On a generalization of the Cahn-Hilliard type equation with logarithmic nonlinearities in island formation

Hussein Fakih; Marwa Badreddine; Hawraa Alsayed; Yahia Awad

doi:10.2478/ijmce-2025-0008

1. Introduction

The significance of the Cahn-Hilliard equation in material science cannot be overstated. This equation effectively captures the crucial qualitative aspects of two-phase systems, particularly in relation to phase separation processes. In the realm of material science, the resulting pattern formation is termed as the microstructure of the material, wielding a substantial influence on diverse material properties such as strength, hardness, and conductivity.

The broad applicability of the Cahn-Hilliard model across various evolutionary stages underscores its versatility. It serves as a robust model for early-stage systems, offering a qualitative description for intermediate times, and continues to be relevant for late-stage systems. Notably, the gradual evolution during late stages often occurs at such a slow pace that pattern formation essentially becomes frozen over the relevant time scales. Consequently, the observed practical behavior reflects the long-term dynamics of the system. For a more in-depth exploration, interested readers are directed to references such as [1,2,3,4,5].

Beyond its fundamental role in material science, the Cahn-Hilliard equation finds an application in modeling a diverse array of phenomena. This extends to areas such as population dynamics [6], bacterial films [7], thin films [8, 9], image processing [10, 11], and even celestial phenomena like the rings of Saturn [12].

In a related study [13], the authors explored a model put forth in [14]:

(1)

\frac{\partial ς}{\partial t} + Δ^{2} ς - Δ f (ς) + η (ς) = 0,

where

(2)

η (s) = s (s - 1)

and

(3)

f (s) = {(s - \frac{1}{2})}^{3} - (s - \frac{1}{2}) .

Furthermore, in [15] the author has analysed the model (1) with a regular nonlinear term (3), but with a general source term, which is given by

(4)

η (s) = α s^{2} + β s + γ,

where α > 0 and β, γ ∈ ℝ. The authors in [13, 15] have proved that the solutions can blow up in finite time and exist globally under strong assumptions on the solutions and not only on the initial data. However, the author in [16] considered the model (1)–(2) but with logarithmic nonlinear terms f and proved the existence of a solution to the problem.

The Cahn-Hilliard equation incorporating a mass source term is expressed as

(5)

\frac{\partial ς}{\partial t} + Δ^{2} ς - Δ f (ς) + η (x, ς) = 0,

where η represents the mass source term. This equation serves as a versatile model with applications in diverse biological contexts, notably in the growth of cancerous tumor and other biological entities. The choice of η determines specific behaviors, for instance, a linear function η(x, s) = αs, α > 0 yields the Cahn-Hilliard-Oono equation, capturing long-range interactions in phase separation (see [17]; see also [18] for the study of the limit dynamics when α approaches zero). A quadratic function η(x, s) = αs(s − 1), α > 0 finds applications in biology [19,20,21,22], particularly in wound healing and tumor growth. Another relevant function, commonly employed in tumor growth scenarios, is

η (x, s) = \frac{α}{2} (s + 1) - β {(1 - s)}^{2} {(1 + s)}^{2} + s (x, t)

, where α and β denote growth and death coefficients. Additionally, the function η(x, ς) = 1_Ω\D(x)ς is associated with image inpainting applications, as documented in [23,24,25,26,27], and further explored with non-regular non-linear terms in [28].

In this paper, we explore the model (1) incorporating the nonlinear function η

(6)

η (s) = s^{2} - p (1 - s)

along with a logarithmic nonlinear term. The model is subject to Neumann boundary conditions. Under certain assumptions, we establish the existence of solutions for the problem. It is noteworthy that, as detailed in the article, the solutions in certain scenarios may experience a finite-time blow-up.

2. Mathematical problem

Let Ω ⊂ ℝⁿ, n = 1, 2 or 3 be a bounded and regular domain with boundary Γ. We consider the following problem

(7)

\partial_{t} ς + Δ^{2} ς - Δ f (ς) + η (ς) = 0,

(8)

\partial_{ν} ς = \partial_{ν} Δ ς = 0, on Γ,

(9)

ς (0, x) = ς_{0} (x), in Ω .

Assume that all constants are equal to one, ν is the outer unit normal vector to Γ, f = F′ as defined below and η(s) = s² − p(1 − s) where p is a strictly positive real number.

Note that

(10)

f^{'} \geq - λ_{1} .

Let us write

F (s) = \frac{λ_{1}}{2} (1 - s^{2}) + F_{1} (s)

with

f_{1} = F_{1}^{'}

and introduce for N ∈ ℕ the approximated function F_1,N ∈ C⁴(ℝ), which is defined by:

(11)

F_{1, N}^{(4)} (s) = \begin{array}{l} F_{1}^{(4)} (1 - \frac{1}{N}); & if s \geq 1 - \frac{1}{N}, \\ F_{1}^{(4)} (s); & if | s | \leq 1 - \frac{1}{N}, \\ F^{(4)} (- 1 + \frac{1}{N}); & if s \leq - 1 + \frac{1}{N} . \end{array}

(12)

F_{1, N}^{(k)} (0) = F_{1}^{(k)} (0), k = 0, 1, 2, 3

and

f_{1} (s) = f (s) + λ_{1} s = \frac{λ_{2}}{2} ln (\frac{1 + s}{1 - s})

Hence,

(13)

F_{1, N} (s) = \begin{array}{l} \sum \frac{1}{k!} F_{1}^{(k)} (1 - \frac{1}{N}) {(s - 1 + \frac{1}{N})}^{k}; & if s \geq 1 - \frac{1}{N}, \\ F_{1} (s); & if | s | \leq 1 - \frac{1}{N}, \\ \sum_{k = 0}^{4} \frac{1}{k!} F_{1}^{(k)} (- 1 + \frac{1}{N}) {(s + 1 - \frac{1}{N})}^{k}; & if s \leq - 1 + \frac{1}{N} . \end{array}

Setting

\begin{matrix} F_{N} (s) = \frac{λ_{1}}{2} (1 - s^{2}) + F_{1, N} (s), \\ f_{1, N} = F_{1, N}^{'} \end{matrix}

and

f_{N} = F_{N}^{'},

there holds

(14)

f_{1, N}^{'} \geq 0, f_{N}^{'} \geq - λ_{1},

(15)

F_{N} \geq - c_{1}, c_{1} \geq 0,

(16)

f_{N} (s) . s \geq c_{2} F_{N} (s) + | f_{1, N} (s) | - c_{3}, c_{2} > 0, c_{3} \geq 0, s \in ℝ,

and

(17)

(f_{N} (s + m) - f_{N} (m)) s \geq c_{4} (s^{4} + m^{2} s^{2}) - c_{5}, c_{4} > 0, c_{5} \geq 0, and s, m \in ℝ .

The constants c_i, i = 1, ⋯ , 5 are independent of N for N large enough. More generally,

(18)

f_{N} (s + m) s \geq c_{6, m} (F_{N} (s + m) + | f_{1, N} (s + m) |) - c_{7, m}, s \in ℝ, m \in (- 1, 1),

where the constants c_6,m and c_7,m are independent of N for sufficiently large N and exhibit continuous and bounded dependence on m.

Finally, we arrive at

Lemma 1

(19)

| η (s + m) - η (m) | \leq c_{8} (s^{2} + | ms |) + c_{9}

and

(20)

| η (s + m) - η (m) |^{2} \leq c_{10} (s^{4} + s^{2} (m^{2} + 1)) + c_{11} .

