s)k_γ−1 2

o

! h^γ−α, which thus furnishes the desired Hölder continuity of theu−integral solution form in (3.7), and thus the proof of the lemma is complete.

Lemma 3.4. Consider the set W :=

(

ψ∈C(I;E_q^γ) : sup

t∈(0,T)

kψ(t)k_γ ≤Ckψ₀k_β )

, (3.25)

and set the left-hand side of the rst equation in (3.7) to F(u)(t). Then (i) F(W)⊂W. That is, F mapsW onto itself.

(ii) The mapping F :E_q^α→Eq^γ is a contraction.

(iii) There exists a unique u ∈ W such that F(u)(t) = u(t) is a solution to (3.1) up to maximal timeT^∗(ku₀k_β) of existence of solutions of (3.2).

Proof. We rst note that we can read the right-hand side of (3.7) in taking the norm of E_q^γ =E_q^β×E_q^γ−β as in the scale spaces product, whereas, by virtue of Lemma3.1, we have thatuχ∇v is well dened in E_q⁰ ∼=L^q(Ω). Therefore, if u∈W, then we nd that

kF(u)(t)k_γ≤Mku₀k_β+M Z t

0

(t−s)^{−12−(γ−β)}kuχ∇vk₀ds

≤Mku₀k_β+χ 2

N eπ

γ+^β₂−¹

2

M Z t

0

(t−s)^{−12−(γ−β)}kuk_βk∇vk_γ−1 2 ds

≤Mku₀k_β+χM C 2

N eπ

γ+^β₂−¹₂

sup

t∈(0,T)

k∇vk_γ−1 2

ku₀k_β×

× Z t

0

(t−s)^{−12−(γ−β)}ds

≤Mku₀k_β+χM C 2

N eπ

γ+^β₂−¹₂

sup

t∈(0,T)

k∇vk_γ−1 2

ku₀k_βT^{−12−(γ−β)}. Thus, for

T = 1

M − 1 C

1

χsup_t∈(0,T)k∇vk_γ−1 2

2 N eπ

^1−2γ−β₂ !_{1−2(γ−β)}² , we obtain that (i) is satised.

To prove (ii), we let u1, u2 ∈W. Then, for 0≤t≤T and using the same initial data, we have that

kF(u₁)(t)− F(u₂)(t)k_γ ≤ Z t

t0

ke^∆(t−s)∇(u₁χ∇v−u₂χ∇v)k_γ ds

≤ M Z t

0

(t−s)^{−12−(γ−β)}k(u₁−u2)χ∇vk_βds

≤ χM 2

N eπ

γ+^β₂−¹₂Z t 0

(t−s)^{−12−(γ−β)}ku₁−u₂k_βk∇vk_γ−1 2 ds

≤ χM 2

N eπ

γ+^β₂−¹₂

T^{12−(γ−β)} sup

t∈(0,T)

k∇vk_γ−1

2 sup

t∈(0,T)

ku₁−u2k_β, so that for

T < 1

χsup_t∈(0,T)k∇vk_γ−1 2

2 N eπ

^1−2γ−β₂ !_{1−2(γ−β)}² , we have that F is a contraction onW.

Thus, viewed together with (i) of this lemma, by the Banach Contraction Mapping Theorem (Theorem 1.1) and Picard's method, or classical continuation allows the extension of the nite existence time to maximal time T^∗ =T(ku₀k_β), yielding the last assertion of the lemma.

It now remains to prove (iii) of the theorem. For this, we observe on the smoothness of solutions as given in the theorem, that (3.15) follows by [30, 67, 68]. Since (3.3) is a C⁺ operator, and Lemma 3.3 holds, u(t) ∈L^σ( ˙I;E_q^α) is Hölder continuous. Consequently, linear non-homogeneous evolution equation results imply the time regularity of the solution component with evenT at∞. In a similar manner, writing the weak form (3.11) as

f(t) =h(uχ∇v)(t),∇ϕi_{q, q}⁰ for any ϕ∈E_q^γ0, (3.26) we conclude that (3.17) also holds, because ∇v ∈ E^γ−

1

q 2 is bounded, and by Lemma 3.3, u ∈ C^ξ( ˙I;E_q^α) for 0 ≤ ξ = γ−α < 1, as a result, imply that (3.26) is Hölder continuous in time. We therefore get by Lemma 3.2.1 and Theorem 3.2.2 in [30] the existence and uniqueness of the solution to (3.1)-(3.2). The converse to the fact that the solution is given by (3.7) is given by Denition 3.1.

