am Applied Mathematics 2152-7393 2152-7385 Scientific Research Publishing 10.4236/am.2026.173009 am-149902 Article Physics Mathematics A Conserved Phase-Field Model with Dirichlet Boundary Condition and Regular Potentials Batangouna Narcisse 1 Institut Supérieur d’Architecture, Urbanisme, Bâtiment et Travaux Publics, Université DENIS SASSOU-N’GUESSO, Kintélé, République du Congo

The author declares no conflicts of interest regarding the publication of this paper.

 03 03 2026 03 2026 17 03 153 163 10 10 2025 28 02 2026 03 03 2026 © 2026 by the authors and Scientific Research Publishing Inc. 2026 This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( https://creativecommons.org/licenses/by/4.0/ ). https://doi.org/10.4236/am.2026.173009

Phase field models are a fundamental tool in describing phase transition phenomena. In this context, our study focuses on a conserved phase field model, subject to Dirichlet boundary conditions and incorporating regular potentials. The main objective of this article is to rigorously establish the existence and uniqueness of solutions for the model under consideration. The Dirichlet boundary conditions impose constraints on the behavior of the phase field at the boundary of the domain.

 A Conserved Phase-Field Model Phase-Field Regular Potential Dirichlet Boundary Conditions 1. Introduction

The Caginalp proposed in [1][2] two phase-field systems, namely,

∂ u ∂ t − Δ u + f ( u ) = θ ∂ θ ∂ t − Δ θ = − ∂ u ∂ t

called non conserved system, and

∂ u ∂ t + Δ 2 u − Δ f ( u ) = − Δ θ ∂ θ ∂ t − Δ θ = − ∂ u ∂ t

called conserved system (in the sense that, when endowed with Dirichlet Boundary conditions, the spatial average of u is conserved).

In this paper, we consider the following model:

∂ u ∂ t + Δ 2 u − Δ f ( u ) = − Δ ( φ − Δ φ ) , ∂ φ ∂ t − Δ ∂ φ ∂ t − Δ φ = − ∂ u ∂ t ,

derived from the theory of heat conduction involving two temperatures (see, for example, [3]-[7] and references therein), in θ = φ − Δ φ , this highlights the complexity of thermal phenomena in non ideal systems. In these materials, thermal dissipation can be influenced by variousmechanics, such as internal structure, the presence of impurities, or interactions between particles. These factors can lead to thermal behaviors that cannot be described by classical models, where a single temperature is generally sufficient [8]-[13].

Indeed the distinction between conductive temperature and thermodynamic temperature allows for a better understanding of heat dynamics in complex materials [14]-[20].

For example in systems where relaxation processes or non-linear effects are present, the two temperatures can diverge, resulting in unexpected effects in heat transfer [21]-[23].

This approach also paves the way for new research on heat management in advanced materials, particularly for applications in engineering and technology where precise control of thermal conductivity is crucial [24]-[26].

If you have any questions or would like to discuss a specific aspect of this topic further, feel free to let me know by considering the linear approximation theory of heat conduction and the total free energy

where Ω is the domain occupied by the material and F is an anti-derivative of f. Finally, the governing equation for a variationnal derivative [27]-[32]. The governing equation for φ is obtained by linearization of the energy equation in the isotropic case, which yields (see [33]-[40])

∂ φ ∂ t − Δ ∂ φ ∂ t − Δ φ = r .

r being the heat supplied from the extermal word, see [41][42], and then, taking

r = − ∂ u ∂ t , we obtain equation (5) and (6). We endow this model with Dirichlet

Boundary Conditions and initial conditions. Then, we are led to the following initial and boundary value problem:

∂ u ∂ t + Δ 2 u − Δ f ( u ) = − Δ ( φ − Δ φ ) , in Ω   , ∂ φ ∂ t − Δ ∂ φ ∂ t − Δ φ = − ∂ u ∂ t , in Ω   , u | Γ = Δ u | Γ = φ | Γ = 0 , on ∂ Ω   , u | t = 0 = u 0 , φ | t = 0 = φ 0 ,

