Dissipative Models in Phase Transitions

Cortona, September 6–11, 2004

Mathematical study of a phase transition model

Gianni Gilardi

Gianni Gilardi

Dipartimento di Matematica “F. Casorati”

Università degli Studi di Pavia Via Ferrata 1, 27100 Pavia, Italy

Tel:  +39 0382 50 56 42
Fax:  +39 0382 50 56 02
E-mail:  gilardi@dimat.unipv.it
Home page: http://www-dimat.unipv.it/˜ gilardi/

Joint work (in progress) with E. Bonetti, P. Colli, and M. Fabrizio.

Physical variables and data (x ∈ Ω ⊂ ℝ^{3} , t ∈ (0, T))

ϑ = ϑ(x, t) absolute temperature
χ = χ(x, t) phase parameter
q = q(x, t) entropy flux
g = g(x, t) given source

2 The model Mathematical model

• entropy balance

∂_{t}(ln ϑ − λ(χ)) + div q = g

λ smooth function

• some constitutive law for q (memory)
• phase dynamics (micro-movements, micro-forces)

μ∂_{t}χ − νΔχ + W′(χ) = −λ′(χ) ϑ

μ, ν > 0, W = generalized double well

• boundary and initial conditions

3 References

For entropy balance, micro-movements, micro-forces • lectures by Frémond and Miranville

• Bonetti–Frémond, MMAS (2003)

• Bonetti–Colli–Frémond, M^{3}AS (2003)

• Frémond, book, Springer, 2001

• Gurtin, Physica D (1996)

• Miranville–Schimperna, to appear

However, phase dynamics is a generalized Allen-Cahn equation

• very wide literature 4 Good model

Such a model is a good one. Indeed

• Second principle of thermodynamics fulfilled • ϑ > 0 even with memory effects 5 Plan The talk will proceed as follows

• More careful description of the system from the mathematical point of view (assumptions on the structure)

• Results (rough statements)

• Existence proof (sketch)

• if some time remains… Plan 5 6 Structure Constitutive law for q

q = −k_{0}∇ϑ − ∫_{−∞}^{t} k(t − s)∇ϑ(x, s) ds k_{0} > 0, k smooth memory kernel

Writing ∫_{−∞}^{t} = ∫_{−∞}^{0} + ∫_{0}^{t} the e.b. becomes

∂_{t}(ln ϑ − λ(χ)) − Δ(k_{0}ϑ + k ∗ ϑ) = R where (a ∗ b)(t) := ∫_{0}^{t} a(t − s) b(s) ds and R (known) accounts for ϑ|_{t<0} Structure 6 7 Structure Phase dynamics Generalized multiple well potential

W(χ) = β̂(χ) + σ(χ) irregular convex part + smooth part

Then, in general, β̂′ becomes a subdifferential

β := ∂β̂

and “=” becomes “∋”

∂_{t}χ − Δχ + β(χ) + σ′(χ) ∋ −λ′(χ) ϑ

(μ = ν = 1 w.l.o.g.)

Assumptions on the structure k_{0} > 0 and k ∈ W^{1,1}(0, T)

β̂: ℝ → [0, +∞]

is convex, proper, l.s.c., β̂(0) = 0 λ, σ : ℝ → ℝ of class C^{2}

growth conditions: lim_{|r|→+∞} |r|^{−2} β̂(r) = +∞ (empty if D(β̂) bdd) λ′ and σ′ Lipschitz (⇒ λ, σ quadratic)

Structure 8 Boundary conditions Dirichlet for ϑ and homogeneous Neumann for χ

ϑ = ϑ_{Γ} and ∂_{n}χ = 0 on Γ = ∂Ω

where ϑ_{Γ} is given, smooth, with inf ϑ_{Γ} > 0

Initial conditions For both ϑ and χ

ϑ(0) = ϑ_{0}  and  χ(0) = χ_{0}

where ϑ_{0}, χ_{0} are given, smooth, with inf ϑ_{0} > 0 10 Summary Formal problem

