On the Caginalp for a Conserve Phase-Field with a Polynomial Potentiel of Order 2<i>p</i> - 1

Our aim in this paper is to study on the Caginalp for a conserved phase-field with a polynomial potentiel of order 2 p - 1. In this part, one treats the conservative version of the problem of generalized phase field. We consider a regular potential, more precisely a polynomial term of the order 2 p - 1 with edge conditions of Dirichlet type. Existence and uniqueness are analyzed. More precisely, we precisely, we prove the existence and uniqueness of solutions.

• A Conserved Phase-Field
• Polynomial Potentiel of Order 2<i>p</i> - 1
• Dirichlet Boundary Conditions
• Maxwell-Cattaneo Law

1. Introduction

The Caginalp phase-field model

∂ u ∂ t − Δ u + f ( u ) = θ (1)

∂ θ ∂ t − Δ θ = − ∂ u ∂ t (2)

proposed in [1] , has been extensively studied (see, e.g., [2] - [7] and [8] ). Here, u denotes the order parameter and θ the (relative) temperature.

Furthermore, all physical constants have been set equal to one. This system models, e.g., melting-solidification phenomena in certain classes of materials.

The Caginalp system can be derived as follows. We first consider the (total) free energy

ψ ( u , θ ) = ∫ Ω ( 1 2 | ∇ u | 2 + f ( u ) − u θ − 1 2 θ 2 ) d x , (3)

where Ω is the domain occupied by the materiel.

We then define the enthalpy H as

H = − ∂ ψ ∂ θ (4)

where ∂ denotes a variational derivative, which gives

H = u + θ . (5)

The governing equations for u and θ are then given by (see [9] )

∂ u ∂ t = − ∂ ψ ∂ u , (6)

∂ H ∂ t + d i v q = 0 , (7)

where q is the thermal flux vector. Assuming the classical Fourier Law

q = − ∇ θ , (8)

we find (1) and (2).

Now, a drawback of the Fourier Law is the so-called “paradox of heat conduction”, namely, it predicts that thermal signals propagate with infinite speed, which, in particular, violates causality (see, e.g. [10] and [11] ). One possible modification, in order to correct this unrealistic feature, is the Maxwell-Cattaneo Law.

( 1 + ∂ ∂ t ) q = − ∇ θ , (9)

In that case, it follows from (7) that

( 1 + ∂ ∂ t ) ∂ H ∂ t − Δ θ = 0 ,

hence,

∂ 2 θ ∂ t 2 + ∂ θ ∂ t − Δ θ = ∂ 2 u ∂ t 2 + ∂ u ∂ t . (10)

This model can also be derived by considering, as in [12] (see also [13] - [20] ), the Caginalp phase-field model with the so-called Gurtin-Pipkin Law

q ( t ) = − ∫ 0 + ∞   k ( s ) ∇ θ ( t − s ) d s . (11)

for an exponentially decaying memory kernel k, namely,

k ( s ) = e − s . (12)

Indeed, differentiating (11) with respect to t and integrating by parts, we recover the Maxwell-Cattaneo Law (9).

Now, in view of the mathematical treatment of the problem, it is more convenient to introduce the new variable

α = ∫ 0 t     θ ( s ) d s ,   θ = ∂ α ∂ t , (13)

and we have, integrating (10) with respect to s ∈ [ 0,1 ] .