As far as the nonlinear term f is concerned, we assume that

f ∈ c 1 ( R ) , f ( 0 ) = 0 f ′ ≥ − c 0 , c 0 ≥ 0 , f ( s ) s ≥ c 1 F ( s ) − c 2 ≥ − c 3 , c 1 > 0 , c 2 , c 3 ≥ 0 , s ∈ R ,

where F ( s ) = ∫ 0 s f ( ξ ) d ξ . In particular, the usual cubic non linear term f ( s ) = s 3 − s satisfies these assumptions. We futher assume that

u 0 ∈ H 0 1 ( Ω ) ∩ H 2 ( Ω )

Remark 1.1.We take here, for simplicity, Dirichlet Boundary Conditions. However, we can obtain the same results for Neumann Boundary Conditions, namely,

∂ u ∂ ν = ∂ Δ u ∂ ν = ∂ φ ∂ ν on Γ

where v denotes the unit outer normal to Γ . To do so, we rewrite, owing to (7) and (8), the equations in the form

∂ u ¯ ∂ t + Δ 2 u ¯ − Δ ( f ( u ) − 〈 f ( u ) 〉 ) = − Δ ( φ ¯ − Δ φ ¯ ) ∂ φ ¯ ∂ t − Δ ∂ φ ¯ ∂ t − Δ φ ¯ = − ∂ u ¯ ∂ t ,

where v ¯ = v − 〈 v 〉 , | 〈 v 0 〉 | ≤ M 1 , | 〈 v 0 〉 | ≤ M 2 , for fixed positive constants M 1 and M 2 . Then, we note that

v → ( ‖ ( − Δ ) − 1 2 v ‖ 2 + 〈 v 〉 2 ) 1 2

here, − Δ denotes the minus Laplace operator with Neumann boundary conditions and acting on functions with null average and where it is understood that (see [43]-[52])

〈 ⋅ 〉 = 1 v o l ( Ω ) 〈 ⋅ , 1 〉 H − 1 ( Ω ) , H 1 ( Ω )

Furthermore

v ↦ ( ‖ v ¯ ‖ 2 + 〈 v 〉 2 ) 1 2 , v ↦ ( ‖ ∇ v ‖ 2 + 〈 v 〉 2 ) 1 2 , v ↦ ( ‖ Δ v ‖ 2 + 〈 v 〉 2 ) 1 2 , v ↦ ( ‖ ∇ Δ v ‖ 2 + 〈 v 〉 2 ) 1 2 ,

are norms in H − 1 ( Ω ) , L 2 ( Ω ) , H 1 ( Ω ) , H 2 ( Ω ) and H 3 ( Ω ) , respectively, which are equivalent to the usual ones.

We further assume that

| f ( s ) | ≤ ε F ( s ) + c ε , ∀ ε > 0 , s ∈ R ,

which allows to deal with term 〈 f ( u ) 〉 .

 2. Notations

We denote by ‖ ⋅ ‖ the usual L 2 -norm (with associated product scalar (.,.) and

set ‖ ⋅ ‖ − 1 = ‖ ( − Δ ) − 1 2 . ‖ , where − Δ denotes the minus Laplace operator with

Dirichlet Boundary Conditions. More generally, ‖ ⋅ ‖ X denote the norm of Banach space X.

Throughout this paper, the same letters c 1 , c 2 and c 3 denote (generally positive) constants which may change from line to line, or even a same line.

 3. The Priori Estimates

The estimates derived in this section are formal, but they can easily be justified within a Galerkin scheme.

We rewrite (7) in the equivalent form

( − Δ ) − 1 ∂ u ∂ t − Δ u + f ( u ) = φ − Δ φ .

We multiply (16) by ∂ u ∂ t and have, integrating over Ω and by parts;

d d t ( ‖ ∇ u ‖ 2 + 2 ∫ Ω F ( u ) d x ) + 2 ‖ ∂ u ∂ t ‖ − 1 2 = 2 ( φ − Δ φ , ∂ u ∂ t )

We then multiply (8) by φ − Δ φ and obtain

d d t ( ‖ φ ‖ 2 + 2 ‖ ∇ φ ‖ 2 + ‖ Δ φ ‖ 2 ) + 2 ‖ ∇ φ ‖ 2 + ‖ Δ φ ‖ 2 = − 2 ( φ − Δ φ , ∂ u ∂ t ) .