Proof

\begin{array}{l} | η (s + m) - η (m) | & = | s^{2} + 2 sm + ps | \\ \leq s^{2} + 2 | ms | + p | s | \\ \leq s^{2} + 2 | ms | + \frac{p^{2}}{2} + \frac{s^{2}}{2} \\ \leq c_{8} (s^{2} + | ms |) + c_{9} \end{array}

and

\begin{array}{l} | η (s + m) - η (m) |^{2} & \leq {(c (s^{2} + | ms |) + c^{'})}^{2} \\ \leq c {((s^{2} + | ms |) + 1)}^{2} \\ \leq c (s^{4} + 2 s^{2} | ms | + m^{2} s^{2} + 2 s^{2} + 2 | ms | + 1) \\ \leq c (s^{4} + 2 s^{2} (\frac{m^{2}}{2} + \frac{s^{2}}{2}) + m^{2} s^{2} + 2 s^{2} + 2 (\frac{1}{2} + \frac{m^{2} s^{2}}{2}) + 1) \\ \leq c_{10} (s^{4} + m^{2} s^{2} + s^{2}) + c_{11} . \end{array}

We now introduce the approximated problem

(21)

\partial_{t} ς_{N} + Δ^{2} ς_{N} - Δ f_{N} (ς_{N}) + η (ς_{N}) = 0,

(22)

\partial_{ν} ς_{N} = \partial_{ν} Δ ς_{N} = 0, on Γ,

(23)

ς_{N} (0, x) = ς_{0} (x), in Ω .

From (17) and (19)–(20) it follows that we have the existence and uniqueness (depending on the regularity of ς₀) of the (at least) local in time solution ς_N of (21)–(23).

2.1. Notations

We denote by (.) the usual L²-scalar product with associated norm ||.||. We also set ||.||₋₁ = ||(−Δ)⁻¹||, where (−Δ)⁻¹ denotes the inverse of the minus Laplace operator associated with Neumann boundary conditions and acting on zero-mean functions.

More generally, we denote by ||.||_X the norm on the Banach space X.

We set $〈 . 〉 = \frac{1}{Vol (Ω)} \int_{Ω} . dx$ , being understood that if ζ ∈ H⁻¹(Ω) = H¹(Ω)′ then

〈 ζ 〉 = \frac{1}{Vol (Ω)} {〈 ζ, 1 〉}_{H^{- 1} (Ω), H^{1} (Ω)} .

We also set, whenever this makes sense

\bar{ζ} = ζ - 〈 ζ 〉 .

We note that:

\begin{matrix} ζ \mapsto {(| | \bar{ζ} | |_{- 1}^{2} + 〈 ζ^{2} 〉)}^{\frac{1}{2}}, \\ ζ \mapsto {(| | \bar{ζ} | |^{2} + 〈 ζ^{2} 〉)}^{\frac{1}{2}}, \\ ζ \mapsto {(| | \nabla ζ | |^{2} + 〈 ζ^{2} 〉)}^{\frac{1}{2}}, \\ ζ \mapsto {(| | Δ ζ | |^{2} + 〈 ζ^{2} 〉)}^{\frac{1}{2}} \end{matrix}

are all norms on H⁻¹(Ω), L²(Ω), H¹(Ω) or H²(Ω) that are equivalent to the usual norms on these spaces. Furthermore, ||.||₋₁ is a norm on {ζ ∈ H⁻¹(Ω),〈ζ〉 = 0} which is equivalent to the usual H⁻¹ norm.

Note that the same letter c (and sometimes also c′ or c″) in this paper stands for (generally positive) constants that are independent of N and may change from line to line. The same applies to constants such as c_δ, $c_{δ}^{'}$ and $c_{δ}^{″}$ which depend on a parameter δ.

3. Applications

In this part, we observe the existence and blow up properties of the results founded by using projected scheme.

3.1. Existence and blow up solutions

In this section, our objective is to establish estimates for u_N that are independent of N. These estimates can be rigorously justified based on the approximated problems. A crucial aspect involves obtaining a uniform (with respect to N) estimate for f_N (ς_N) in L²(Ω × (0, T)), where T > 0 is independent of N. This is essential for passing to the limit in the nonlinear term and obtaining a solution to the singular initial problem. Notably, achieving this goal necessitates a uniform (with respect to N) strict separation property for 〈ς_N〉 from the singular points −1 and 1. It is worth mentioning that such a strict separation property is straightforward for the original Cahn-Hilliard equation, given the conservation of the spatial average of the order parameter, provided that the same property holds for the initial datum.

We assume that ς₀ ∈ H¹(Ω) ∩ L^∞(Ω), with |ς₀(x)| < 1 almost everywhere in Ω, and

(24)

| 〈 ς_{0} 〉 | \leq 1 - 2 δ, δ \in (0, \frac{1}{2}],

given. If we first integrate (21) over Ω, we find due to (22),

(25)

\frac{d 〈 ς_{N} 〉}{dt} + 〈 η (ς_{N}) 〉 = 0 .

In fact, we have that

\frac{d ς_{N}}{dt} + Δ^{2} ς_{N} - Δ f_{N} (ς_{N}) + η (ς_{N}) = 0 .

Noting that

\int_{Ω} Δ^{2} ς dx = - \int_{Ω} \nabla Δ ς . \nabla 1 dx + \int_{Γ} \frac{\partial Δ ς}{\partial ν} . ς ds = 0 .

Furthermore,

\int_{Ω} Δ f_{N} (ς_{N}) dx = - \int_{Ω} \nabla f_{N} (ς_{N}) . \nabla 1 dx + \int_{Γ} \frac{\partial f_{N} (ς_{N})}{\partial ν} 1 ds = 0,

hence

\frac{d}{dt} \int_{Ω} ς_{N} dx + \int_{Ω} η (ς_{N}) dx = 0,

which yields

\frac{d 〈 ς_{N} 〉}{dt} + 〈 η (ς_{N}) 〉 = 0 .

On the other hand, setting ς_N = 〈ς_N〉 + ζ_N, (ζ_N) = 0 yields

\frac{d 〈 ς_{N} 〉}{dt} + 〈 ς_{N}^{2} - p (1 - ς_{N}) 〉 = 0

and

\frac{d 〈 ς_{N} 〉}{dt} + 〈 {(〈 ς_{N} 〉 + ζ_{N})}^{2} - p (1 - ς_{N}) 〉 = 0 .

Therefore,

\frac{d 〈 ς_{N} 〉}{dt} + 〈 {〈 ς_{N} 〉}^{2} + 2 〈 ς_{N} 〉 〈 ζ_{N} 〉 + ζ_{N}^{2} - p (1 - ς_{N}) 〉 = 0,

so that

\frac{d 〈 ς_{N} 〉}{dt} + {〈 ς_{N} 〉}^{2} + 2 〈 ς_{N} 〉 〈 ζ_{N} 〉 + 〈 ζ_{N}^{2} 〉 - p (1 - 〈 ς_{N} 〉) = 0,

it then follows

\frac{d 〈 ς_{N} 〉}{dt} + {〈 ς_{N} 〉}^{2} - p (1 - 〈 ς_{N} 〉) = - 〈 ζ_{N}^{2} 〉 .

Hence, we get

(26)

\frac{d 〈 ς_{N} 〉}{dt} + η 〈 ς_{N} 〉 = - 〈 ζ_{N}^{2} 〉 .

Furthermore, ζ_N is a solution to:

(27)

\partial_{t} ζ_{N} + Δ^{2} ζ_{N} - Δ f_{N} (ς_{N}) + \bar{η (ς_{N})} = 0,

(28)

\partial_{ν} ζ_{N} = \partial_{ν} Δ ζ_{N} = 0, on Γ,

(29)

ζ_{N} |_{t = 0} = ζ_{0} (x), in Ω .

