## 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

### Keywords:

limit parabolic equation
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### References:

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