Summing (17) and (18), we find the differential equality

d d t E 1 + 2 ‖ ∂ u ∂ t ‖ − 1 2 + 2 ‖ ∇ φ ‖ 2 + ‖ Δ φ ‖ 2 = 0

where

(20)

satisfies

E 1 ≥ c ( ‖ u ‖ H 1 2 + ‖ φ ‖ H 2 2 + ∫ Ω F ( u ) d x )

Next, we multiply (7) by u and have, integrating over Ω

1 2 d d t ‖ u ‖ − 1 2 + ‖ ∇ u ‖ 2 + ∫ Ω F ( u ) d x = ( φ − Δ φ , u )

and by Poincare inequalities ‖ u ‖ − 1 ≤ c ‖ ∇ u ‖

d d t ‖ u ‖ − 1 2 + c ( ‖ ∇ u ‖ 2 + ∫ Ω F ( u ) d x ) ≤ c ′ ‖ ∇ φ ‖ 2 + c ″

Multipling also (8) by ∂ φ ∂ t , we have

d d t ‖ ∇ φ ‖ 2 + ‖ ∂ φ ∂ t ‖ 2 + ‖ ∇ ∂ φ ∂ t ‖ 2 ≤ ‖ ∂ u ∂ t ‖ 2

Summing (19), γ 1 (23) and (24), where γ 1 > 0 , we find a differential inequality of the form

d d t E 2 + c ( E 2 + ‖ ∂ u ∂ t ‖ − 1 2 + ‖ ∇ ∂ u ∂ t ‖ 2 + ‖ ∂ φ ∂ t ‖ 2 + ‖ ∇ ∂ φ ∂ t ‖ ) ≤ c ′ , c > 0.

where

E 2 = E 1 + ‖ ∇ φ ‖ 2 + γ 1 ‖ ∇ u ‖ 2

satisfies

E 2 ≥ c ( ‖ u ‖ H 1 2 + ‖ φ ‖ H 2 2 + ∫ Ω F ( u ) d x )

We multiply (7) by − Δ u and have, owing to (12), where ‖ ∇ u ‖ − 1 ≤ c ‖ ∇ u ‖

d d t ‖ ∇ u ‖ − 1 2 + ‖ Δ u ‖ 2 ≤ c 0 ‖ ∇ u ‖ 2 + ‖ ∇ φ ‖ 2 + ‖ Δ φ ‖ 2

Summing (25) and γ 2 times (28), where γ 2 > 0 is small enough, we obtain a differential inequality of the form

d d t E 3 + c ( E 3 + ‖ Δ u ‖ 2 + ‖ ∂ u ∂ t ‖ − 1 2 + ‖ ∇ ∂ u ∂ t ‖ 2 + ‖ ∂ φ ∂ t ‖ 2 + ‖ ∇ ∂ φ ∂ t ‖ ) ≤ c ″

where

E 3 = E 2 + γ 2 ‖ ∇ u ‖ − 1 2

satisfies

E 3 ≥ c ( ‖ u ‖ H 1 2 + ‖ φ ‖ H 2 2 + ∫ Ω F ( u ) d x )

Multiply (16) by − Δ ∂ u ∂ t and integrate over. We have

(30)

We deduce the following inequality

d d t ‖ Δ u ‖ 2 + 2 ‖ ∂ u ∂ t ‖ 2 ≤ 2 ∫ Ω | f ′ ( u ) | | ∇ u | | ∇ ∂ u ∂ t | d x + 2 ( ∇ φ , ∇ ∂ u ∂ t ) + 2 ( Δ φ , Δ ∂ u ∂ t ) .