∂_{t}(ln ϑ − λ(χ)) − Δ(k_{0}ϑ + k ∗ ϑ) = R ∂_{t}χ − Δχ + β(χ) + σ′(χ) ∋ −λ′(χ) ϑ ϑ = ϑ_{Γ} and ∂_{n}χ = 0 on Γ ϑ(0) = ϑ_{0} and χ(0) = χ_{0} 10 Summary Formal problem

∂_{t}(ln ϑ − λ(χ)) − Δ(k_{0}ϑ + k ∗ ϑ) = R

∂_{t}χ − Δχ + β(χ) + σ′(χ) ∋ −λ′(χ) ϑ
ϑ = ϑ_{Γ} and ∂_{n}χ = 0 on Γ
ϑ(0) = ϑ_{0} and χ(0) = χ_{0}

Results

• well posedness

Formal problem

∂_{t}(ln ϑ − λ(χ)) − Δ(k_{0}ϑ + k ∗ ϑ) = R
∂_{t}χ − Δχ + β(χ) + σ′(χ) ∋ −λ′(χ) ϑ
ϑ = ϑ_{Γ} and ∂_{n}χ = 0 on Γ
ϑ(0) = ϑ_{0} and χ(0) = χ_{0}

Results

• well posedness

• continuous dependence

∂_{t}(ln ϑ − λ(χ)) − Δ(k_{0}ϑ + k ∗ ϑ) = R
∂_{t}χ − Δχ + β(χ) + σ′(χ) ∋ −λ′(χ) ϑ
ϑ = ϑ_{Γ} and ∂_{n}χ = 0 on Γ
ϑ(0) = ϑ_{0} and χ(0) = χ_{0}

Results

• well posedness

• continuous dependence

• some more regularity 10 Summary

Formal problem

∂_{t}(ln ϑ − λ(χ)) − Δ(k_{0}ϑ + k ∗ ϑ) = R
∂_{t}χ − Δχ + β(χ) + σ′(χ) ∋ −λ′(χ) ϑ
ϑ = ϑ_{Γ} and ∂_{n}χ = 0 on Γ
ϑ(0) = ϑ_{0} and χ(0) = χ_{0}

Results

• well posedness

• continuous dependence

• some more regularity

• long time behaviour ϑ ∈ L^{2}(0, T; V) and ϑ > 0 a.e. in Q ln ϑ ∈ L^{∞}(0, T; H) ∩ H^{1}(0, T; H^{−1}(Ω)) χ ∈ L^{2}(0, T; H^{2}(Ω)) ∩ H^{1}(0, T; H) ξ ∈ L^{2}(Q) ϑ ∈ L^{2}(0, T; V) and ϑ > 0 a.e. in Q ln ϑ ∈ L^{∞}(0, T; H) ∩ H^{1}(0, T; H^{−1}(Ω)) χ ∈ L^{2}(0, T; H^{2}(Ω)) ∩ H^{1}(0, T; H) ξ ∈ L^{2}(Q)

• Q := Ω × (0, T), V := H^{1}(Ω), H := L^{2}(Ω) ϑ ∈ L^{2}(0, T; V) and ϑ > 0 a.e. in Q ln ϑ ∈ L^{∞}(0, T; H) ∩ H^{1}(0, T; H^{−1}(Ω)) χ ∈ L^{2}(0, T; H^{2}(Ω)) ∩ H^{1}(0, T; H) ξ ∈ L^{2}(Q)

• Q := Ω × (0, T), V := H^{1}(Ω), H := L^{2}(Ω) ∂_{t}(ln ϑ − λ(χ)) − Δ(k_{0}ϑ + k ∗ ϑ) = R in H^{−1} ∂_{t}χ − Δχ + ξ + σ′(χ) = −λ′(χ)ϑ a.e. in Q ξ ∈ β(χ) a.e. in Q ϑ = ϑ_{Γ} and ∂_{n}χ = 0 on Γ (ln ϑ)(0) = ln ϑ_{0} and χ(0) = χ_{0} 12 Results

Theorem 1. Assume k_{0} > 0 and

• the above smoothness of k, β̂, λ, σ

• the above growth conditions on β̂, λ, σ

• smooth R, ϑ_{Γ} , ϑ_{0} , χ_{0} . Then, there exists a unique solution.

