Asymptotic behaviour for a phase field model in higher order sobolev spaces

Bernal, Aníbal Rodríguez; Casas, Ángela Jiménez

doi:10.5209/rev_rema.2002.v15.n1.16964

Cited by 14 publications

(13 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As in the previous examples to get uniqueness we need to assume stronger assumptions on the nonlinear term as, for example, the monotonicity condition ∂ ∂r f (x, r) ≥ −C [Kalantarov, 1991] (see also [Bates & Zheng, 1992, Brochet et al, 1993, Jiménez-Casas & Rodríguez-Bernal, 2002). …”

Section: 1(f) Phase-field Equationsmentioning

confidence: 99%

Recent Developments in Dynamical Systems: Three Perspectives

Balibrea

Caraballo

Kloeden

et al. 2010

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

The aim of this paper is to give an account of some problems considered in past years in the setting of Dynamical Systems, some new research directions and also state some open problems Keywords: Topological entropy, Li-Yorke chaos, Lyapunov exponent, continua, nonautonomous systems, difference equations, ordinary and partial differential equations, set-valued dynamical system, global attractor, non-autonomous and random pullback attractors.

show abstract

Section: 1(f) Phase-field Equationsmentioning

confidence: 99%

Recent Developments in Dynamical Systems: Three Perspectives

Balibrea

Caraballo

Kloeden

et al. 2010

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

show abstract

“…there is a strong solution ∞ ∈ H 2 ( ) to problem (11) and the corresponding constant ∞ given by (12) such that the trajectory ( (t), t (t), (t), q(t)) strongly converges to 3 as t goes to infinity. More precisely, there exists ∈ (0, 1 2 ) and a positive constant C such that, for all t 0,…”

Section: Theorem 11mentioning

confidence: 99%

“…Then ∞ is a critical point of the functional in H 1 ( ). Conversely, if ∞ is a critical point of the functional in H 1 ( ), then ∞ ∈ H 2 ( ) and it is a strong solution to problem (11).…”

Section: Lemma 31mentioning

confidence: 99%

See 1 more Smart Citation

Convergence to equilibrium for a fully hyperbolic phase‐field model with Cattaneo heat flux law

Jiang¹

2008

Math Methods in App Sciences

View full text Add to dashboard Cite

SUMMARYWe consider a fully hyperbolic phase-field model in this paper. Our model consists of a damped hyperbolic equation of second order with respect to the phase function (t), which is coupled with a hyperbolic system of first order with respect to the relative temperature (t) and the heat flux vector q(t). We prove the well-posedness of this system subject to homogeneous Neumann boundary condition and no-heat flux boundary condition. Then, we show that this dynamical system is a dissipative one. Finally, using the celebrated Łojasiewicz-Simon inequality and by constructing an auxiliary functional, we prove that the solution of this problem converges to an equilibrium as time goes to infinity. We also obtain an estimate of the decay rate to equilibrium.

show abstract

“…The latter paper also contains global existence and uniqueness results with various boundary conditions, as well as the first detailed analysis of the asymptotic behavior of the solutions. Furthermore, equations (1.2) were studied as a dissipative dynamical system in [4,5,2,31] and in [30,32]. In addition, it is worth recalling [40] for convergence to equilibria, and [38,9] for singular φ.…”

mentioning

confidence: 99%

Asymptotic behavior of a nonisothermal Ginzburg-Landau model

Grasselli

Zheng

2008

Quart. Appl. Math.

View full text Add to dashboard Cite

Abstract. We analyze a system of nonlinear parabolic equations which describes the evolution of an order parameter f and the relative temperature v in a superconducting material occupying a bounded domain Ω ⊂ R n , n ≤ 3. Here the dependent variable f is subject to the homogeneous Neumann boundary condition, while v is equal to a timedependent Dirichlet datumũ on the boundary. Therefore, the corresponding dynamical system is nonautonomous. Our main goal is to analyze the asymptotic behavior of its solutions. We first show that the system has a bounded absorbing set and a global attractor which are uniform with respect to a sufficiently general class ofũ. Then, we give sufficient conditions onũ which ensure the convergence of a given trajectory to a single stationary state and we estimate the convergence rate. Finally, we demonstrate the existence of an exponential attractor of finite fractal dimension for quasi-periodic or stabilizing boundary dataũ.

show abstract

Asymptotic behaviour for a phase field model in higher order sobolev spaces

Abstract: In this paper we analyze the long time behavior of a phasefield model by showing the existence of global compact attractors in the strong norm of high order Sobolev spaces.

Cited by 14 publications

References 20 publications

Recent Developments in Dynamical Systems: Three Perspectives

Recent Developments in Dynamical Systems: Three Perspectives

Convergence to equilibrium for a fully hyperbolic phase‐field model with Cattaneo heat flux law

Asymptotic behavior of a nonisothermal Ginzburg-Landau model

Contact Info

Product

Resources

About