Existence of Global Attractor for a Hyperbolic Phase Field System of Caginalp Type with Polynomial Growth Potential

Abstract

Aims/ objectives: We are interested in a hyperbolic phase field system of Caginalp type, parameterized by for which the solution is a function defined on (0; T)×Ω. We show the existence of the global attractor for a hyperbolic phase field system of Caginalp type, with homogenous conditions Dirichlet on the boundary, this system is governed by a polynomial growth potential, in a bounded and smooth domain. the hyperbolic phase field system of Caginalp type is based on a thermomecanical theory of deformable continua. Note that the global attractor is the smallest compact set in the phase space, which is invariant by the semigroup and attracts all bounded sets of initial data, as time goes to infinity. So the global attractor allows to make description of asymptotic behaviour about dynamic system. Study Design: Propagation study of waves. Place and Duration of Study: Departement of mathematics (group of research called G.R.A.F.E.D.P), Sciences Faculty and Technical of Marien NGOUABI University PO Box 69, between October 2015 and July 2016. Methodology: To show the existence of the global attractor about the perturbed damped hyperbolic system, with initial conditions and homogenous conditions Dirichlet on the boundary, we proceed by proving the dissipativity and regularity of the semigoup associated to the system, and we then split the semigroup such that we have the sum of two continuous operators, where the first tends uniformly to zero when the time goes to infinity, and the second is regularizing. Results: We show the existence of global attractor, about a hyperbolic phase field system of Caginalp type, governed by polynomial growth potential. Conclusion: All the procedures explained in the methodology being demonstrated, we can assert the existence of the smallest compact set of the phase space, invariant by the semigroup and which attracts all the bounded sets of initial data from a some time.