12 Results

Theorem 1. Assume k_{0} > 0 and

• the above smoothness of k, β̂, λ, σ

• the above growth conditions on β̂, λ, σ

• smooth R, ϑ_{Γ} , ϑ_{0} , χ_{0} . Then, there exists a unique solution.

Theorem 2. The solution continuously depends on the data in the appropriate topologies.

12 Results

Theorem 1. Assume k_{0} > 0 and

• the above smoothness of k, β̂, λ, σ

• the above growth conditions on β̂, λ, σ

• smooth R, ϑ_{Γ} , ϑ_{0} , χ_{0} . Then, there exists a unique solution.

Theorem 2. The solution continuously depends on the data in the appropriate topologies.

• Moreover, an error estimate holds.

13 Results Theorem 3. If data are a little smoother, then the solution satisfies

|χ|^{r} ∈ L^{2}(0, T; V) and χ ∈ L^{∞}(0, T; L^{r}(Ω)) ϑ^{p} ∈ L^{2}(0, T; V) and ϑ ∈ L^{∞}(0, T; L^{2p−1}(Ω))

for 1 ≤ r < ∞ and 1 ≤ p < 2.

Theorem 3. If data are a little smoother, then the solution satisfies

|χ|^{r} ∈ L^{2}(0, T; V) and χ ∈ L^{∞}(0, T; L^{r}(Ω))

ϑ^{p} ∈ L^{2}(0, T; V) and ϑ ∈ L^{∞}(0, T; L^{2p−1}(Ω))

for 1 ≤ r < ∞ and 1 ≤ p < 2.

• Even further regularity could be proved (derivatives: ∂_{t}(ph.d.))

14 Results Long time behaviour (in progress) Assume k “small” as t → +∞ and set

k_{∞} := ∫_{0}^{+∞} k(s) ds

Assume k to be “generalized positive”, i.e.,

\begin{matrix}∫_{0}^{t} (k_{0}v^{2}(s) + (k ∗ v)(s) v(s)) ds \\ ≥ α ∫_{0}^{t} v^{2}(s) ds (α > 0, ∀ t, ∀ v) \end{matrix}

14 Results Long time behaviour (in progress) Assume k “small” as t → +∞ and set

k_{∞} := ∫_{0}^{+∞} k(s) ds

Assume k to be “generalized positive”, i.e., ∫_{0}^{t} (k_{0}v^{2}(s) + (k ∗ v)(s) v(s)) ds

≥ α ∫_{0}^{t} v^{2}(s) ds (α > 0, ∀ t, ∀ v)
• This implies
k_{0} + k_{∞} ≥ α, hence k_{0} + k_{∞} > 0

Results 14 15 Results

Theorem 4. Assume

• the above conditions on k

• ϑ_{Γ} and R suitably reach asymptotic values ϑ_{Γ,∞} and R_{∞}

• R_{∞} ≥ 0 (or a weaker assumption) Then the ω − limit of any trajectory is non empty and made just of solutions (ϑ_{∞}, χ_{∞}) of

− (k_{0} + k_{∞})Δϑ_{∞} = R_{∞}
− Δχ_{∞} + β(χ_{∞}) + σ′(χ_{∞}) ∋ −λ′(χ_{∞})ϑ_{∞}
ϑ_{∞} = ϑ_{Γ,∞} and ∂_{n}χ_{∞} = 0 on Γ

• Standard method: regularization, a priori estimates, convergence using compactness and monotonicity • Standard method:

regularization, a priori estimates, convergence using compactness and monotonicity

Regularization

\begin{aligned}ln ϑ & ≈ Ln_{ε}(ϑ_{ε}) \\ β(χ) & ≈ β_{ε}(χ_{ε}) \\ λ(χ) & ≈ λ_{ε}(χ_{ε}) \\ ϑ_{0} & ≈ ϑ_{0,ε} \end{aligned}

17 Existence

Regularized problem ∂_{t}(Ln_{ε}(ϑ_{ε}) − λ_{ε}(χ_{ε})) − Δ(k_{0}ϑ_{ε} + k ∗ ϑ_{ε}) = R ∂_{t}χ_{ε} − Δχ_{ε} + β_{ε}(χ_{ε}) + σ′(χ_{ε}) = −λ′_{ε}(χ_{ε})ϑ_{ε} ϑ_{ε} = ϑ_{Γ} and ∂_{n}χ_{ε} = 0 on Γ ϑ_{ε}(0) = ϑ_{0,ε} and χ_{ε}(0) = χ_{0}.