∂ 2 α ∂ t 2 + ∂ α ∂ t − Δ α = − ∂ u ∂ t (14)

where

α ( t , x ) = ∫ 0 t     T ( τ , x ) d τ + α 0 ( x ) (15)

is the conductive thermal displacement. Noting that T = ∂ α ∂ t , we finally deduce

from (33) and (36)-(37) the following variant of the Caginalp phase-field system (see [17] ):

∂ u ∂ t − Δ u + f ( u ) = ∂ α ∂ t (16)

∂ 2 α ∂ t 2 + ∂ α ∂ t − Δ α = − ∂ u ∂ t (17)

In this paper, we consider the following conserved phase-field model:

∂ u ∂ t + Δ 2 u − Δ f ( u ) = − Δ ∂ α ∂ t (18)

∂ 2 α ∂ t 2 + ∂ α ∂ t − Δ α = − ∂ u ∂ t (19)

These equations are known as the conserved phase-field model (see [21] - [30] ) based on type II heat conduction and with two temperatures (see [3] and [4] ), conservative in the sense that, when endowed with Neumann boundary conditions, the spatial average of the order parameter is a conserved quantity. Indeed, in that case, integrating (18) over the spatial domain Ω , we have the conservation of mass,

〈 u ( t ) 〉 = 〈 u ( 0 ) 〉 ,   t ≥ 0 (20)

〈 ⋅ 〉 = 1 v o l Ω ∫ Ω     d x (21)

denotes the spatial average. Furthermore, integrating (19) over, we obtain

〈 α ( t ) 〉 = 〈 α ( 0 ) 〉 ,   t ≥ 0 (22)

Our aim in this paper is to study the existence and uniqueness of solution of (17)-(39). We consider here only one type of boundary condition, namely, Dirichlet (see [31] [32] [33] ).

2. Setting of the Problem

We consider the following initial and boundary value problem

∂ u ∂ t + Δ 2 u − Δ f ( u ) = − Δ ∂ α ∂ t (23)

∂ 2 α ∂ t 2 + ∂ α ∂ t − Δ α = − ∂ u ∂ t (24)

u | Γ = Δ u | Γ = α | Γ = 0 ,   on     ∂ Ω , (25)

u | t = 0 = u 0 ,     α | t = 0 = α 0 ,     ∂ α ∂ t = α 1 (26)

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

f ∈ C ∞ ( R ) , f ( 0 ) = 0 (27)

Consider the following polynomial potential of order 2p − 1

f ( s ) = ∑ i = 1 2 p − 1     a i s i , p ∈ N ∗ , p ≥ 2 ; a 2 p − 1 = 2 p b 2 p ≥ 0 (28)

The function f satisfies the following properties

1 2 a 2 p − 1 s 2 p − c 1 ≤ f ( s ) s ≤ 3 2 a 2 p − 1 s 2 p + c 1 , (29)

f ′ ( s ) ≥ 1 2 a 2 p − 1 s 2 p − 2 − c 2 ≥ − k , ∀ s ∈ R , k ≥ 0 (30)

where

F ( s ) = ∫ 0 s     f ( τ ) d τ (31)

such as

1 4 p a 2 p − 1 s 2 p − c 3 ≤ F ( s ) ≤ 3 4 p a 2 p − 1 s 2 p + c 3 (32)

Remark 2.1. 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 (33)

where v denotes the unit outer normal to Γ . To do so, we rewrite, owing to (23) and (24), the equations in the form

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

∂ 2 φ ¯ ∂ t 2 + ∂ 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

where, here, − Δ denotes the minus Laplace operator with Neumann boundary conditions and acting on functions with null average and where it is understood that

〈 ⋅ 〉 = 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

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

We further assume that

| f ( s ) | ≤ ε F ( s ) + c ε ,     ∀ ε > 0 ,     s ∈ R , (34)

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

3. Notations

We denote by ‖   ⋅   ‖ the usual L²-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.