Thanks to use f ′ ( s ) , we find the estimate

Since u ∈ L ∞ ( 0 , T , H 0 1 ( Ω ) ) , then the estimate (31) implies

d d t ‖ Δ u ‖ 2 + 2 ‖ ∂ u ∂ t ‖ 2 ≤ c ‖ Δ u ‖ 2 + ‖ ∇ ∂ u ∂ t ‖ + 2 ( ∇ φ , ∇ ∂ u ∂ t ) + 2 ( Δ φ , Δ ∂ u ∂ t )

Multiplying (8) by Δ 2 φ and integrating over Ω , we get

d d t ( ‖ Δ φ ‖ 2 + ‖ ∇ Δ φ ‖ 2 ) + 2 ‖ ∇ Δ φ ‖ 2 = − 2 ( Δ φ , Δ ∂ u ∂ t )

Summing (32) and (33), we obtain

d d t E 4 + 2 ∂ u ∂ t + 2 ‖ ∇ Δ φ ‖ 2 ≤ c ‖ Δ u ‖ 2 + 2 ‖ ∇ ∂ u ∂ t ‖ 2 + ‖ ∇ φ ‖ 2

where

E 4 = ‖ Δ u ‖ 2 + ‖ Δ φ ‖ 2 + ‖ ∇ Δ φ ‖ 2 γ 3 d d t E 5 + c ( E 5 + ‖ ∂ u ∂ t ‖ − 1 2 + ‖ ∇ ∂ u ∂ t ‖ 2 + ‖ ∂ φ ∂ t ‖ + ‖ ∇ ∂ φ ∂ t ‖ 2 ) ≤ c ‖ ∇ φ ‖ 2 + c ″

where

E 5 = E 3 + γ 3 E 4

satisfies

We multiply (8) by − Δ ∂ φ ∂ t and have, integrating over Ω and by parts,

d d t ‖ Δ φ ‖ 2 + ‖ Δ ∂ φ ∂ t ‖ 2 + 2 ‖ ∇ ∂ φ ∂ t ‖ 2 ≤ ‖ ∂ u ∂ t ‖ 2

hence, noting that ‖ ∂ u ∂ t ‖ 2 ≤ c ‖ ∂ u ∂ t ‖ − 1 ‖ ∇ ∂ u ∂ t ‖ ,

d d t ‖ Δ φ ‖ 2 + ‖ Δ ∂ φ ∂ t ‖ 2 + 2 ‖ ∇ ∂ φ ∂ t ‖ 2 ≤ c ‖ ∂ u ∂ t ‖ − 1 + 1 2 ‖ ∇ ∂ u ∂ t ‖ 2

Summing (35) and γ 4 (37), we fond a differential inequality of the form

d d t E 6 + c ( E 6 + ‖ ∂ u ∂ t ‖ − 1 2 + ‖ ∇ ∂ u ∂ t ‖ 2 + ‖ ∂ φ ∂ t ‖ + ‖ ∇ ∂ φ ∂ t ‖ 2 + ‖ Δ ∂ φ ∂ t ‖ 2 ) ≤ c ‖ ∇ φ ‖ 2 + c ″

where

E 6 = E 5 + γ 4 ‖ Δ φ ‖ 2

satisfies

 4. Existence and Uniqueness of Solutions

Theorem 4.1. (Existence) We assume ( u 0 , φ 0 ) ∈ ( H 0 1 ∩ H 2 ) × ( H 0 1 ∩ H 3 ) , the problem (7)-(10) possesses at least one solution ( u , φ ) such that

u ∈ L ∞ ( R + ; H 0 1 ∩ H 2 ) ∩ L 2 ( 0 , T ; H 0 1 ∩ H 2 ) ∂ u ∂ t ∈ L 2 ( 0 , T ; H 0 1 )

Theorem 4.2. The system (7)-(10) possesses a unique solution with the above regularity.

Proof. There only remains to prove the uniqueness.

Let ( u ( 1 ) , φ ( 1 ) ) and ( u ( 2 ) , φ ( 2 ) ) be two solutions (7)-(10) with initial data

( u 0 ( 1 ) , φ 0 1 ) and ( u 0 ( 2 ) , φ 0 2 ) , respectively. We set

( u , φ ) = ( u ( 1 ) , φ ( 1 ) ) − ( u ( 2 ) , φ ( 2 ) )

and

( u 0 , φ 0 ) = ( u 0 ( 1 ) , φ 0 1 ) − ( u 0 ( 2 ) , φ 0 2 )