ln_{ε}, β_{ε} = Yosida regularizations Ln_{ε}(r) := εr + ln_{ε}(r) λ_{ε} Lipschitz regularization of λ ϑ_{0,ε} regularized initial temperature 18 Existence

Regularized problem ∂_{t}(Ln_{ε}(ϑ_{ε}) − λ_{ε}(χ_{ε})) − Δ(k_{0}ϑ_{ε} + k ∗ ϑ_{ε}) = R ∂_{t}χ_{ε} − Δχ_{ε} + β_{ε}(χ_{ε}) + σ′(χ_{ε}) = −λ′_{ε}(χ_{ε})ϑ_{ε} ϑ_{ε} = ϑ_{Γ} and ∂_{n}χ_{ε} = 0 on Γ ϑ_{ε}(0) = ϑ_{0,ε} and χ_{ε}(0) = χ_{0}.

Simplifying trick: replace 1st eq’n and 1st b.c. by ∂_{t}(Ln_{ε}(ϑ_{ε}) − λ_{ε}(χ_{ε})) − Δ(k_{0}u_{ε} + k ∗ u_{ε}) = R u_{ε} := ϑ_{ε} − ϑ_{H} vanishes on Γ ϑ_{H} := harmonic extension of ϑ_{Γ}.

• what needed for ϑ_{H} becomes HP on ϑ_{Γ} 19 Existence Technical troubles

• Regularized problem: not obvious it has a solution Technical troubles

• Regularized problem: not obvious it has a solution, hence Faedo-Galerkin discretization Technical troubles

• Regularized problem: not obvious it has a solution, hence Faedo-Galerkin discretization

• Discretized problem: far from its normal form Technical troubles

• Regularized problem: not obvious it has a solution, hence Faedo-Galerkin discretization

• Discretized problem: far from its normal form, hence implicit function theorem Technical troubles

• Regularized problem:

not obvious it has a solution, hence Faedo-Galerkin discretization

• Discretized problem:

far from its normal form, hence implicit function theorem

• Just a local discrete solution Technical troubles

• Regularized problem:

not obvious it has a solution, hence Faedo-Galerkin discretization

• Discretized problem:

far from its normal form, hence implicit function theorem

• Just a local discrete solution, hence a priori estimates for global, etc.

Technical troubles

• Regularized problem:

not obvious it has a solution, hence Faedo-Galerkin discretization

• Discretized problem:

far from its normal form, hence implicit function theorem

• Just a local discrete solution, hence a priori estimates for global, etc.

• Formal estimates ≠ “true” estimates Technical troubles

• Regularized problem:

not obvious it has a solution, hence Faedo-Galerkin discretization

• Discretized problem:

far from its normal form, hence implicit function theorem

• Just a local discrete solution a priori estimates for global, etc.

• Formal estimates ≠ “true” estimates

• Here, just formal estimates on the limit problem Goal: a priori estimates ensuring good convergences, in particular

• weak convergence for (ϑ_{ε}, ln_{ε}(ϑ_{ε})) strong for one component

• weak convergence for (χ_{ε}, β_{ε}(χ_{ε})) strong for one component

in order to apply the theory of maximal monotone operators to the induced ln and β in L^{2}(Q)

21 Formal estimates First a priori estimate ∂_{t}(ln ϑ − λ(χ)) − Δ(k_{0}u + k ∗ u) = R (1) ∂_{t}χ − Δχ + β(χ) + σ′(χ) “=” −λ′(χ)ϑ (2)

u := ϑ − ϑ_{H}
u = 0 and ∂_{n}χ = 0 on Γ
ϑ(0) = ϑ_{0} and χ(0) = χ_{0}

Test as follows ∫∫_{Ω×(0,t)}((1) × u + (2) × ∂_{t}χ)

• convolution to the rhs controlled via Gronwall Formal estimates 21 22 Formal estimates How to treat the log