To prove the generation of a perturbed analytic semigroup, we have the following lemma.

Lemma 3.5. The operator in (3.19) is an innitesimal generator of a perturbed analytic semigroup in scale spaces Zq^β+α, and the strong solution of the theorem coincides with that generated by (3.19).

Proof. We rstly observe from what has been proven up to now that v ∈ L^∞_γ−α(0, T;Eq^γ) and lim sup

t→∞ t^γ−αkv(t)k_γ ≤ Mkv₀k_α using (3.24) with σ = ∞, while still with (3.24), we obtain lim sup

t→∞

t^α−βku(t)k_α ≤Mku₀k_β, and the assertion should follow. More precisely, to complete ideas, we prove that (3.19) is well dened, continuously, coercive, strictly monotone and is a sectorial operator inE_q⁰ ∼=L^q(Ω).

To this end, deneb:Zq^β+α×Zq^β+α7→Rby b(U,Υ) =

Z

Ω

∇v∇ϕ+λ Z

Ω

vϕ+ Z

Ω

∇u∇ψ−χ Z

Ω

u∇v∇ψ−a Z

Ω

uϕ, (3.27)

where Υ = (ϕ, ψ)^>, and note that since Lemma 3.1-(3.8) is assumed, continuity of the mapping (3.27) is clear. We therefore need to prove only the coercivity, (since to apply Browder-Minty theorem strictly monotonicity can be easily deduced).

Thus, taking Υ =U, we nd that b(U, U)≥ k∇vk²_α−1

2

+k∇uk²_β−1 2

−χ 2

N eπ

α+^β₂−¹

2

kuk_βk∇vk_α−1 2

k∇uk_β−1 2+ +λkvk²_α− a

2qkuk²_β− a 2qkvk²_α

≥ 1− χ 2q

2 N eπ

α+^β₂−¹₂!

k∇vk²_α−1 2

+ 1− χ q

2 N eπ

α+^β₂−¹₂

− a 2q

!

×

× k∇uk²_β−1 2

+

λ− a 2q

kvk²_α

≥ 1− χ 2q

2 N eπ

α+^β₂−¹

2

+

λ− a 2q

!

k∇vk²_α−1 2 + + 1−χ

q 2

N eπ

α+^β₂−¹

2

− a 2q

!

k∇uk_β−1 2

≥ 1−χ q

2 N eπ

α+^β₂−¹₂

− a 2q

!

kUk²_β+α,

(3.28)

implying the coercivity of (3.27), using (3.18). Thus, (3.19) is uniquely invertible using Browder-Minty's theorem, and is a sectorial operator inE_q⁰ ∼=L^q(Ω), since

(A+µ)^−γP˜

0 = sup

kUk≤1







(A+µ)^−γP˜(U) 0

kUk₀







≤ C

µ^γ(a+kPk_γ,0)

≤ C µ^γ a+

2 N eeπ

γ−¹

2

! ,

for any0≤γ <1satisfying Lemma 3.1-(3.8), for someC ∈R⁺\{0}, |π−argµ| ≥ϑ, ϑ < ^π₂, and the conclusion of the lemma is obtained using Corollary 1.4.5 in [30]. Clearly, (3.21) and (3.16) imply (3.20). So, the proof of the lemma is complete.

To complete the proof of part (iv) of the theorem, if suces to note that sinceγ−¹₂ ≥ _2q^N, we haveE^γ−

1

q 2 ⊂L^∞(Ω)by virtue of (3.5), and Theorem 3.2-(3.16) imply that∇v∈L^∞(Ω) is bounded for allt >0. As u∈E_q⁰ ∼=L^q(Ω), q > ^N₂ and 1≥γ−¹₂ > ^N_2q, viewing the weak form (3.26) in L^q as well as the equation in elliptic form by passing ut to the right hand side, using [67], we getu∈L^∞(Ω)is bounded for allt >0. The rest is trivial or immediate.

Thus, Theorem 3.2 has been established.