Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation. (English) Zbl 0752.35034

Hale, Jack K.; Raugel, Geneviève

J. Dyn. Differ. Equations 2, No. 1, 19-67 (1990).

Assume \(\varepsilon\) is a small positive parameter, \(\Omega\) is a bounded, smooth domain or a convex polyhedral domain in \(\mathbb{R}^ n\), \(n=1,2,3\), and consider the following problem: \[ \varepsilon u_{tt}+u_ t-\Delta u=-f(u)-g\quad\text{ in }\Omega\times(0,\infty),\quad u=0\quad\text{ on }\partial\Omega; \]
\[ u(0,x)=u_ 0(x),\quad u_ t(0,x)=u_ 1(x),\tag{1} \] where \(g\in L^ 2(\Omega)\), \((u_ 0,u_ 1)\in X_ 0\equiv H^ 1_ 0(\Omega)\times L^ 2(\Omega)\), and \(f\in L^ 2(\mathbb{R},\mathbb{R})\), \(\;\varlimsup_{| y|\to\infty}(-f(y)/y)\leq 0\), and, for \(n\geq 2\), there is a constant \(c_ 0>0\) such that \(| f''(y)|\leq c_ 0(| y|^ \gamma+1)\) for \(y\in\mathbb{R}\), where \(0\leq\gamma<\infty\) if \(n=2\), and \(0\leq\gamma\leq 1\) if \(n=3\).
The authors obtain some relationships between the solutions of the hyperbolic problem (1) and the limit parabolic problem \((\varepsilon=0)\) \[ u_ t-\Delta u=-f(u)-g\text{ in }\Omega\times(0,\infty),\quad u=0\text{ on }\partial\Omega,\quad u(0,x)=u_ 0(x).\tag{2} \] More precisely under certain conditions on \(f\), the hyperbolic equation given by (1) has a global attractor \(A_ \varepsilon\) in \(H^ 1_ 0(\Omega)\times L^ 2(\Omega)\). For \(\varepsilon=0\), the parabolic equation also has a global attractor which can be naturally embedded into a compact set \(A_ 0\) in \(H^ 1_ 0(\Omega)\times L^ 2(\Omega)\).
If all of the equilibrium points of the parabolic equation are hyperbolic, the authors prove that the sets \(A_ \varepsilon\) are lower semicontinuous at \(\varepsilon=0\). Furthermore, an estimate of the symmetric distance between \(A_ 0\) and \(A_ \varepsilon\) is given.

Reviewer: Th.M.Rassias (Athens)

MSC:

35L55	Higher-order hyperbolic systems
37C70	Attractors and repellers of smooth dynamical systems and their topological structure
35B25	Singular perturbations in context of PDEs
58J37	Perturbations of PDEs on manifolds; asymptotics

Keywords:

limit parabolic equation

References:

[1]	Babin, A. V., and Vishik, M.I. (1983). Regular attractors of semigroups of evolutionary equations,J. Math. Pures Appl. 62, 441-491. · Zbl 0565.47045
[2]	Babin, A. V., and Vishik, M. I. (1985).Attracteurs maximaux dans les équations aux dérivées partielles (Collége de France Séminaire, 1984), Pitman, Boston.
[3]	Babin., A. V., and Vishik, M. I. (1987). Uniform asymptotic solutions of a singularly perturbed evolutionary equation,Uspekhi Mat. Nauk 42, 231-232 [in Russian].
[4]	Billotti, J., and LaSalle, J. P. (1971). Periodic dissipative processes,Bull. Am. Math. Soc. (N.S.) 6, 1082-1089. · Zbl 0274.34061
[5]	Ghidaglia, J. M., and Témam, R. (1987). Attractors for damped hyperbolic equations,J. Math. Pures Appl. 66, 273-319. · Zbl 0572.35071
[6]	Hale, J. K. (1980).Ordinary Differential Equations, 2nd ed., Kreiger. · Zbl 0433.34003
[7]	Hale, J. K. (1988).Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, Rhode Island. · Zbl 0642.58013
[8]	Hale, J. K., and Raugel, G. (1988). Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation,J. Diff. Eq. 73, 197-214. · Zbl 0666.35012
[9]	Hale, J. K., and Raugel, G. (to appear). Lower semicontinuity of the attractor for gradient systems,Annali Mat. Pura Appl., to appear. · Zbl 0752.35034
[10]	Hale, J. K., and Scheurle, J. (1985). Smoothness of bounded solutions of nonlinear evolution equations,J. Diff. Eq. 56, 142-163. · Zbl 0541.34020
[11]	Hale, J. K., Lin, X.-B., and Raugel, G. (1988). Upper semicontinuity of attractors and partial differential equations,Math. Comp. 50, 89-123. · Zbl 0666.35013
[12]	Haraux, A. (1985).Two Remarks on Dissipative Hyperbolic Systems (Collége de France Séminaire, 1984), Pitman, Boston.
[13]	Henry, D. (1981).Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin. · Zbl 0456.35001
[14]	Henry, D. (1986). Generic properties of equilibrium solutions by perturbation of the boundary, Preprint, Centre de Recerca Matematica Institut d’Estudis Catalans, 37.
[15]	Lions, J. L. (1973).Perturbations singulières dans les problémes aux limites et en contrôle optimal, Springer-Verlag, Berlin. · Zbl 0268.49001
[16]	Lions, J. L., and Magenes, E. (1968).Problèmes aux limites nonhomogènes et applications, Dunod, Paris. · Zbl 0165.10801
[17]	Mora, X. (1987). Finite dimensional attracting invariant manifolds for damped semilinear wave equations. In Diaz, J. L., and Lions, P. L. (eds.),Nonlinear Partial Differential Equations. II, Longman, Essex, England. · Zbl 0642.35061
[18]	Mora, I., and Sola-Morales, J. (1989). The singular limit dynamics of semilinear damped wave equations,J. Diff. Eq. 88, 262-307. · Zbl 0699.35177
[19]	Palis, J., and de Melo, W. (1982).Geometric Theory of Dynamical Systems, Springer-Verlag, Berlin. · Zbl 0491.58001