∫∫_{Ω×(0,t)}(∂_{t} ln ϑ) u = ∫∫_{Ω×(0,t)}(∂_{t} ln ϑ)(ϑ − ϑ_{H})

= ∫∫_{Ω×(0,t)}(∂_{t} ln ϑ)(ϑ − ϑ_{H}) ± ϑ_{∗} ∫_{Ω} ln^{−} ϑ(t)

lhs ∫_{Ω}(ϑ(t) + ϑ_{∗} ln^{−} ϑ(t)) = OK if ϑ_{∗} > 0
rhs ∫∫_{Ω×(0,t)}(∂_{t} ln ϑ) ϑ_{H} + ϑ_{∗} ∫_{Ω} ln^{−} ϑ(t)

23 Formal estimates rhs ∫∫_{Ω×(0,t)}(∂_{t} ln ϑ) ϑ_{H} + ϑ_{∗} ∫_{Ω} ln^{−} ϑ(t) = ∫_{Ω}(ϑ_{∗} − ϑ_{H}(t)) ln^{−} ϑ(t) + ∫_{Ω} ln^{+} ϑ(t) ϑ_{H}(t) − ∫_{Ω} i.v. − ∫∫_{Ω×(0,t)}(ln ϑ)∂_{t}ϑ_{H} rhs ∫∫_{Ω×(0,t)}(∂_{t} ln ϑ) ϑ_{H} + ϑ_{∗} ∫_{Ω} ln^{−} ϑ(t)

= ∫_{Ω}(ϑ_{∗} − ϑ_{H}(t)) ln^{−} ϑ(t) + ∫_{Ω} ln^{+} ϑ(t) ϑ_{H}(t)

− ∫_{Ω} i.v. − ∫∫_{Ω×(0,t)}(ln ϑ)∂_{t}ϑ_{H}

• Assume ϑ_{Γ}(t) ≥ ϑ_{∗}, whence ϑ_{∗} − ϑ_{H}(t) ≤ 0, assume ϑ_{Γ} smooth, whence ϑ_{H} smooth, and use Gronwall 24 Formal estimates One obtains

∥ϑ, ln ϑ∥_{L^{∞}(0,T;L^{1}(Ω))}  (useless)

+ ∥u∥_{L^{2}(0,T;V)} + ∥χ∥_{L^{∞}(0,T;V)} + ∥∂_{t}χ∥_{L^{2}(0,T;H)} ≤ c

(V = H^{1}(Ω) and H = L^{2}(Ω))

24 Formal estimates One obtains

∥ϑ, ln ϑ∥_{L^{∞}(0,T;L^{1}(Ω))}  (useless)

+ ∥u∥_{L^{2}(0,T;V)} + ∥χ∥_{L^{∞}(0,T;V)} + ∥∂_{t}χ∥_{L^{2}(0,T;H)} ≤ c

(V = H^{1}(Ω) and H = L^{2}(Ω))

Main consequence ∥∂_{t} ln ϑ∥_{L^{2}(0,T;H^{−1}(Ω))} ≤ c 24 Formal estimates One obtains

∥ϑ, ln ϑ∥_{L^{∞}(0,T;L^{1}(Ω))}  (useless)

+ ∥u∥_{L^{2}(0,T;V)} + ∥χ∥_{L^{∞}(0,T;V)} + ∥∂_{t}χ∥_{L^{2}(0,T;H)} ≤ c

(V = H^{1}(Ω) and H = L^{2}(Ω))

Main consequence

∥∂_{t} ln ϑ∥_{L^{2}(0,T;H^{−1}(Ω))} ≤ c

• A good estimate for ln ϑ is still needed. Formal estimates 24 25 Formal estimates Second a priori estimate ∂_{t}χ − Δχ + β(χ) + σ′(χ) “=” −λ′(χ)ϑ (2) Test (2) by β(χ) and obtain

∥β(χ)∥_{L^{2}(0,T;H)} ≤ c.