Since (21) gives,

\begin{array}{r} \frac{\partial (ζ_{N} + 〈 ς_{N} 〉)}{\partial t} + Δ^{2} ς_{N} - Δ f_{N} (ς_{N}) + η (ς_{N}) = 0 \\ \frac{\partial ζ_{N}}{\partial t} + \frac{\partial 〈 ς_{N} 〉}{\partial t} + Δ^{2} ς_{N} - Δ f_{N} (ς_{N}) + η (ς_{N}) = 0 \\ \frac{\partial ζ_{N}}{\partial t} - 〈 g (ς_{N}) 〉 + Δ^{2} ζ_{N} - Δ f_{N} (ς_{N}) + η (ς_{N}) = 0 \\ \frac{\partial ζ_{N}}{\partial t} + Δ^{2} ζ_{N} - Δ f_{N} (ς_{N}) + \bar{η (ς_{N})} = 0 . \end{array}

\frac{\partial ς_{N}}{\partial ν} = \frac{\partial ζ_{N}}{\partial ν} and \frac{\partial Δ ς_{n}}{\partial ν} = \frac{\partial Δ ζ_{n}}{\partial ν} .

These equations can be written equivalently, if we multiply by −(Δ)⁻¹, as:

(30)

{(- Δ)}^{- 1} \partial_{t} ζ_{N} - Δ ζ_{N} + \bar{f_{N} (ς_{N})} + {(- Δ)}^{- 1} \bar{η (ς_{N})} = 0, in Ω \times (0, T),

(31)

\partial_{ν} ζ_{N} = 0, on Γ,

(32)

ζ_{N} |_{t = 0} = ζ_{0}, in Ω .

Multiplying (30) by ζ_N and integrating over Ω and by parts, we have:

(33)

\frac{1}{2} \frac{d}{dt} | | ζ_{N} | |_{- 1}^{2} + ∥ \nabla ζ_{N} ∥^{2} + [\bar{f_{N} (ς_{N})}, ζ_{N}] + [η (ς_{N}), {(- Δ^{- 1})}^{- 1} ζ_{N}] = 0 .

Since

\begin{array}{l} \int_{Ω} {(- Δ)}^{- 1} \frac{\partial ζ_{N}}{\partial t} ζ_{N} dx & = \frac{1}{2} \frac{d}{dt} \int_{Ω} {(- Δ)}^{- 1} ζ_{N}^{2} dx \\ = \frac{1}{2} \frac{d}{dt} | | ζ_{N} | |_{- 1}^{2} \end{array}

and

\begin{array}{l} \int_{Ω} - Δ ζ_{N} ζ_{N} dx & = \int_{Ω} \nabla ζ_{N} \cdot \nabla ζ_{N} dx - \int_{Γ} \frac{\partial ζ_{N}}{\partial ν} ζ_{N} ds \\ = | | \nabla ζ_{N} | |^{2} . \end{array}

Note that

[\bar{f_{N} (ς_{N})}, ζ_{N}] = [f_{N} (ς_{N}) - f_{N} (〈 ς_{N} 〉), ζ_{N}]

and taking s = ζ_N, m = 〈ς_N〉 in (17), we get

(34)

[\bar{f_{N} (ς_{N})}, ζ_{N}] \geq c_{4} \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx - c .

Indeed, from (17), we have

\begin{array}{l} (f_{N} (s + m) - f_{N} (m)) s \geq c_{4} (s^{4} + m^{2} s^{2}) - c_{5} \\ (f_{N} (ς_{N}) - f_{N} 〈 ς_{N} 〉) ζ_{N} \geq c_{4} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) - c_{5}, \end{array}

\begin{array}{l} [f_{N} (ς_{N}) - f_{N} 〈 ς_{N} 〉, ζ_{N}] & = \int_{Ω} (f_{N} (ς_{N}) - f_{N} 〈 ς_{N} 〉) ζ_{N} dx \\ \geq c_{4} \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx - c_{5} . \end{array}

Furthermore,

\begin{array}{l} | [\bar{η (ς_{N})}, (- Δ^{- 1}) ζ_{N}] | & = | [η (ς_{N}) - η (〈 ς_{N} 〉), (- Δ^{- 1}) ζ_{N}]] | \\ \leq c | | ζ_{N} | | | | η (ς_{N}) - g (〈 ς_{N} 〉) | |, \end{array}

by Cauchy Schwartz and using ||ζ_N||₋₁ ≤ c||ζ_N|| which results after reapplying Lemma (1) with s = ζ_N and m = 〈ς_N〉 and Young’s inequality:

(35)

\begin{array}{l} | [\bar{η (ς_{N})}, (- Δ^{- 1}) ζ_{N}] | & \leq \frac{c_{4}}{4} \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c | | ζ_{N} | |^{2} \\ \leq \frac{c_{4}}{4} \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c . \end{array}

Indeed, we have that

\begin{array}{l} | [\bar{η (ς_{N})}, (- Δ^{- 1}) ζ_{N}] | & \leq c | | ζ_{N} | | | | η (ς_{N}) - η (〈 ς_{N} 〉) | | \\ \leq c (\frac{| | ζ_{N} | |^{2}}{2} + \frac{| | η (ς_{N}) - η (〈 ς_{N} 〉) | |^{2}}{2}) \\ \leq \frac{c}{2} (| | ζ_{N} | |^{2} + c_{9} \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2} + ζ_{N}^{2}) dx + c_{10}) \\ \leq \frac{c_{4}}{2} \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c | | ζ_{N} | |^{2} + c^{'} \\ \leq \frac{c_{4}}{2} \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c . \end{array}

It follows from (33)–(35), that

(36)

\frac{d}{dt} | | ζ_{N} | |_{- 1}^{2} + c (| | ζ_{N} | |_{H^{1} (Ω)}^{2} + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx) \leq c^{'}, c > 0 .

Since

\begin{array}{l} \frac{1}{2} \frac{d}{dt} | | ζ_{N} | |_{- 1}^{2} + | | \nabla ζ_{N} | |^{2} & = - [\bar{f_{N} (ς_{N})}, ζ_{N}] - [\bar{η (ς_{N})}, - Δ^{- 1} ζ_{N}] \\ \leq - c_{4} \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c + \frac{c_{4}}{2} \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c . \end{array}

Next, multiplying (30) by −Δζ_N and integrating over Ω, we obtain

(37)

\frac{1}{2} \frac{d}{dt} | | ζ_{N} | |^{2} + | | Δ ζ_{N} | |^{2} + [\bar{f_{N} (ς_{N})}, - Δ ζ_{N}] + [\bar{η (ς_{N})}, ζ_{N}] = 0 .

In fact, we have

\begin{array}{l} \int_{Ω} - Δ^{- 1} \frac{\partial ζ_{N}}{\partial t} (- Δ ζ_{N}) dx & = \int_{Ω} \frac{\partial ζ_{N}}{\partial t} ζ_{N} dx \\ = \frac{1}{2} \frac{d}{dt} \int_{Ω} ζ_{N}^{2} dx \\ = \frac{1}{2} \frac{d}{dt} | | ζ_{N} | |^{2} . \end{array}

Furthermore, owing to (14)

(38)

\begin{array}{l} [\bar{f_{N} (ς_{N})}, - Δ ζ_{N}] & = [f_{N} (ς_{N}), - Δ ζ_{N}] \\ = - \int_{Ω} f_{N} (ς_{N}) Δ ζ_{N} dx \\ = \int_{Ω} \nabla f_{N} (ς_{N}) . \nabla ζ_{N} dx \\ = \int_{Ω} f^{'} (ς_{N}) \nabla ς_{N} \nabla ζ_{N} dx \\ = [f^{'} (ς_{N}) \nabla ς_{N}, \nabla ζ_{N}] \\ \geq - λ_{1} | | \nabla ζ_{N} | |^{2} \end{array}

and owing once more to Lemma (1), we have

(39)

\begin{array}{l} | [\bar{η (ς_{N})}, ζ_{N}] | & = | [η (ς_{N}) - η (〈 ς_{N} 〉), ζ_{N}] | \\ \leq \frac{1}{2} | | Δ ζ_{N} | |^{2} + c \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c^{'} . \end{array}

Indeed,

\begin{array}{l} | [η (ς_{N}) - η (〈 ς_{N} 〉), ζ_{N}] | & \leq | | η (ς_{N}) - η (〈 ς_{N} 〉) | | | | ζ_{N} | | \\ \leq \frac{| | ζ_{N} | |^{2}}{2} + \frac{| | η (ς_{N}) - η (〈 ς_{N} 〉) | |^{2}}{2} \\ \leq \frac{1}{2} | | Δ ζ_{N} | |^{2} + c \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c^{'} . \end{array}

From (37)–(39) it follows that

(40)

\frac{d}{dt} | | ζ_{N} | |^{2} + | | Δ ζ_{N} | |^{2} \leq c (| | ζ_{N} | |_{H^{1} (Ω)}^{2} + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx) + c^{'} .