4. A Priori Estimates

The estimates derived in this subsection will be formal, but they can easily be justified within a Galerkin scheme. We rewrite (23) in the equivalent form

( − Δ ) − 1 ∂ u ∂ t − Δ u + f ( u ) = ∂ α ∂ t . (35)

We multiply (35) 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 ( ∂ u ∂ t , ∂ α ∂ t ) (36)

We then multiply (24) by ∂ α ∂ t to obtain

d d t ( ‖ ∇ α ‖ 2 + ‖ ∂ α ∂ t ‖ 2 ) + 2 ‖ ∂ α ∂ t ‖ 2 = − 2 ( ∂ u ∂ t , ∂ α ∂ t ) (37)

Summing (36) and (37), we find the differential inequality of the form

d d t ( ‖ ∇ u ‖ 2 + 2 ∫ Ω     F ( u ) d x + ‖ ∇ α ‖ 2 + ‖ ∂ α ∂ t ‖ 2 ) + 2 ‖ ∂ u ∂ t ‖ − 1 2 + 2 ‖ ∂ α ∂ t ‖ 2 = 0 (38)

Integrating from 0 to t with t ∈ [ 0 ; T ] we obtain

∫ 0 t ( d d t ‖ ∇ u ‖ 2 + 2 ∫ Ω     F ( u ) d x + ‖ ∇ α ( s ) ‖ 2 + ‖ ∂ α ( s ) ∂ t ‖ 2 ) d s   + 2 ∫ ‖ ∂ α ( s ) ∂ t ‖ 2 d s + 2 ∫ ‖ ∂ u ( s ) ∂ t ‖ − 1 2 d s = 0

of (35) we deduce

F ( u 0 ) ≤ 3 4 p a 2 p − 1 u 0 2 p + c 3

which involves

2 ∫ Ω   F ( u 0 ) d x ≤ 3 2 p a 2 p − 1 ‖ u 0 ‖ L 2 p 2 p + 2 c 3 | Ω |

still of (35) we have

3 4 p a 2 p − 1 u 0 2 p − c 3 ≤ F ( u )

which involves

1 2 p a 2 p − 1 ‖ u 0 ‖ L 2 p 2 p − 2 c 3 | Ω | ≤ F ( u )

where

E ( t ) + 2 ∫ 0 t ( ‖ ∂ α ( s ) ∂ t ‖ 2 + ‖ ∂ u ( s ) ∂ t ‖ − 1 2 ) d s ≤ C

with

E ( t ) = ‖ ∇ u ( t ) ‖ 2 + 1 2 p a 2 p − 1 ‖ u ( t ) ‖ L 2 p 2 p + ‖ ∂ α ( t ) ∂ t ‖ 2 + ‖ ∇ α ( t ) ‖ 2 (39)

and C = ‖ ∇ u 0 ‖ 2 + 3 2 p a 2 p − 1 ‖ u 0 ‖ L 2 p 2 p + ‖ α 1 ‖ 2 + ‖ ∇ α 0 ‖ 2 + C 3 .

Finally, we conclude that u ∈ L ∞ ( R ∗ ; H 0 1 ( Ω ) ∩ L 2 p ( Ω ) ) ; α ∈ L 2 ( 0, T ; H − 1 ( Ω ) ) ;

∂ u ∂ t ∈ L 2 ( 0, T ; H − 1 ( Ω ) ) ; ∂ α ∂ t ∈ L ∞ ( R + ∗ ; L 2 ( Ω ) ) ∩ L 2 ( 0, T ; L 2 ( Ω ) ) ∀ T > 0

Theorem 4.1. (Existence) We assume ( u 0 , α 0 , α 1 ) ∈ ( H 0 1 ( Ω ) ∩ L 2 p ( Ω ) ) × H 0 1 ( Ω ) × L 2 ( Ω ) then the system (18)-(19) possesses at least one solution ( u , α ) such that

u ∈ L ∞ ( R ∗ ; H 0 1 ( Ω ) ∩ L 2 p ( Ω ) ) ; α ∈ L 2 ( 0, T ; H − 1 ( Ω ) )

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

∀ T > 0

Theorem 4.2. (Uniqueness) Let the assumptions of Theorem 4.1 hold. Then, the system (18)-(19) possesses a unique solution ( u , α ) such that

u ∈ L ∞ ( R ∗ ; H 0 1 ( Ω ) ∩ L 2 p ( Ω ) ) ; α ∈ L 2 ( 0, T ; H − 1 ( Ω ) )