We deduce

∥Δχ∥_{L^{2}(0,T;H)} ≤ c

whence also

∥χ∥_{L^{2}(0,T;H^{2}(Ω))} ≤ c

Summary for χ ∥χ_{ε}∥_{L^{2}(0,T;H^{2}(Ω))} + ∥∂_{t}χ_{ε}∥_{L^{2}(0,T;H)} ≤ c

Hence, using λ′ and σ′ Lipschitz ∥β_{ε}(χ_{ε})∥_{L^{2}(0,T;H)} ≤ c

we identify the limits involving χ 27 Formal estimates Third a priori estimate Recall

(1) ∂_{t}(ln ϑ − λ(χ)) − Δ(k_{0}u + k ∗ u) = R

Integrate in time:

∫^{•}(1) dt  \{ \begin{matrix}ln ϑ − k_{0}(1 ∗ Δu) \\ = k ∗ (1 ∗ Δu) + λ(χ) \\ + 1 ∗ R + ln ϑ_{0} − λ(χ_{0}) \end{matrix}

Test ∫^{•}(1) dt by −Δu = −Δϑ. We obtain ∥1 ∗ Δu∥_{L^{∞}(0,T;H)} ≤ c ⇒ ∥ln ϑ∥_{L^{∞}(0,T;H)} ≤ c 28 Conclusion Summary for ϑ

∥ϑ∥_{L^{2}(0,T;V)} ≤ c  old
∥∂_{t} ln ϑ∥_{L^{2}(0,T;H^{−1}(Ω))} ≤ c old
∥ln ϑ∥_{L^{∞}(0,T;H)} ≤ c  new

28 Conclusion Summary for ϑ

∥ϑ∥_{L^{2}(0,T;V)} ≤ c  old
∥∂_{t} ln ϑ∥_{L^{2}(0,T;H^{−1}(Ω))} ≤ c old
∥ln ϑ∥_{L^{∞}(0,T;H)} ≤ c  new

True estimates

∥ϑ_{ε}∥_{L^{2}(0,T;V)} ≤ c  old
∥∂_{t} Ln_{ε} ϑ_{ε}∥_{L^{2}(0,T;H^{−1}(Ω))} ≤ c old
∥Ln_{ε} ϑ_{ε}∥_{L^{∞}(0,T;H)} ≤ c  new

29 Conclusion Limit of ln_{ε} ϑ_{ε}

∥ϑ_{ε}∥_{L^{2}(0,T;V)} ≤ c  ϑ_{ε} →^{w} ϑ
∥∂_{t} Ln_{ε}(ϑ_{ε})∥_{L^{2}(0,T;H^{−1}(Ω))} ≤ c ln_{ε}(ϑ_{ε}) →^{s} ℓ
∥Ln_{ε} ϑ_{ε}∥_{L^{∞}(0,T;H)} ≤ c

and we should deduce ϑ > 0 and ℓ = ln ϑ.

29 Conclusion Limit of ln_{ε} ϑ_{ε}

∥ϑ_{ε}∥_{L^{2}(0,T;V)} ≤ c  ϑ_{ε} →^{w} ϑ
∥∂_{t} Ln_{ε}(ϑ_{ε})∥_{L^{2}(0,T;H^{−1}(Ω))} ≤ c ln_{ε}(ϑ_{ε}) →^{s} ℓ
∥Ln_{ε} ϑ_{ε}∥_{L^{∞}(0,T;H)} ≤ c

and we should deduce ϑ > 0 and ℓ = ln ϑ, i.e., ∫∫_{Ω×(0,T)} ϑ_{ε} ln_{ε}(ϑ_{ε})

tends to ∫∫_{Ω×(0,T)} ϑℓ ???

29 Conclusion Limit of ln_{ε} ϑ_{ε}

∥ϑ_{ε}∥_{L^{2}(0,T;V)} ≤ c  ϑ_{ε} →^{w} ϑ
∥∂_{t} Ln_{ε}(ϑ_{ε})∥_{L^{2}(0,T;H^{−1}(Ω))} ≤ c ln_{ε}(ϑ_{ε}) →^{s} ℓ
∥Ln_{ε} ϑ_{ε}∥_{L^{∞}(0,T;H)} ≤ c

and we do deduce ϑ > 0 and ℓ = ln ϑ. Indeed ∫∫_{Ω×(0,T)} ϑ_{ε} ln_{ε}(ϑ_{ε}) = ∫∫_{Ω×(0,T)}(u_{ε} + ϑ_{H}) ln_{ε}(ϑ_{ε})