By (37), we have

\begin{array}{l} \frac{1}{2} \frac{d}{dt} | | ζ_{N} | |^{2} + | | Δ ζ_{N} | |^{2} & = [\bar{f_{N} (ς_{N})}, - Δ ζ_{N}] - [\bar{η (ς_{N})}, ζ_{N}] \\ \leq λ_{1} | | \nabla ζ_{N} | |^{2} + \frac{1}{2} | | Δ ζ_{N} | |^{2} + c \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c^{'}, \end{array}

so,

\frac{d}{dt} | | ζ_{N} | |^{2} + | | Δ ζ_{N} | |^{2} \leq c (| | \nabla ζ_{N} | |^{2} + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx) + c^{'} .

Finally, summing (36) and γ₁ times (40), where γ₁ > 0 is small enough and independent of N, we find

(41)

\frac{d}{dt} (| | ζ_{N} | |_{- 1}^{2} + γ_{1} | | ζ_{N} | |^{2}) + c (| | ζ_{N} | |_{H^{2} (Ω)}^{2} + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx) \leq c^{'}, c > 0 .

Indeed, we have (36)+γ₁(40) gives

\frac{d}{dt} (| | ζ_{N} | |^{2} + γ_{1} | | ζ_{N} | |^{2}) + c | | ζ_{N} | |_{H^{1} (Ω)} + γ_{1} | | Δ ζ_{N} | |^{2} + c \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx \leq c^{'},

then

\frac{d}{dt} (| | ζ_{N} | |^{2} + γ_{1} | | ζ_{N} | |^{2}) + c (| | ζ_{N} | |_{H^{2} (Ω)} + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx) \leq c^{'} .

We now come back to (25)–(26). Noting that g(s) ≥ ps − p, we have

\begin{matrix} \frac{d 〈 ς_{N} 〉}{dt} + η (〈 ς_{N} 〉) = - 〈 ζ_{N}^{2} 〉 \leq 0 \\ \frac{d 〈 ς_{N} 〉}{dt} \leq - η (〈 ς_{N} 〉) \leq - p 〈 ς_{N} 〉 + p \\ \frac{d 〈 ς_{N} 〉}{dt} + p 〈 ς_{N} 〉 \leq p . \end{matrix}

Consider the ODE:

\frac{d 〈 ς_{N} 〉}{dt} + p 〈 ς_{N} 〉 = p,

then

\frac{d 〈 ς_{N} 〉}{dt} = - p (〈 ς_{N} 〉 - 1),

hence

\frac{d 〈 ς_{N} 〉}{(〈 ς_{N} 〉 - 1)} = - p dt,

we infer

ln | 〈 ς_{N} 〉 - 1 | = - p t + k,

〈 ς_{N} 〉 = c e^{- p t} + 1,

but at t = 0, 〈ς_N〉 = 〈ς₀〉, hence c = 〈ς₀〉 − 1.

Finally,

(42)

〈 ς_{N} (t) 〉 \leq (〈 ς_{0} 〉 - 1) e^{- p t} + 1,

as long as it exists. In particular,

(43)

〈 ς_{N} (t) 〉 < 1 .

Note that it follows from (41) that:

\frac{d}{dt} (| | ζ_{N} | |_{- 1}^{2} + γ_{1} | | ζ_{N} | |^{2}) + c (| | ζ_{N} | |_{- 1}^{2} + γ_{1} | | ζ_{N} | |^{2}) \leq c^{'}, c > 0 .

Indeed, we have that

\begin{array}{l} c (| | ζ_{N} | |_{H^{2} (Ω)}^{2} + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx) & \geq c | | ζ_{N} | |_{H^{(} Ω)}^{2} - \frac{1}{2} | | Δ ζ_{N} | |^{2} - c^{'} \\ \geq c | | Δ ζ_{N} | |^{2} - c^{'} \\ \geq c | | ζ_{N} | |^{2} - c^{'} \\ \geq \frac{c}{2} | | ζ_{N} | |^{2} + \frac{c}{2} | | ζ_{N} | |^{2} - c^{'} \\ \geq c (| | ζ_{N} | |_{- 1}^{2} + γ_{1} | | ζ_{N} | |^{2}) - c^{'}, \end{array}

but by (41)

\begin{matrix} \frac{d}{dt} (| | ζ_{N} | |_{- 1}^{2} + γ_{1} | | ζ_{N} | |^{2}) + c (| | ζ_{N} | |_{- 1}^{2} + γ_{1} | | ζ_{N} | |^{2}) \\ \leq \frac{d}{dt} (| | ζ_{N} | |_{- 1}^{2} + γ_{1} | | ζ_{N} | |^{2}) + c (| | ζ_{N} | |_{H^{2} (Ω)}^{2} + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx) + c^{'} \\ \leq 2 c^{'} . \end{matrix}

Which yields

(44)

| | ζ_{N} (t) | |^{2} \leq c e^{- c^{'} t} | | ζ_{0} | |^{2} + c^{'}, c^{″} > 0 .

In fact,

\frac{d}{dt} (| | ζ_{N} | |_{- 1}^{2} + γ_{1} | | ζ_{N} | |^{2}) + c (| | ζ_{N} | |_{- 1}^{2} + γ_{1} | | ζ_{N} | |^{2}) \leq c^{'}, c > 0,

then by Gronwall lemma

\begin{matrix} | | ζ_{N} | |_{- 1}^{2} + γ_{1} | | ζ_{N} | |^{2} \leq (| | ζ_{0} | |_{- 1}^{2} + γ_{1} | | ζ_{0} | |^{2}) + c^{″} \\ γ_{1} | | ζ_{N} | |^{2} \leq e^{- ct} (| | ζ_{0} | |_{- 1}^{2} + γ_{1} | | ζ_{0} | |^{2}) + c^{″} \\ | | ζ_{N} | |^{2} \leq \frac{1}{γ_{1}} e^{- ct} (| | ζ_{0} | |_{- 1}^{2} + γ_{1} | | ζ_{0} | |^{2}) + \frac{c^{″}}{γ_{1}}, \end{matrix}

but

| | ζ_{0} | |_{- 1}^{2} \leq q | | ζ_{0} | |^{2}

, then

\begin{array}{l} | | ζ_{N} | |^{2} & \leq \frac{1}{γ_{1}} e^{- ct} (q | | ζ_{0} | |^{2} + γ_{1} | | ζ_{0} | |^{2}) + \frac{c^{″}}{γ_{1}} \\ \leq c e^{- c^{'} t} | | ζ_{0} | |^{2} + c^{″} . \end{array}

As long as it exists, in particular

〈 ζ_{N}^{2} 〉 = k | | ζ_{N}^{2} (t) | | \leq c | | ζ_{0} | |^{2} + c^{'}, since e^{- c^{'} t} \leq 1,

and ζ₀ = ς₀ − 〈ς₀〉 and |〈ς₀〉| ≤ 1 − 2δ, so

(45)

〈 ζ_{N}^{2} (t) 〉 \leq c (ς_{0}, δ),

as long as it exists.