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

∀ T > 0

Let ( u ( 1 ) , α ( 1 ) , ∂ α ( 1 ) ∂ t ) and ( u ( 2 ) , α ( 2 ) , ∂ α ( 2 ) ∂ t ) be two solutions (23)-(25) with initial data ( u 0 ( 1 ) , α 0 ( 1 ) , α 1 ( 1 ) ) and ( u 0 ( 2 ) , α 0 ( 2 ) , α 1 ( 2 ) ) , respectively. We set

( u , α , ∂ α ∂ t ) = ( u ( 1 ) , α ( 1 ) , ∂ α ( 1 ) ∂ t ) − ( u ( 2 ) , α ( 2 ) , ∂ α ( 2 ) ∂ t )

and

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

Then, ( u , α ) satisfies

∂ u ∂ t + Δ 2 u − Δ ( f ( u ( 1 ) ) − f ( u ( 2 ) ) ) = − Δ ∂ α ∂ t (40)

∂ 2 α ∂ t 2 + ∂ α ∂ t − Δ α = − ∂ u ∂ t (41)

u | Γ = Δ u | Γ = α | Γ = 0 ,   on     ∂ Ω , (42)

u | t = 0 = u 0 ,     α | t = 0 = α 0 ,     ∂ α ∂ t = α 1 (43)

We multiply (40) by ( − Δ ) − 1 ∂ u ∂ t , we have

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

d d t ‖ ∇ u ‖ 2 + 2 ‖ ∂ u ∂ t ‖ − 1 2 = − 2 ( f ( u ( 1 ) ) − f ( u ( 2 ) ) , ∂ u ∂ t ) + 2 ( ∂ u ∂ t , ∂ α ∂ t ) . (44)

We multiply by (41) by ∂ α ∂ t , we have

d d t ( ‖ ∇ α ‖ 2 + ‖ ∂ α ∂ t ‖ 2 ) + 2 ‖ ∂ α ∂ t ‖ 2 = − 2 ( ∂ u ∂ t , ∂ α ∂ t ) (45)

Now summing (44) and (45) we obtain

d d t ( ‖ ∇ u ‖ 2 + ‖ ∇ α ‖ 2 + ‖ ∂ α ∂ t ‖ 2 ) + 2 ‖ ∂ u ∂ t ‖ − 1 2 + 2 ‖ ∂ α ∂ t ‖ 2 = − 2 ( f ( u ( 1 ) ) − f ( u ( 2 ) ) , ∂ u ∂ t ) (46)

We know that

f ( u 1 ) − f ( u 2 ) = ∑ k = 1 2 p − 1     a k ( u ( 1 ) k ) − ∑ k = 1 2 p − 1     a k ( u ( 2 ) k ) = ∑ k = 1 2 p − 1     a k ( u ( 1 ) k − u ( 2 ) k )

which involves

| f ( u 1 ) − f ( u 2 ) | ≤ ∑ k = 1 2 p − 1 | a k | | u ( 1 ) k − u ( 2 ) k | ≤ ∑ k = 1 2 p − 1 | a k | | u ( 1 ) − u ( 2 ) | | u ( 1 ) | k − 1 + ∑ j = 1 k − 2 | u ( 1 ) | k − 1 − j | u ( 2 ) | j + | u ( 2 ) | k − 1 .

Based on Young’s inequality, we have

∑ j = 1 k − 2 | u ( 1 ) | k − 1 − j | u ( 2 ) | j ≤ ∑ j = 1 k − 2 ( k − j − 1 k − 1 | u ( 1 ) | k − 1 + j k − 1 | u ( 2 ) | k − 1 )

with p = k − 1 k − j − 1 and q = k − 1 j such as 1 p + 1 q = 1 . So

∑ j = 1 k − 2 | u ( 1 ) | k − 1 − j | u ( 2 ) | j ≤ 1 k − 1 ∑ j = 1 k − 2 ( k − 1 ) | u ( 1 ) | k − 1 + 1 k − 1 ∑ j = 1 k − 2     j ( | u ( 2 ) | k − 1 − | u ( 1 ) | k − 1 ) .