tends to ∫∫_{Ω×(0,T)}(u + ϑ_{H})ℓ = ∫∫_{Ω×(0,T)} ϑℓ maybe

THE END

31 Long time behaviour Assume k ∈ L^{1}(0, +∞) (actually more). Set

k_{∞} := ∫_{0}^{+∞} k(s) ds

Assume k to be “generalized positive”, i.e.,

∫_{0}^{t} (k_{0}v^{2}(s) + (k ∗ v)(s) v(s)) ds

≥ α ∫_{0}^{t} v^{2}(s) ds (α > 0, ∀ t, ∀ v)
• This implies
k_{0} + k_{∞} ≥ α, hence k_{0} + k_{∞} > 0

32 Long time behaviour

Theorem 4. Assume

• the above conditions on k

• ϑ_{Γ} and R suitably reach asymptotic values ϑ_{Γ,∞} and R_{∞}

• R_{∞} ≥ 0 (or a weaker assumption) Then the ω − limit of any trajectory is non empty and made just of solutions (ϑ_{∞}, χ_{∞}) of

− (k_{0} + k_{∞})Δϑ_{∞} = R_{∞}
− Δχ_{∞} + β(χ_{∞}) + σ′(χ_{∞}) ∋ −λ′(χ_{∞})ϑ_{∞}
ϑ_{∞} = ϑ_{Γ,∞} and ∂_{n}χ_{∞} = 0 on Γ

33 Long time behaviour Equations (+ B.C. + I.C.) ∂_{t}(ln ϑ − λ(χ)) − Δ(k_{0}u + k ∗ u) = R (1) ∂_{t}χ − Δχ + β(χ) + σ′(χ) ∋ −λ′(χ)ϑ (2) − (k_{0} + k_{∞})Δu_{∞} = R_{∞} (1_{∞}) First a priori estimate Test as follows ∫∫_{Ω×(0,t)}(((1) − (1_{∞})) × (u − u_{∞}) + (2) × ∂_{t}χ)

34 Long time behaviour The convolution term

∫∫_{Ω×(0,t)} k_{0}v^{2} + conv. + adj. with v = u − u_{∞}

Play with the assumption on k

∫_{0}^{t} (k_{0}v^{2}(s) + (k ∗ v)(s) v(s)) ds ≥ α ∫_{0}^{t} v^{2}(s) ds

Adjusting terms OK with a suitable HP on k at ∞ 35 Long time behaviour The log term ∫∫_{Ω×(0,t)} ∂_{t} ln ϑ · (u − u_{∞}) = ∫∫_{Ω×(0,t)} ∂_{t}(ϑ − u_{∞} ln ϑ) − ∫∫_{Ω×(0,t)} ∂_{t} ln ϑ · ϑ_{H} = ∫_{Ω}(ϑ(t) − u_{∞} ln ϑ(t)) − ∫_{Ω} ln ϑ(t) · ϑ_{H}(t)

+ c + ∫∫_{Ω×(0,t)} ln ϑ · ∂_{t}ϑ_{H}_{∈L^{1}(L^{∞})}

= ∫_{Ω}(ϑ(t) − \underbrace{(u_{∞} + ϑ_{H}(t))}_{≥ϑ_{∗}>0 by HP} ln ϑ(t)) + Gronwall

Long time behaviour 35 36 Long time behaviour One obtains (among other bounds)

∥u − u_{∞}∥_{L^{2}(0,∞;V)}
+ ∥∂_{t}χ∥_{L^{2}(0,∞;H)} + ∥χ∥_{L^{∞}(0,∞;V)}
≤ c

In particular, the trajectory of χ is bounded in V 37 Long time behaviour Second a priori estimate Test as follows

∫∫_{Ω×(0,t)}(((1) − (1_{∞})) × ∂_{t}u + \frac{d(2)}{dt} × ∂_{t}χ)

(or with min {1, t} ∂_{t}u…) and obtain

∥u − u_{∞}∥_{L^{∞}(0,∞;V)} + … ≤ c

whence, the trajectory of u is bounded in V