Let y_± be the solution of the Ricatti ODE’s

(46)

y_{+}^{'} + η (y_{+}) = 0, y_{+} (0) = 〈 ς_{0} 〉

(47)

y_{-}^{'} + η (y_{-}) = - c (ς_{0}, δ), y_{-} (0) = 〈 ς_{0} 〉,

where c(ς₀, δ) is the constant in (45). Then it follows from the comparison principle that, as long as this makes sense

(48)

y_{-} (t) \leq 〈 ς_{N} (t) 〉 \leq y_{+} (t) .

Indeed, we have that

\frac{d 〈 ς_{N} 〉}{dt} + g (〈 ς_{N} 〉) = - 〈 ζ_{N}^{2} 〉 \leq 0,

so, 〈ς_N (t)〉 ≤ y₊(t), also

〈 ζ_{N}^{2} 〉 \leq c (ς_{0}, δ),

hence

\frac{d 〈 ς_{N} 〉}{dt} + η (〈 ς_{N} 〉) \geq - c (ς_{0}, δ),

so 〈ς_N (t)) ≥ y₋(t).

In particular, it follows that (at least) a local in times solution exists on some [0, T ], where T > 0 is independent of N. Note also that

y_{+} (t) = \frac{y_{2} - c y_{1} e^{\frac{t}{c_{0}}}}{1 - c e^{\frac{t}{c_{0}}}},

with

\begin{matrix} y_{1} = \frac{p + \sqrt{p^{2} + 4 p}}{2}, \\ y_{2} = \frac{p - \sqrt{p^{2} + 4 p}}{2} \end{matrix}

and

c = \frac{y_{2} - 〈 ς_{0} 〉}{y_{1} - 〈 ς_{0} 〉} .

y₊(t) is the solution of the ODE: y′ + η(y) = 0, so y′ + y² − p(1 − y) = 0, so

\begin{matrix} \frac{dy}{dt} & = p + py - y^{2} \\ \int \frac{dy}{p + py - y^{2}} & = \int dt . \end{matrix}

We have

\begin{matrix} p + py - y^{2} = 0 : \\ Δ = p^{2} - 4 (p) (- 1) = p^{2} + 4 p, \end{matrix}

hence the roots of the quadratic equation are

\begin{matrix} y_{1} = \frac{- p - \sqrt{p^{2} + 4 p}}{- 2} = y_{1} = \frac{p + \sqrt{p^{2} + 4 p}}{2}, \\ y_{2} = \frac{- p + \sqrt{p^{2} + 4 p}}{- 2} = y_{2} = \frac{p - \sqrt{p^{2} + 4 p}}{2}, \end{matrix}

\begin{matrix} \int \frac{- dy}{(y - y_{1}) (y - y_{2})} = t + c, \\ \frac{1}{(y - y_{1}) (y - y_{2})} = \frac{A}{y - y_{1}} + \frac{B}{y - y_{2}} . \end{matrix}

Where

\begin{matrix} A = lim_{y \to y_{1}} \frac{1}{y - y_{2}} = \frac{1}{y_{1} - y_{2}} = \frac{1}{\sqrt{p^{2} + 4 p}} = c_{0}, \\ B = lim_{y \to y_{2}} \frac{1}{y - y_{1}} = \frac{1}{y_{2} - y_{1}} = \frac{- 1}{\sqrt{p^{2} + 4 p}} = - c_{0} . \end{matrix}

We infer

\begin{matrix} \int (\frac{- c_{0}}{y - y_{1}} + \frac{c_{0}}{y - y_{2}}) dt = t + c, \\ c_{0} (ln | y - y_{2} | - ln | y - y_{1} |) = t + ln c, \\ c_{0} ln | \frac{y - y_{2}}{y - y_{1}} | = t + ln c, \\ ln | \frac{y - y_{2}}{y - y_{1}} |^{c_{0}} = t + ln c, \\ {(\frac{y - y_{2}}{y - y_{1}})}^{c_{0}} = c e^{t}, \\ \frac{y - y_{2}}{y - y_{1}} = c e^{\frac{t}{c_{0}}}, \\ y - y_{2} = c e^{\frac{t}{c_{0}}} (y - y_{1}), \\ y (1 - c e^{\frac{t}{c_{0}}}) = y_{2} - c y_{1} e^{\frac{t}{c_{0}}}, \\ y = \frac{y_{2} - c y_{1} e^{\frac{t}{c_{0}}}}{1 - c e^{\frac{t}{c_{0}}}} . \end{matrix}

But y|_t=0 = 〈ς₀〉, so

\begin{matrix} 〈 ς_{0} 〉 = \frac{y_{2} - c y_{1}}{1 - c}, \\ y_{2} - c y_{1} = 〈 ς_{0} 〉 (1 - c), \\ y_{2} - 〈 ς_{0} 〉 = c (y_{1} - 〈 ς_{0} 〉), \\ c = \frac{y_{2} - 〈 ς_{0} 〉}{y_{1} - 〈 ς_{0} 〉} . \end{matrix}

We assume from now on that t ∈ [0, T ], where T is as above, we again multiply (30) by ζ_N and we have

\frac{1}{2} \frac{d}{dt} | | ζ_{N} | |_{- 1}^{2} + | | \nabla ζ_{N} | |^{2} + [f_{N} (ς_{N}), ζ_{N}] \leq c \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c^{'},

since by (33)

(49)

\begin{matrix} \frac{1}{2} \frac{d}{dt} | | ζ_{N} | |_{- 1}^{2} + | | \nabla ζ_{N} | |^{2} + [\bar{f_{N} (ς_{N})}, ζ_{N}] & = - [\bar{η (ς_{N})}, - Δ^{- 1} ζ_{N}] \\ \leq c \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c^{'} . \end{matrix}

Which yields, employing (18) with s = ζ_N and m = 〈ς_N〉,

(50)

\frac{d}{dt} | | ζ_{N} | |_{- 1}^{2} + c_{δ} (| | f_{N} (ς_{N}) | |_{L^{1} (Ω)} + \int_{Ω} F_{N} (ς_{N}) dx) \leq c^{'} \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c_{δ}^{″}, c_{δ > 0} .

In fact, we know from (18) that

\begin{array}{l} f_{N} (ς_{N}) ζ_{N} & \geq c_{δ} (F_{N} (ς_{N}) + | f_{1, N} (ς_{N}) |) - c_{δ}^{'} \\ \int_{Ω} f_{N} (ς_{N}) ζ_{N} dx & \geq c_{δ} (\int_{Ω} F_{N} (ς_{N}) dx + \int_{Ω} | f_{1, N} (ς_{N}) | dx) - c_{δ}^{'}, \end{array}

\begin{matrix} \frac{1}{2} \frac{d}{dt} | | ζ_{N} | |_{- 1}^{2} + c_{δ} (| | f_{1, 1} (ς_{N}) | |_{L^{1} (Ω)} + \int_{Ω} F_{N} (ς_{N}) dx) - c_{δ}^{'} \\ \leq c \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c^{'}, \end{matrix}

but f₁(ς_N) = f_N (ς_N) + λς_N. We then infer that

\frac{d}{dt} | | ζ_{N} | |_{- 1}^{2} + c_{δ} (| | f_{N} (ς_{N}) | |_{L^{1} (Ω)} + \int_{Ω} F_{N} (ς_{N}) dx) \leq c^{'} \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c_{δ}^{″}, c_{δ > 0} .