As

∑ j = 1 k − 2     j = ( k − 2 ) ( k − 1 ) 2

then

∑ j = 1 k − 2 | u ( 1 ) | k − 1 − j | u ( 2 ) | j ≤ ( k − 2 ) | u ( 1 ) | k − 1 + k − 2 2 | u ( 2 ) | k − 1 − k − 2 2 | u ( 1 ) | k − 1 ≤ k − 2 2 ( | u ( 1 ) | k − 1 + | u ( 2 ) | k − 1 ) .

We know that

∀ k ∈ N ; k − 2 ≤ k then k − 2 2 ≤ k 2 ≤ k

∑ j = 1 k − 2 | u ( 1 ) | k − 1 − j | u ( 2 ) | j ≤ k ( | u ( 1 ) | k − 1 + | u ( 2 ) | k − 1 )

which gives

| f ( u 1 ) − f ( u 2 ) | ≤ ∑ j = 1 k − 2 | a k | | u ( 1 ) − u ( 2 ) | ( ( k + 1 ) | u ( 1 ) | k − 1 + ( k + 1 ) | u ( 2 ) | k − 2 ) ≤ | u | ∑ j = 1 k − 2 ( k + 1 ) | a k | ( | u ( 1 ) | k − 1 + | u ( 2 ) | k − 1 )

∃   k > 0 such as

( k + 1 ) | a k | ≤ k ; ∀   k ∈ 1,2, ⋯ ,2 p − 1

so

| f ( u 1 ) − f ( u 2 ) | ≤ | u | k ∑ k = 1 k − 2 ( | u ( 1 ) | k − 1 + | u ( 2 ) | k − 1 ) .

Based on Young’s inequality, we have ∀   k ≥ 2

| u ( 1 ) | k − 1 ≤ k − 1 2 p − 2 ( | u ( 1 ) | k − 1 ) 2 p − 2 k − 1 + 2 p − k − 1 2 p − 2

and

| u ( 2 ) | k − 1 ≤ k − 1 2 p − 2 ( | u ( 2 ) | k − 1 ) 2 p − 2 k − 1 + 2 p − k − 1 2 p − 2

that involve

| f ( u 1 ) − f ( u 2 ) | ≤ | u | k 2 p − 2 ∑ k = 1 2 p − 1 ( ( k − 1 ) ( | u ( 1 ) | 2 p − 2 + | u ( 2 ) | 2 p − 2 ) + 2 ( 2 p − k − 1 2 p − 2 ) ) ≤ c | u | ( | u ( 1 ) | 2 p − 2 + | u ( 2 ) | 2 p − 2 + 1 ) .

We finally

∫ Ω | f ( u 1 ) − f ( u 2 ) | | ∂ u ∂ t | d x ≤ c ∫ Ω | u | ( | u ( 1 ) | 2 p − 2 + | u ( 2 ) | 2 p − 2 + 1 ) | ∂ u ∂ t | d x . (47)

The second member of (45) is increased in R n for n = 1 , 2 , 3 .

If n = 1; u i ∈ H 0 1 ( Ω ) ⊂ H 1 ( Ω ) = W 1 , 2 ( Ω ) for i = 1 , 2 .

