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Introduction
The significance of the Cahn-Hilliard equation in materials science cannot be overstated. This equation effectively captures crucial qualitative aspects of two-phase systems, particularly in relation to phase separation processes. In the realm of materials science, the resulting pattern formation is termed 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 materials science, the Cahn-Hilliard equation finds 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]:
\frac{{\partial \varsigma }}{{\partial t}} + {\Delta ^2}\varsigma - \Delta f(\varsigma ) + \eta (\varsigma ) = 0,
where
\eta (s) = s(s - 1)
and
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
\eta (s) = \alpha {s^2} + \beta s + \gamma ,
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
\frac{{\partial \varsigma }}{{\partial t}} + {\Delta ^2}\varsigma - \Delta f(\varsigma ) + \eta (x,\varsigma ) = 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
\eta (x,s) = \frac{\alpha }{2}(s + 1) - \beta {(1 - s)^2}{(1 + s)^2} + s(x,t)
, where α and β denote growth and death coefficients. Additionally, the function η(x, ς) = 1Ω\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\eta (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.
Mathematical problem
Let Ω ⊂ ℝn, n = 1, 2 or 3 be a bounded and regular domain with boundary Γ. We consider the following problem
{\partial _t}\varsigma + {\Delta ^2}\varsigma - \Delta f(\varsigma ) + \eta (\varsigma ) = 0,{\partial _\nu }\varsigma = {\partial _\nu }\Delta \varsigma = 0,\;{\rm{on}}\;\Gamma ,\varsigma (0,x) = {\varsigma _0}(x),\;{\rm{in}}\;\Omega .
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
{f^\prime} \ge - {\lambda _1}.
Let us write
F(s) = \frac{{{\lambda _1}}}{2}\left( {1 - {s^2}} \right) + {F_1}(s)
with
{f_1} = F_1^\prime
and introduce for N ∈ ℕ the approximated function F1,N ∈ C4(ℝ), which is defined by:
F_{1,N}^{(4)}(s) = \left\{ {\begin{array}{*{20}{l}}{F_1^{(4)}\left( {1 - \frac{1}{N}} \right);}&{{\rm{if}}\;s \ge 1 - \frac{1}{N},}\\{F_1^{(4)}(s);}&{{\rm{if}}\;|s| \le 1 - \frac{1}{N},}\\{{F^{(4)}}\left( { - 1 + \frac{1}{N}} \right);}&{{\rm{if}}\;s \le - 1 + \frac{1}{N}.}\end{array}} \right.F_{1,N}^{(k)}(0) = F_1^{(k)}(0),\quad k = 0,1,2,3
and
{f_1}(s) = f(s) + {\lambda _1}s = \frac{{{\lambda _2}}}{2}\ln \left( {\frac{{1 + s}}{{1 - s}}} \right)
Hence,
{F_{1,N}}(s) = \left\{ {\begin{array}{*{20}{l}}{\sum \frac{1}{{k!}}F_1^{(k)}\left( {1 - \frac{1}{N}} \right){{\left( {s - 1 + \frac{1}{N}} \right)}^k};}&{{\rm{if}}\;s \ge 1 - \frac{1}{N},}\\{{F_1}(s);}&{{\rm{if}}\;|s| \le 1 - \frac{1}{N},}\\{\sum\limits_{k = 0}^4 \frac{1}{{k!}}F_1^{(k)}\left( { - 1 + \frac{1}{N}} \right){{\left( {s + 1 - \frac{1}{N}} \right)}^k};}&{{\rm{if}}\;s \le - 1 + \frac{1}{N}.}\end{array}} \right.
Setting
\begin{array}{*{20}{c}}{{F_N}(s) = \frac{{{\lambda _1}}}{2}(1 - {s^2}) + {F_{1,N}}(s),}\\{{f_{1,N}} = F_{1,N}^\prime}\end{array}
and
{f_N} = F_N^\prime,
there holds
f_{1,N}^\prime \ge 0,\;\;\;\;\;f_N^\prime \ge - {\lambda _1},{F_N} \ge - {c_1},\;\;\;\;{c_1} \ge 0,{f_N}(s).s \ge {c_2}{F_N}(s) + |{f_{1,N}}(s)| - {c_3},\;\;\;\;\;{c_2} > 0,{c_3} \ge 0,s \in \mathbb{R},
and
\left( {{f_N}(s + m) - {f_N}(m)} \right)s \ge {c_4}\left( {{s^4} + {m^2}{s^2}} \right) - {c_5},\;\;\;\;{c_4} > 0,{c_5} \ge 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{and}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} s,m \in \mathbb{R}.
The constants ci, i = 1, ⋯ , 5 are independent of N for N large enough. More generally:
{f_N}\left( {s + m} \right)s \ge {c_{6,m}}\left( {{F_N}(s + m) + |{f_{1,N}}(s + m)|} \right) - {c_{7,m}},{\kern 1pt} {\kern 1pt} s \in \mathbb{R},{\kern 1pt} m \in ( - 1,1),
where the constants c6,m and c7,m are independent of N for sufficiently large N and exhibit continuous and bounded dependence on m.
Finally, we arrive at
Lemma 1
|\eta (s + m) - \eta (m)| \le {c_8}\left( {{s^2} + |ms|} \right) + {c_9}and|\eta (s + m) - \eta (m){|^2} \le {c_{10}}\left( {{s^4} + {s^2}({m^2} + 1)} \right) + {c_{11}}.