Summing (41) and γ₂× (50), where γ₂ > 0 is small enough and independent of N and δ, we obtain a differential inequality of the form

(51)

\begin{matrix} \frac{d E_{1, N}}{dt} + c_{δ} (E_{1, N} + | | ζ_{N} | |_{H^{2} (Ω)}^{2}) + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + | | f_{N} (ς_{N}) | |_{L^{1} (Ω)} \\ + \int_{Ω} F_{N} (ς_{N}) dx \leq c_{δ}^{'}, c_{δ} > 0, \end{matrix}

where

E_{1, N} = (1 + γ_{2}) | | ζ_{N} | |_{- 1}^{2} + γ_{1} | | ζ_{N} | |^{2}

In fact, (41)+γ₂(50) give

\begin{matrix} \frac{d}{dt} (| | ζ_{N} | |_{- 1}^{2} + γ_{1} | | ζ_{N} | |^{2}) + c (| | ζ_{N} | |_{H^{2} (Ω)}^{2} + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx) \\ + γ_{2} \frac{d}{dt} | | ζ_{N} | |_{- 1}^{2} + γ_{2} c_{δ} (| | f_{N} (ς_{N}) | |_{L^{1} (Ω)} + \int_{Ω} F_{N} (ς_{N}) dx) \\ \leq c^{'} + γ_{2} c^{'} \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + γ_{2} c_{δ}^{″}, c_{δ} > 0 . \end{matrix}

Hence,

\begin{matrix} \frac{d E_{1, N}}{dt} + c_{δ} (| | ζ_{N} | |_{H^{2} (Ω)}^{2} + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + | | f_{N} (ς_{N}) | |_{L^{1} (Ω)} + \int_{Ω} F_{N} (ς_{N}) dx) \\ \leq c_{δ}^{'}, c_{δ} > 0, \end{matrix}

but

\begin{array}{l} E_{1, N} & = (1 + γ_{2}) | | ζ_{N} | |_{- 1}^{2} + γ_{1} | | ζ_{N} | |^{2} \\ \leq (1 + γ_{2}) k | | ζ_{N} | |_{H^{2} (Ω)}^{2} + γ_{1} k^{'} | | ζ_{N} | |_{H^{2} (Ω)}^{2}, \end{array}

so,

E_{1, N} \leq k | | ζ_{N} | |_{H^{2} (Ω)}^{2},

then

\begin{matrix} \frac{d E_{1, N}}{dt} + c_{δ} (E_{1, N} + | | ζ_{N} | |_{H^{2} (Ω)}^{2} + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx) + | | f_{N} (ς_{N}) | |_{L^{1} (Ω)} \\ + \int_{Ω} F_{N} (ς_{N}) dx \leq c_{δ}^{'}, c_{δ} > 0 . \end{matrix}

If we note that |〈ς_N〉 ≤ 1, we see that

\frac{d {〈 ς_{N} 〉}^{2}}{dt} = 2 〈 ς_{N} 〉 (- 〈 ζ_{N}^{2} 〉 - {〈 ς_{N} 〉}^{2} + p (1 - 〈 ς_{N} 〉)) \leq c | | ζ_{N} | |^{2},

which results in the following

(52)

\frac{d {〈 ς_{N} 〉}^{2}}{dt} + {〈 ς_{N} 〉}^{2} \leq c | | ζ_{N} | |^{2} + c^{'},

since

\begin{array}{l} \frac{d {〈 ς_{N} 〉}^{2}}{dt} & = 2 〈 ς_{N} 〉 \frac{d}{dt} 〈 ς_{N} 〉 \\ = 2 〈 ς_{N} 〉 (- g (〈 ς_{N} 〉) - 〈 ζ_{N}^{2} 〉) \\ = 2 〈 ς_{N} 〉 (- {〈 ς_{N} 〉}^{2} + p (1 - 〈 ς_{N} 〉) - 〈 ζ_{N}^{2} 〉) \\ = - 2 {〈 ς_{N} 〉}^{3} + 2 p 〈 ς_{N} 〉 - 2 p {〈 ς_{N} 〉}^{2} - 2 〈 ς_{N} 〉 {〈 ζ_{N} 〉}^{2} \\ \leq c | | ζ_{N} | |^{2} + c^{'}, \end{array}

\begin{array}{l} \frac{d {〈 ς_{N} 〉}^{2}}{dt} + {〈 ς_{N} 〉}^{2} & \leq c | | ζ_{N} | |^{2} + c^{'} + 1 \\ \leq c | | ζ_{N} | |^{2} + c^{″} . \end{array}

Summing (51) and γ₃ × (52), where γ₃ > 0 (independent of N) is small enough, we get

(53)

\begin{matrix} \frac{d E_{2, N}}{dt} + c_{δ} (E_{2, N} + | | ς_{N} | |_{H^{2} (Ω)}^{2} + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx) + | | f_{N} (ς_{N}) | |_{L^{1} (Ω)} \\ + \int_{Ω} F_{N} (ς_{N}) dx \leq c_{δ}^{'}, c_{δ} > 0, \end{matrix}

where

E_{2, N} = E_{1, N} + γ_{3} {〈 ς_{N} 〉}^{2},

since (51) +γ₃ (52) give

\begin{matrix} \frac{d}{dt} E_{1, N} + c_{δ} (E_{1, N} + | | ζ_{N} | |_{H^{2} (Ω)}^{2} + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + | | f_{N} (ς_{N}) | |_{L^{1} (Ω)} + \int_{Ω} F_{N} (ς_{N}) dx) \\ + γ_{3} \frac{d}{dt} {〈 ς_{N} 〉}^{2} + γ^{3} {〈 ς_{N} 〉}^{2} \\ \leq c_{δ}^{'} + c γ_{3} | | ζ_{N} | |^{2} + γ_{3} c^{'}, \end{matrix}

but

| | ζ_{N} | |^{2} = \int_{Ω} ζ_{N}^{2} dx \leq c \int_{Ω} ζ_{N}^{4} dx,

so we get (53).

E_2,N satisfies

c | | ς_{N} | |^{2} \leq E_{2, N} \leq c^{'} | | ς_{N} | |^{2}, c, c^{'} > 0 .

Indeed, we have

\begin{array}{l} E_{2, N} & = E_{1, N} + γ_{3} {〈 ς_{N} 〉}^{2} \\ = (1 + γ_{2}) | | ζ_{N} | |_{- 1}^{2} + γ_{1} | | ζ_{N} | |^{2} + γ_{3} {〈 ς_{N} 〉}^{2}, \end{array}

but ||ζ_N||₋₁ ∼ ||ζ_N||_L²(Ω), since 〈ζ_N〉 = 0, so ∃c₁, c₂ > 0, such that

\begin{matrix} c_{1} | | ζ_{N} | |_{L^{2} (Ω)}^{2} \leq | | ζ_{N} | |_{- 1}^{2} \leq c_{2} | | ζ_{N} | |_{L^{2} (Ω)}^{2} . \\ γ_{3} {〈 ς_{N} 〉}^{2} + (1 + γ_{2}) c_{1} | | ζ_{N} | |_{L^{2} (Ω)}^{2} + γ_{1} | | ζ_{N} | |^{2} \leq E_{2, N} \\ \leq (1 + γ_{2}) c_{2} | | ζ_{N} | |_{L^{2} (Ω)}^{2} + γ_{1} | | ζ_{N} | |^{2} + γ_{3} {〈 ς_{N} 〉}^{2} \\ c_{3} (| | ζ_{N} | |_{L^{2} (Ω)}^{2} + {〈 ς_{N} 〉}^{2}) \leq E_{2, N} \leq c_{4} (| | ζ_{N} | |_{L^{2} (Ω)}^{2} + {〈 ς_{N} 〉}^{2}), \end{matrix}

then

c | | ς_{N} | |^{2} \leq E_{2, N} \leq c^{'} | | ς_{N} | |^{2} .