Thanks to the continuous injection H 1 ( Ω ) ⊂ C ( Ω ¯ ) , then is C > 0 , by applying Holder’s inegality, we get

∫ Ω | u | ( | u ( 1 ) | 2 p − 2 + | u ( 1 ) | 2 p − 2 + 1 ) | ∂ u ∂ t | d x ≤ C ‖ u ‖ ‖ ∂ u ∂ t ‖ ,

which involves using the compact injection H 1 ( Ω ) ⊂ L 2 ( Ω ) , we have

∫ Ω | f ( u 1 ) − f ( u 2 ) | | ∂ u ∂ t | d x ≤ C ‖ u ‖ H 1 ‖ ∂ u ∂ t ‖ (48)

If n = 2 then H 1 ( Ω ) ⊂ L q ( Ω ) , ∀   q ∈ [ 1, ∞ [ .

Based on Holder’s inequality, we have

∫ Ω | u | ( | u ( 1 ) | 2 p − 2 + | u ( 1 ) | 2 p − 2 + 1 ) | ∂ u ∂ t | d x ≤ C ‖ u ‖ L 3 ‖ ∂ u ∂ t ‖ .

Finally

∫ Ω | f ( u 1 ) − f ( u 2 ) | | ∂ u ∂ t | d x ≤ C ‖ u ‖ H 1 ‖ ∂ u ∂ t ‖

If n = 3, then H 1 ( Ω ) ⊂ L q ( Ω ) with q ∈ [ 1,6 ]

In this case, we also

∫ Ω | u | ( | u ( 1 ) | 2 p − 2 + | u ( 1 ) | 2 p − 2 + 1 ) | ∂ u ∂ t | d x ≤ C ‖ u ‖ L 6 ‖ ∂ u ∂ t ‖ .

So

∫ Ω | f ( u 1 ) − f ( u 2 ) | | ∂ u ∂ t | d x ≤ C ‖ u ‖ H 1 ‖ ∂ u ∂ t ‖ .

We notice that in R n for n = 1 , 2 , 3 , we have

∫ Ω | f ( u 1 ) − f ( u 2 ) | | ∂ u ∂ t | d x ≤ C ‖ u ‖ H 1 ‖ ∂ u ∂ t ‖ .

Using Young’s inequality, we have

∫ Ω | f ( u 1 ) − f ( u 2 ) | | ∂ u ∂ t | d x ≤ C ‖ u ‖ H 1 2 + ‖ ∂ u ∂ t ‖ 2 (49)

Inserting (49) into (46), we find

d d t E 2 + 2 ‖ ∂ u ∂ t ‖ − 1 2 + 2 ‖ ∂ α ∂ t ‖ 2 ≤ c ′ ‖ u ‖ H 1 2 + ‖ ∂ u ∂ t ‖ 2

and recalling the interpolation inequality ‖ ∂ u ∂ t ‖ 2 ≤ c ‖ ∂ u ∂ t ‖ − 1 ‖ ∇ ∂ u ∂ t ‖

with E 2 = ‖ ∇ u ‖ 2 + ‖ ∇ α ‖ 2 + ‖ ∂ α ∂ t ‖ 2

Finally

d d t E 2 + c ″ ‖ ∂ u ∂ t ‖ − 1 2 + 2 ‖ ∂ α ∂ t ‖ 2 ≤ C E 2 ,   C > 0 (50)

Theorem 4.3. (Second theorem of the solution’s existence) The existence and uniqueness of the solution (23)-(25) problem being proven, now we seek the solution of (23)-(25) with more regularity.

Assume ( u 0 , α 0 , α 1 ) ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) ∩ L 2 p ( Ω ) × ( u 0 , α 0 , α 1 ) ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) ∩ L 2 p ( Ω ) × H 0 1 ( Ω ) , then the (23)-(24) system admits a unique ( u , α ) solution such as

u ∈ L ∞ ( 0, T ; H 2 ( Ω ) ∩ H 0 1 ( Ω ) ) , α ∈ L ∞ ( 0, T ; H 2 ( Ω ) ∩ H 0 1 ( Ω ) ) ,

∂ α ∂ t ∈ L ∞ ( 0, T ; H 2 ( Ω ) ∩ H 0 1 ( Ω ) ) ∩ L 2 ( 0, T ; H 2 ( Ω ) ∩ H 0 1 ( Ω ) ) ,

and

∂ u ∂ t ∈ L 2 ( 0, T ; H − 1 ( Ω ) )

