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

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.

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
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