In the next step, we multiply (30) by

\frac{d ς_{N}}{dt}

, then integrate over Ω and have

(54)

| | \frac{d ζ_{N}}{dt} | |_{- 1}^{2} + \frac{1}{2} \frac{d}{dt} | | \nabla ς_{N} | |^{2} + [f_{N} (ς_{N}), \frac{d ς_{N}}{dt}] + [ς_{N}^{2} - p (1 - ς_{N}), {(- Δ)}^{- 1} \frac{d ς_{N}}{dt}] = 0 .

Indeed, (30) results in

{(- Δ)}^{- 1} \frac{d ς_{N}}{dt} - Δ ζ_{N} + \bar{f_{N} (ς_{N})} + {(- Δ)}^{- 1} \bar{η (ς_{N})} = 0,

but

\begin{matrix} \int_{Ω} - Δ ζ_{N} \frac{d ζ_{N}}{dt} dx = \\ = \frac{1}{2} \frac{d}{dt} (\int_{Ω} \nabla ζ_{N} \cdot \nabla ζ_{N} dx - \int_{Γ} ζ_{N} \frac{\partial ζ_{N}}{\partial ν} ds) \\ = \frac{1}{2} \frac{d}{dt} | | \nabla ζ_{N} | |^{2} . \end{matrix}

Furthermore

(55)

\begin{array}{l} [f_{N} (ς_{N}), \frac{\partial ζ_{N}}{\partial t}] & = [f_{N} (ς_{N}), \frac{\partial ς_{N}}{\partial t}] - [f_{N} (ς_{N}), \frac{\partial 〈 ς_{N} 〉}{\partial t}] \\ = \frac{d}{dt} \int_{Ω} F_{N} (ς_{N}) dx + [f_{N} (ς_{N}), {〈 ζ_{N} 〉}^{2} + {〈 ς_{N} 〉}^{2} - p (1 - 〈 ς_{N} 〉)] \\ = \frac{d}{dt} \int_{Ω} F_{N} (ς_{N}) dx + Vol (Ω) 〈 f_{N} (ς_{N}) 〉 [{〈 ζ_{N} 〉}^{2} + {〈 ς_{N} 〉}^{2} - p (1 - 〈 ς_{N} 〉)] \\ \leq | | f_{N} (ς_{N}) | |_{L^{1} (Ω)} (| | ζ_{N} | |_{L^{2} (Ω)}^{2} + 1) . \end{array}

We also note that

(56)

\begin{array}{l} | [ς_{N}^{2} - p (1 - ς_{N}), {(- Δ)}^{- 1} \frac{\partial ζ_{N}}{\partial t}] | & = [ς_{N}^{2} - 〈 ς_{N}^{2} 〉 + p (ς_{N} - 〈 ς_{N} 〉), {(- Δ)}^{- 1} \frac{\partial ζ_{N}}{dt}] \\ \leq \frac{1}{2} | | \frac{\partial ζ_{N}}{\partial t} | |_{- 1}^{2} + c \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c^{'} . \end{array}

It follows from (54)–(56) that

(57)

\begin{matrix} \frac{d}{dt} (| | \nabla ζ_{N} | |^{2} + 2 \int_{Ω} F_{N} (ς_{N}) dx) + | | \frac{\partial ζ_{N}}{\partial t} | |_{- 1}^{2} \\ \leq c (| | f_{N} (ς_{N}) | |_{L^{1} (Ω)} (| | ζ_{N} | |^{2} + 1) + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx) . \end{matrix}

By (54)

\begin{matrix} | | \frac{\partial ζ_{N}}{\partial t} | |_{- 1}^{2} + \frac{1}{2} \frac{d}{dt} | | \nabla ζ_{N} | |^{2} = - [f_{N} (ς_{N}), \frac{\partial ζ_{N}}{\partial t}] - [ς_{N}^{2} - p (1 - ς_{N}), {(- Δ)}^{- 1} \frac{\partial ς_{N}}{\partial t}] \\ \leq - \frac{d}{dt} \int_{Ω} F_{N} (ς_{N}) dx + c | | f_{N} (ς_{N}) | |_{L^{1} (Ω)} (| | ζ_{N} | |^{2} + 1) + \frac{1}{2} | | \frac{\partial ζ_{N}}{\partial t} | |_{- 1}^{2} + c \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx, \end{matrix}

then

\begin{matrix} \frac{1}{2} | | \frac{\partial ζ_{N}}{\partial t} | |_{- 1}^{2} + \frac{1}{2} \frac{d}{dt} | | \nabla ζ_{N} | |^{2} + \frac{d}{dt} \int_{Ω} F_{N} (ς_{N}) dx \\ \leq c (| | f_{N} (ς_{N}) | |_{L^{1} (Ω)} (| | ζ_{N} | |^{2} + 1) + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx), \end{matrix}

and next we multiply by 2 and get (57). It follows from (53) that ς_N is bounded in L^∞(0, T ; L²(Ω))∩L²(0, T ; H²(Ω)), and

\int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx

is bounded in L¹(0, T) which implies that ς_N is bounded in L⁴((0, T) × Ω) and f_N (ς_N) is bounded in L¹((0, T) × Ω) independently of N. It therefore follows from (57) that ς_N is also bounded in L^∞(0, T ; H¹(Ω)) and

\frac{\partial ς_{N}}{\partial t}

is bounded in L²(0, T ; H⁻¹(Ω)) independently of N (note that F_N is bounded on [−1, 1], independently of N).

We finally multiply (30) by $\bar{f_{N} (ς_{N})}$ and integrate over Ω and have

| | \bar{f_{N} (ς_{N})} | |^{2} - [Δ ζ_{N}, \bar{f_{N} (ς_{N})}] + [{(- Δ)}^{- 1} \frac{\partial ζ_{N}}{\partial t}, \bar{f_{N} (ς_{N})}] + [{(- Δ)}^{- 1} \bar{η (ς_{N})}, \bar{f_{N} (ς_{N})}] = 0,

noting that

\begin{array}{l} [Δ ζ_{N}, \bar{f_{N} (ς_{N})}] & = [Δ ζ_{N}, f_{N} (ς_{N})] \\ = - [f_{N}^{'} (ς_{N}) \nabla ς_{N}, \nabla ς_{N}] \\ \leq λ_{1} | | \nabla ς_{N} | |^{2} . \end{array}

Indeed, we have that

\begin{array}{l} [Δ ζ_{N}, \bar{f_{N} (ς_{N})}] & = [Δ ζ_{N}, f_{N} (ς_{N}) - 〈 f_{N} (ς_{N}) 〉] \\ = [Δ ζ_{N}, f_{N} (ς_{N})] \\ = \int_{Ω} Δ ς_{N} f_{N} (ς_{N}) dx \\ = - \int_{Ω} \nabla ς_{N} \cdot \nabla f_{N} (ς_{N}) dx \\ = - \int_{Ω} \nabla ς_{N} \cdot \nabla (ς_{N}) f_{N}^{'} (ς_{N}) dx \\ \leq λ_{1} | | \nabla ς_{N} | |^{2} . \end{array}

Proceeding as above, we get the inequality

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| | \bar{f_{N} (ς_{N})} | |^{2} \leq c (| | ς_{N} | |_{H^{1} (Ω)}^{2} + \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + | | \frac{\partial ζ_{N}}{\partial t} | |_{- 1}^{2}) .