Theorems of existence (23) and uniqueness (24) being proven then u ∈ L ∞ ( 0, T ; H 2 ( Ω ) ∩ L 2 p ( Ω ) ) , α ∈ L ∞ ( 0, T ; H 0 1 ( Ω ) ) , ∂ α ∂ t ∈ L ∞ ( 0, T ; L 2 ( Ω ) ) ∩ L 2 ( 0, T ; L 2 ( Ω ) ) and ∂ u ∂ t ∈ L ∞ ( 0, T ; H − 1 ( Ω ) ) , ∀ T > 0 .

We multiply (23) by ( − Δ ) − 1 ∂ u ∂ t and have, integrating over Ω , we have

d d t ( ‖ ∇ u ‖ 2 + 2 ∫ Ω     F ( u ) d x ) + 2 ‖ ∂ u ∂ t ‖ − 1 2 = 2 ( ∂ u ∂ t , ∂ α ∂ t ) (51)

Multiplying (24) by ∂ α ∂ t , we have

d d t ( ‖ ∇ α ‖ 2 + ‖ ∂ α ∂ t ‖ 2 ) + 2 ‖ ∂ α ∂ t ‖ 2 = − 2 ( ∂ u ∂ t , ∂ α ∂ t ) (52)

Now summing (51) and (52) we obtain

d d t ( ‖ ∇ u ‖ 2 + 2 ∫ Ω     F ( u ) d x + ‖ ∇ α ‖ 2 + ‖ ∂ α ∂ t ‖ 2 ) + 2 ‖ ∂ u ∂ t ‖ − 1 2 + 2 ‖ ∂ α ∂ t ‖ 2 = 0 (53)

where

E 3 = ‖ ∇ u ‖ 2 + 2 ∫ Ω     F ( u ) d x + ‖ ∇ α ‖ 2 + ‖ ∂ α ∂ t ‖ 2

finally

‖ ∇ u ( t ) ‖ 2 + c ‖ u ( t ) ‖ L 2 p 2 p + ‖ ∇ α ( t ) ‖ 2 + ‖ ∂ α ∂ t ‖ 2 + 2 ∫ 0 t ( ‖ ∂ α ( s ) ∂ t ‖ 2 + ‖ ∂ u ( s ) ∂ t ‖ − 1 2 ) d s ≤ c 1 .

We infer that

u ∈ L ∞ ( 0, T ; H 2 ( Ω ) ∩ L 2 p ( Ω ) ) , α ∈ L ∞ ( 0, T ; H 0 1 ( Ω ) ) ,

∂ α ∂ t ∈ L ∞ ( 0, T ; L 2 ( Ω ) ) ∩ L 2 ( 0, T ; L 2 ( Ω ) ) and ∂ u ∂ t ∈ L ∞ ( 0, T ; H − 1 ( Ω ) ) .

We multiply (24) by ∂ 2 α ∂ t 2 , we have

d d t ( ‖ ∂ α ∂ t ‖ 2 + ‖ ∇ α ‖ 2 ) + ‖ ∂ 2 α ∂ t 2 ‖ 2 ≤ ‖ ∂ u ∂ t ‖ 2 .

We infer from this that ∂ 2 α ∂ t 2 ∈ L 2 ( 0, T ; L 2 ( Ω ) ) .

5. Conclusion

In this work we have studied the existence and uniqueness of the solution of a conservative-type Caginalp system with Dirichlet-type boundary conditions. Finally we have also succeeded in this work to establish the existence theorems of the solution of this system with low regularity and more regularity. As a perspective, we plan to study this problem in a bounded or unbounded domain with different types of potentials and Neumann-type conditions.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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