Proof
\begin{array}{*{20}{l}}{|\eta (s + m) - \eta (m)|}&{ = \;|{s^2} + 2sm + ps|}\\{}&{ \le {s^2} + 2|ms| + p|s|}\\{}&{ \le {s^2} + 2|ms| + \frac{{{p^2}}}{2} + \frac{{{s^2}}}{2}}\\{}&{ \le {c_8}\left( {{s^2} + |ms|} \right) + {c_9}}\end{array}
and
\begin{array}{*{20}{l}}{|\eta (s + m) - \eta (m){|^2}}&{ \le \;{{\left( {c\left( {{s^2} + |ms|} \right) + {c^\prime}} \right)}^2}}\\{}&{ \le c{{\left( {\left( {{s^2} + |ms|} \right) + 1} \right)}^2}}\\{}&{ \le c\left( {{s^4} + 2{s^2}|ms| + {m^2}{s^2} + 2{s^2} + 2|ms| + 1} \right)}\\{}&{ \le c\left( {{s^4} + 2{s^2}\left( {\frac{{{m^2}}}{2} + \frac{{{s^2}}}{2}} \right) + {m^2}{s^2} + 2{s^2} + 2\left( {\frac{1}{2} + \frac{{{m^2}{s^2}}}{2}} \right) + 1} \right)}\\{}&{ \le {c_{10}}\left( {{s^4} + {m^2}{s^2} + {s^2}} \right) + {c_{11}}.}\end{array}
We now introduce the approximated problem
{\partial _t}{\varsigma _N} + {\Delta ^2}{\varsigma _N} - \Delta {f_N}({\varsigma _N}) + \eta ({\varsigma _N}) = 0,{\partial _\nu }{\varsigma _N} = {\partial _\nu }\Delta {\varsigma _N} = 0,\;{\rm{on}}\;\Gamma ,{\varsigma _N}(0,x) = {\varsigma _0}(x),\;{\rm{in}}\;\Omega .
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 ςN of (21)–(23).
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
\langle {\kern 1pt} .{\kern 1pt} \rangle = \frac{1}{{{\kern 1pt} {\rm{Vol}}{\kern 1pt} (\Omega )}}\int_\Omega .dx
, being understood that if ζ ∈ H−1(Ω) = H1(Ω)′ then
\langle \zeta \rangle = \frac{1}{{{\kern 1pt} {\rm{Vol}}{\kern 1pt} (\Omega )}}{\langle \zeta ,1\rangle _{{H^{ - 1}}(\Omega ),{H^1}(\Omega )}}.
We also set, whenever this makes sense
\overline \zeta = \zeta - \langle \zeta \rangle .
We note that:
\begin{array}{*{20}{c}}{\zeta \mapsto {{\left( {||\overline \zeta ||_{ - 1}^2 + \langle {\zeta ^2}\rangle } \right)}^{\frac{1}{2}}},}\\{\zeta \mapsto {{\left( {||\overline \zeta |{|^2} + \langle {\zeta ^2}\rangle } \right)}^{\frac{1}{2}}},}\\{\zeta \mapsto {{\left( {||\nabla \zeta |{|^2} + \langle {\zeta ^2}\rangle } \right)}^{\frac{1}{2}}},}\\{\zeta \mapsto {{\left( {||\Delta \zeta |{|^2} + \langle {\zeta ^2}\rangle } \right)}^{\frac{1}{2}}}}\end{array}
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 cδ,
c_\delta ^\prime
and
c_\delta ^{''}
which depend on a parameter δ.
Applications
In this part, we observe the existence and blow up properties of the results founded by using projected scheme.
Existence / blow up solutions
In this section, our objective is to establish estimates for uN 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 fN (ςN) 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 〈ςN〉 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.
This results in a uniform (with respect to N) estimate for
\overline {{f_N}({\varsigma _N})}
in L2((0, T) × Ω). From (18) it follows that
|\langle {f_N}({\varsigma _N})\rangle | \le {c_\delta }\parallel {\zeta _N}\parallel {\kern 1pt} ||\overline {{f_N}({\varsigma _N})} || + c_\delta ^\prime,
we find a uniform (with respect to N) estimate for fN (ςN) in L2((0, T) × Ω).
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 \varsigma }}{{\partial t}} \in {L^2}(0,T;{H^{ - 1}}(\Omega ))
.
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
|\varsigma (x,t)| \le 1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{a.e.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} (x,t) \in \Omega \times [0,{T^ * }).
Furthermore, 〈ς〉 satisfies
\frac{{d\langle \varsigma \rangle }}{{dt}} + \langle {\varsigma ^2} - p(1 - \varsigma )\rangle = 0.
Using the fact that
\frac{{d\langle \varsigma \rangle }}{{dt}} + p\langle \varsigma \rangle = - \langle {\varsigma ^2} - 1\rangle ,
which yields
\langle \varsigma (t)\rangle = {e^{ - pt}}\langle {\varsigma _0}\rangle - {e^{ - pt}}\int_0^t {e^{ps}}\langle {\varsigma ^2} - 1\rangle dx.
Noting that
|\langle {\varsigma ^2} - 1\rangle | \le 2,
this yields
|\langle \varsigma (t)\rangle | \le {e^{ - pt}}\langle {\varsigma _0}\rangle + 1 - {e^{ - pt}},\;\;\;\;\;t \in [0,{T^ * }).
In particular, it follows from last inequality that
|\varsigma (x,t)| \le 1,\;\;\;\;\;t \in [0,{T^ * }).
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 λ0χΩ\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.
Declarations
Conflict of interest
Not applicable.
Funding
Not applicable.
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.
Acknowledgement
The authors wish to thank the referees for their careful reading of the article and useful comments.
Data availability statement
All data that support the findings of this study are included within the article.
Using of AI tools
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