Indeed, we have that

\begin{array}{l} | [- Δ^{- 1} \frac{\partial ζ_{N}}{\partial t}, \bar{f_{N} (ς_{N})}] | & \leq | | - Δ^{- 1} \frac{\partial ζ_{N}}{\partial t} | | | | \bar{f_{N} (ς_{N})} | | \\ \leq \frac{ε}{2} | | \frac{\partial ζ_{N}}{\partial t} | |_{- 1}^{2} + \frac{1}{2 ε} | | \bar{f_{N} (ς_{N})} | |^{2} - | [- Δ^{- 1} \bar{η (ς_{N})}, \bar{f_{N} (ς_{N})}] | \\ \leq ∥ - Δ^{- 1} \bar{η (ς_{N})} ∥ | | \bar{f_{N} (ς_{N})} | | \\ \leq \frac{ε}{2} | | - Δ^{- 1} \bar{η (ς_{N})} | |^{2} + \frac{1}{2 ε} | | \bar{f_{N} (ς_{N})} | |^{2} \\ \leq c | | \bar{η (ς_{N})} | |^{2} + | | \bar{f_{N} (ς_{N})} | |^{2} \\ \leq c \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c^{'} + \frac{1}{2 ε} | | \bar{f_{N} (ς_{N})} | |^{2} . \end{array}

Therefore

\begin{matrix} | | \bar{f_{N} (ς_{N})} | |^{2} = [Δ ζ_{N}, \bar{f_{N} (ς_{N})}] - [{(- Δ)}^{- 1} \frac{\partial ζ_{N}}{\partial t}, \bar{f_{N} (ς_{N})}] \\ \leq λ_{1} | | \nabla ς_{N} | |^{2} + \frac{ε}{2} | | \frac{\partial ζ_{N}}{\partial t} | |_{- 1}^{2} + \frac{1}{2 ε} | | \bar{f_{N} (ς_{N})} | |^{2} + c \int_{Ω} (ζ_{N}^{4} + {〈 ς_{N} 〉}^{2} ζ_{N}^{2}) dx + c^{'} + \frac{1}{2 ε} | | \bar{f_{N} (ς_{N})} | |^{2}, \end{matrix}

and we get (58).

This results in a uniform (with respect to N) estimate for $\bar{f_{N} (ς_{N})}$ in L²((0, T) × Ω). From (18) it follows that

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| 〈 f_{N} (ς_{N}) 〉 | \leq c_{δ} ∥ ζ_{N} ∥ | | \bar{f_{N} (ς_{N})} | | + c_{δ}^{'},

we find a uniform (with respect to N) estimate for f_N (ς_N) in L₂((0, T) × Ω).

Besides, owing to (18) with 〈ς_N〉 = m and ζ_N = s,

\begin{matrix} f_{N} (ς_{N}) - ς_{N} \geq c_{6} (F_{N} (ς_{N}) + | f_{1, N} (ς_{N}) |) - c_{7} \\ | f_{1, N} (ς_{N}) | \leq \frac{1}{c_{6}} f_{N} (ς_{N}) ζ_{N} - F_{N} (ς_{N}) + \frac{c_{7}}{c_{6}} \end{matrix}

\begin{array}{l} | 〈 f_{1, N} (ς_{N}) 〉 | & \leq 〈 | f_{1, N} (ς_{N}) | 〉 \\ \leq \frac{1}{c_{6}} \frac{1}{Vol (Ω)} \int_{Ω} f_{N} (ς_{N}) ζ_{N} dx - \frac{1}{Vol (Ω)} \int_{Ω} F_{N} (ς_{N}) dx + \frac{c_{7}}{c_{6}}, \end{array}

but F_N (ς_N) ≤ c f (ς_N)ς_N + | f_1,N (ς_N)| and f_N (ς_N) = f_1,N (ς_N) − λ₁ς_N.

\begin{array}{l} | 〈 f_{N} (ς_{N}) 〉 | & \leq | 〈 f_{1, N} (ς_{N}) 〉 | + λ_{1} | 〈 ς_{N} 〉 | \\ \leq | 〈 f_{1, N} (ς_{N}) 〉 | + c | | ς_{N} | |_{L^{2} (Ω)} \\ \leq c | | ς_{N} | |_{L^{2} (Ω)} + | | \bar{f_{N} (ς_{N})} | | . \end{array}

3.2. Existence of solutions

We have the following theorems.

Theorem 2

We assume that ς₀ ∈ H¹(Ω) is such that |〈ς₀〉| < 1 and −1 < ς₀(x) < 1 a.e., x ∈ Ω, then there exists T = T (ς₀) > 0 and a solution of (7)–(9) on [0, T ] such that ς ∈ C([0, T ]; H¹(Ω)) ∩ L²(0, T ; H²(Ω)) ∩ L⁴((0, T) × Ω) and $\frac{\partial ς}{\partial t} \in L^{2} (0, T; H^{- 1} (Ω))$ .

Furthermore, −1 < ς(x, t) < 1 a.e. (x, t) ∈ Ω × (0, T).

The proof of this theorem is standard due to the uniform estimates obtained in the previous section.

Theorem 3

Assume that the same assumptions apply as in Theorem 2, then the solution ς is global in time.

Proof

Consider [0, T^∗), such that T^∗ > 0 is the maximal time interval in which the solution ς is given in Theorem 2 exists, so that

| ς (x, t) | \leq 1 a.e. (x, t) \in Ω \times [0, T^{*}) .

Furthermore, 〈ς〉 satisfies

\frac{d 〈 ς 〉}{dt} + 〈 ς^{2} - p (1 - ς) 〉 = 0 .

Using the fact that

\frac{d 〈 ς 〉}{dt} + p 〈 ς 〉 = - 〈 ς^{2} - 1 〉,

which yields

〈 ς (t) 〉 = e^{- pt} 〈 ς_{0} 〉 - e^{- pt} \int_{0}^{t} e^{ps} 〈 ς^{2} - 1 〉 dx .

Noting that

| 〈 ς^{2} - 1 〉 | \leq 2,

this yields

| 〈 ς (t) 〉 | \leq e^{- pt} 〈 ς_{0} 〉 + 1 - e^{- pt}, t \in [0, T^{*}) .

In particular, it follows from the last inequality that

| ς (x, t) | \leq 1, t \in [0, T^{*}) .

4. Conclusions

In this study, we examined a variation of the Cahn-Hilliard equation featuring a logarithmic nonlinear term and a proliferation term given by η(s) = s² − p(1 − s). The model, subject to Neumann boundary conditions, captures interactions between liquid and gas, with p denoting the gas pressure. Specifically, the model finds application in understanding the formation of islands. We successfully demonstrated the existence of a solution to the problem. Notably, our challenge stemmed from the singularities in the nonlinear terms. Constructing approximated problems posed difficulty, as we could not rule out the possibility of solutions to these approximated problems experiencing blowup in finite time. Consequently, deriving uniform estimates for the approximated problems in the presence of singularities became a more intricate task.

It is noteworthy that our future work will delve into the exploration of the Cahn-Hilliard equation incorporating a fidelity term of the form λ₀χ_Ω\D(x)(u−h). Here, λ₀ stands as a suitably large constant, and D represents the inpainting model. The results indicate that inpainting in this scenario is faster and more efficient compared to a model with a regular polynomial nonlinear term.

5. Declarations

5.1. Conflict of interest

Not applicable.

5.2. Funding

Not applicable.

5.3. Author’s contribution

H.F.-Data Curation, Conceptualization, Design, Formal Analysis, Project Administration. M.B.-Data Curation, Conceptualization, Design. H.A.-Writing - Original Draft, Investigation. Y.A.-Writing - Original Draft, Resources. All authors reviewed the results and approved the final version of the manuscript.

5.4. Acknowledgement

The authors wish to thank the referees for their careful reading of the article and useful comments.

5.5. Data availability statement

All data that support the findings of this study are included within the article.

5.6. Using of AI tools

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

On a generalization of the Cahn-Hilliard type equation with logarithmic nonlinearities in island formation

Full Article

1. Introduction

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2. Mathematical problem

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Lemma 1

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Proof

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2.1. Notations

3. Applications

3.1. Existence and blow up solutions

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3.2. Existence of solutions

Theorem 2

Theorem 3

Proof

4. Conclusions

5. Declarations

5.1. Conflict of interest

5.2. Funding

5.3. Author’s contribution

5.4. Acknowledgement

5.5. Data availability statement

5.6. Using of AI tools