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ISSN 1088-6850(e) ISSN 0002-9947(p)

Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity

Author(s): Meihua Yang; Chunyou Sun
Journal: Trans. Amer. Math. Soc. 361 (2009), 1069-1101.
MSC (2000): Primary 37L05, 35B40, 35B41
Posted: September 29, 2008
MathSciNet review: 2452835
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Abstract: This paper is dedicated to analyzing the dynamical behavior of strongly damped wave equations with critical nonlinearity in locally uniform spaces. After proving the global well-posedness, we first establish the asymptotic regularity of the solutions which appears to be optimal and the existence of a bounded (in $H^2_{lu}(\mathbb{R}^N)\times H^1_{lu}(\mathbb{R}^N)$ ) subset which attracts exponentially every initial $H^1_{lu}(\mathbb{R}^N)\times L^2_{lu}(\mathbb{R}^N)$ -bounded set with respect to the $H^1_{lu}(\mathbb{R}^N)\times L^2_{lu}(\mathbb{R}^N)$ -norm. Then, we show there is a $(H ^1_{lu}(\mathbb{R}^N)\times L^2_{lu}(\mathbb{R}^N), H^1_\rho(\mathbb{R}^N)\times H^1_\rho(\mathbb{R}^N))$ -global attractor, which reflects the strongly damped property of $\Delta u_t$ to some extent.

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