Global Attractor for Caginalp Hyperbolic Field-phase System with Singular Potential

This article is devoted to the study of the Caginalp hyperbolic phase-field system with singular potentials. We first prove the existence and uniqueness of solutions for Caginalp hyperbolic phase-field system with logarithmic potential. We then prove the existence of global attractor. One of main difficulties is to prove that the solutions are strictly separated from singular values of the potential.


Introduction
The global attractor is a compact invariant and smallest set which attracts the bounded sets of phase space.It presents two major default: it can attract the trajectories slowly and it can be sensitive to perturbations.In this article, we are interested in the study of the following Caginalp hyperbolic phase-field system in a smooth and bounded domain Ω ⊂ R n , 1 ≤ n ≤ 3, with homogenous Dirichlet conditions u = α = 0 on ∂Ω, and initial conditions where ϵ > 0 is a relaxation parameter, u = u(t, x), the order parameter and α = α(t, x) are unknown functions, f is a given singular potential function.
Recently, above Caginalp hyperbolic phase-field system endowed with homogenous Dirichlet boundary conditions with a regular potentials, is studied in (Moukoko 2014(Moukoko , 2015)), in order to prove the existence and uniqueness of solutions, existence of: global attractor, exponential attractors and the robust family of exponential attractors.
Here we are interested in the Caginalp hyperbolic system with homogenous Dirichlet boundary conditions and logarithmic potential.We prove the existence and uniqueness of solutions, as well as regularity.The main difficulties in this article is to prove that the order parameter u is strictly separated from the singular values of the potential.

Method
In this section we brief on method needed to prove our two main results of the next section.
We first prove the existence of the solution which are separated from the singular values ±1 of the singular potential f .We replace the logarithmic potential by a regular function and prove that the solution of the resulting system is also the solution of initial system.We define two phase spaces. in order to end, we first prove the existence of the bounded absorbing set in each phase space, and owing to (Miranville & Zelik, 2008), we prove the existence of global attractor.Although in our study, we have to use classical methods of functional analysis applied in the theory of Partial Differential Equations.

Results
We first give some estimates which allow us to determine a first phase space.
Multiplying (1) by 2∂ t u and (2) by 2∂ t α, integrating over Ω and adding the two resulting equations, we have where which implies, by integrating between 0 and t where K is a positive constant.
In our study there are two main results; we prove the existence and uniqueness of solution and the existence of global attractor.

Existence and Uniqueness of Solution
Proof.In order to prove this Theorem, we first show that all solution (u, α) of system (1) − (2) is such that u is separated from the singular points of f , i.e., there exists δ ∈ (0, 1) depending of T such that ∥u∥ L ∞ ((0,T )×Ω) < δ.In the second time, we study the auxiliary problem of the system (1) − (2) .Finally, we show that the solution of auxiliary problem is also the solution of system (1) − (2).
We now have to prove the existence of a solution (u δ , α) of the following auxiliary system obtained by replacing logarithmic function f by the C 1 (R)-regular function f δ in (1), f δ being defined by where δ > 0 very near of 1 and such that [−δ, δ] ⊂ (−1, 1), with homogenous Dirichlet conditions and initial conditions in (3) − (4).
We have the continuous dependence with respect to initial conditions, hence the uniqueness of solution.
In order to prove the existence of global attractor we seek the solution with more regularity.
Theorem 4. Let the assumptions of Theorem 1 hold, ϵ < 1 and (u, α) the solution of system (1) Then, the following estimate is valid where positive constants C, β and monotonic function Q are independent of ϵ.
Proof.We first determine standard energy of the initial and boundary value problem for singularly perturbed damped hyperbolic equation ( 1) .This equation can be written in the following form Applying corollary 5.2 of appendix of (Grasselli, Miranville, Pata & Zelik 2007) to equation ( 41) where κ = 2, we have where positive constants C and β are independent of ϵ.
To determine estimate of the last term of second member of (42), we begin by finding an estimate of ∥ h u,α (s) ∥ 2 We now determine estimates of ∥ f (u) ∥ H 2 and ∥ f ′ (u)∂ t u ∥.We have where C is independent of ϵ.We have, allow for estimate (7), Inserting ( 44) into (42), we obtain, thanks to estimate (35), where positives constants C and β are independent of ϵ.Combining above estimate and estimate (35), we obtain the result.This achieves the proof.
This corollary is the straightforward consequence of above Theorem.
In the sequence, we note Theorem 5. Let the assumptions of Theorem 3 hold, ϵ < 1 and (u, α) the solution of system (1) where the positive constants C, β and the monotonic function Q are independent of ϵ.
This corollary is the straightforward consequence of above Theorem.

Discussion
In this article we have proven as in (Moukoko, 2014(Moukoko, , 2015) ) that the Caginalp hyperbolic phase-field system with singular potential, has a unique solution and the global attractor.It remains to study the existence of exponential attractors and the robust family of exponential attractors for Caginalp hyperbolic phase-field system with singular potential.We can also complete this work by studying Caginalp hyperbolic phase-field system with other types of boundary conditions and regular or singular potential.