Then, ( u , φ ) satisfies

∂ u ∂ t + Δ 2 u − Δ ( f ( u ( 1 ) ) − f ( u 2 ) ) = − Δ ( φ − Δ φ ) , in Ω   , ∂ φ ∂ t − Δ ∂ φ ∂ t − Δ φ = − ∂ u ∂ t , in Ω   , u | Γ = Δ u | Γ = φ | Γ = 0 , on ∂ Ω   , u | t = 0 = u 0 , φ | t = 0 = φ 0 ,

Multiplying (38) by ( − Δ ) − 1 ∂ u ∂ t , we have

d d t ‖ u ‖ 2 + 2 ‖ ∂ u ∂ t ‖ − 1 2 + 2 ( f ( u ( 1 ) ) − f ( u 2 ) , ∂ u ∂ t ) = 2 ( ∂ u ∂ t , φ − Δ φ )

Multiplying (39) by φ − Δ φ , we obtain

d d t ( ‖ φ ‖ 2 + 2 ‖ ∇ φ ‖ 2 + ‖ Δ φ ‖ 2 ) + 2 ‖ ∇ φ ‖ 2 + 2 ‖ Δ φ ‖ 2 = − 2 ( ∂ u ∂ t , φ − Δ φ )

Summing (42) and (43), we find

d d t ( ‖ u ‖ 2 + ‖ φ ‖ 2 + 2 ‖ ∇ φ ‖ 2 + ‖ Δ φ ‖ 2 ) + 2 ‖ ∂ u ∂ t ‖ − 1 2 + 2 ‖ ∇ φ ‖ 2 + 2 ‖ Δ φ ‖ 2 = − 2 ( f ( u ( 1 ) ) − f ( u 2 ) , ∂ u ∂ t )

we have

2 | ( f ( u ( 1 ) ) − f ( u 2 ) , ∂ u ∂ t ) | ≤ 2 ‖ ( − Δ ) 1 2 ( f ( u ( 1 ) ) − f ( u ( 2 ) ) ) ‖ ‖ ( − Δ ) − 1 2 ∂ u ∂ t ‖ ≤ ‖ ( − Δ ) 1 2 ( f ( u ( 1 ) ) − f ( u ( 2 ) ) ) ‖ 2 + ‖ ( − Δ ) − 1 2 ∂ u ∂ t ‖ 2 ≤ Q ‖ u ‖ 2 + ‖ ∂ u ∂ t ‖ − 1 2

where, here and below

Q = Q ( ‖ u 0 ( 1 ) ‖ H 0 , ‖ φ 0 ( 1 ) ‖ H 2 , ‖ u 0 ( 2 ) ‖ H 0 , ‖ φ 0 ( 2 ) ‖ H 2 )

There

d d t ( ‖ u ‖ 2 + ‖ φ ‖ 2 + 2 ‖ ∇ φ ‖ 2 + ‖ Δ φ ‖ 2 ) + ‖ ∂ u ∂ t ‖ − 1 2 + 2 ‖ ∇ φ ‖ 2 + 2 ‖ Δ φ ‖ 2 ≤ Q ‖ u ‖ 2

In particular

d d t E 6 + c ( E 6 + ‖ ∂ u ∂ t ‖ − 1 2 ) ≤ Q ‖ u ‖ 2

It thus follows form (45) and Gronwall’s lemma that

‖ u ‖ H 0 2 + ‖ φ ‖ H 2 2 ≤ c e Q t ( ‖ u 2 ‖ H 0 2 + ‖ φ 0 ‖ H 2 2 ) , t > 0

hence the uniqueness, as well as the continuous dependence with respect to the initial data. □

 5. Conclusions

Our study focuses on the mathematical analysis of the phase field model proposed by Caginalp. We specifically concentrate on the questions of existence and uniqueness of solutions, considering a polynomial potential under Dirichlet boundary conditions.

In this context, it is pertinent to explore a numerical approach to complement our theoretical analysis.

Implementing numerical methods, such as finite element methods or finite difference methods, could allow us to solve the partial differential equations associated with the model. Furthermore, numerical simulation serves as a valuable tool for visualizing and studying the dynamic behaviors of the system, particularly phase transitions and equilibrium configurations.

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