Hyperbolic relaxation of reaction-diffusion equations with dynamic boundary conditions
Authors:
Ciprian G. Gal and Joseph L. Shomberg
Journal:
Quart. Appl. Math. 73 (2015), 93-129
MSC (2010):
Primary 35B41; Secondary 35L20, 35K57
DOI:
https://doi.org/10.1090/S0033-569X-2015-01363-5
Published electronically:
January 29, 2015
MathSciNet review:
3322727
Full-text PDF Free Access
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Abstract: Under consideration is the hyperbolic relaxation of a semilinear reaction-diffusion equation \begin{equation*} \varepsilon u_{tt}+u_{t}-\Delta u+f(u)=0 \end{equation*} on a bounded domain $\Omega \subset \mathbb {R}^{3}$ with $\varepsilon \in (0,1]$ and the prescribed dynamic condition \begin{equation*} \partial _{\mathbf {n}}u+u+u_{t}=0 \end{equation*} on the boundary $\Gamma :=\partial \Omega$. We also consider the limit parabolic problem ($\varepsilon =0$) with the same dynamic boundary condition. Each problem is well-posed in a suitable phase space where the global weak solutions generate a Lipschitz continuous semiflow which admits a bounded absorbing set. Because of the nature of the boundary condition, fractional powers of the Laplace operator are not well-defined. The precompactness property required by the hyperbolic semiflows for the existence of the global attractors is gained through the approach of Pata and Zelik (2006). In this case, the optimal regularity for the global attractors is also readily established. In the parabolic setting, the regularity of the global attractor is necessary for the semicontinuity result. After fitting both problems into a common framework, a proof of the upper-semicontinuity of the family of global attractors is given at $\varepsilon =0$. Finally, we also establish the existence of a family of exponential attractors.
References
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- Ciprian G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, J. Differential Equations 253 (2012), no. 1, 126–166. MR 2917404, DOI https://doi.org/10.1016/j.jde.2012.02.010
- Ciprian G. Gal, Sharp estimates for the global attractor of scalar reaction-diffusion equations with a Wentzell boundary condition, J. Nonlinear Sci. 22 (2012), no. 1, 85–106. MR 2878653, DOI https://doi.org/10.1007/s00332-011-9109-y
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- Ciprian G. Gal and Maurizio Grasselli, Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. B 18 (2013), no. 6, 1581–1610. MR 3038770, DOI https://doi.org/10.3934/dcdsb.2013.18.1581
- Ciprian G. Gal and Maurizio Grasselli, On the asymptotic behavior of the Caginalp system with dynamic boundary conditions, Commun. Pure Appl. Anal. 8 (2009), no. 2, 689–710. MR 2461570 (2010b:35200), DOI https://doi.org/10.3934/cpaa.2009.8.689
- Ciprian G. Gal and Mahamadi Warma, Well posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions, Differential Integral Equations 23 (2010), no. 3-4, 327–358. MR 2588479 (2011a:35279)
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- Maurizio Grasselli and Vittorino Pata, Asymptotic behavior of a parabolic-hyperbolic system, Commun. Pure Appl. Anal. 3 (2004), no. 4, 849–881. MR 2106302 (2005h:35150), DOI https://doi.org/10.3934/cpaa.2004.3.849
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- Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371 (89g:58059)
- L. Herrera and D. Pavón, Hyperbolic theories of dissipation: Why and when do we need them?, Phys. A 307 (2002), 121–130.
- Olga Ladyzhenskaya, Attractors for semigroups and evolution equations, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1991. MR 1133627 (92k:58040)
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693 (41 \#4326)
- J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York, 1972. MR 0350177 (50 \#2670)
- M. Meyries, Maximal regularity in weighted spaces, nonlinear boundary conditions, and global attractors, Ph.D. thesis, Karlsruhe Institute of Technology (KIT), 2010.
- Albert J. Milani and Norbert J. Koksch, An introduction to semiflows, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 134, Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2106597 (2005i:37094)
- Alain Miranville, Vittorino Pata, and Sergey Zelik, Exponential attractors for singularly perturbed damped wave equations: a simple construction, Asymptot. Anal. 53 (2007), no. 1-2, 1–12. MR 2343457 (2008k:35327)
- Delio Mugnolo, Abstract wave equations with acoustic boundary conditions, Math. Nachr. 279 (2006), no. 3, 299–318. MR 2200667 (2007b:35209), DOI https://doi.org/10.1002/mana.200310362
- Delio Mugnolo and Silvia Romanelli, Dirichlet forms for general Wentzell boundary conditions, analytic semigroups, and cosine operator functions, Electron. J. Differential Equations (2006), No. 118, 20 pp. (electronic). MR 2255233 (2007e:35047)
- Vittorino Pata and Sergey Zelik, A remark on the damped wave equation, Commun. Pure Appl. Anal. 5 (2006), no. 3, 609–614. MR 2217604 (2006m:35254)
- Amnon Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486 (85g:47061)
- Luminiţa Popescu and Aníbal Rodriguez-Bernal, On a singularly perturbed wave equation with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 2, 389–413. MR 2056289 (2005e:35149), DOI https://doi.org/10.1017/S0308210500003279
- James C. Robinson, Infinite-dimensional dynamical systems, An introduction to dissipative parabolic PDEs and the theory of global attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. MR 1881888 (2003f:37001a)
- Joseph L. Shomberg and Sergio Frigeri, Global attractors for damped semilinear wave equations with a Robin-acoustic boundary perturbation, in preparation.
- Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. MR 953967 (89m:58056)
- André Vicente, Wave equation with acoustic/memory boundary conditions, Bol. Soc. Parana. Mat. (3) 27 (2009), no. 1, 29–39. MR 2576356 (2011a:35310), DOI https://doi.org/10.5269/bspm.v27i1.9066
- Hao Wu and Songmu Zheng, Convergence to equilibrium for the damped semilinear wave equation with critical exponent and dissipative boundary condition, Quart. Appl. Math. 64 (2006), no. 1, 167–188. MR 2211383 (2007a:35112)
- Songmu Zheng, Nonlinear evolution equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 133, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2088362 (2006a:35001)
- Chaosheng Zhu, Global attractor of the weakly damped wave equation with nonlinear boundary conditions, Commun. Korean Math. Soc. 27 (2012), no. 1, 97–106. MR 2919015, DOI https://doi.org/10.4134/CKMS.2012.27.1.097
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Additional Information
Ciprian G. Gal
Affiliation:
Department of Mathematics, Florida International University, Miami, Florida 33199
Email:
cgal@fiu.edu
Joseph L. Shomberg
Affiliation:
Department of Mathematics and Computer Science, Providence College, Providence, Rhode Island 02918
Email:
jshomber@providence.edu
Keywords:
Dynamic boundary condition,
semilinear reaction diffusion equation,
hyperbolic relaxation,
damped wave equation,
singular perturbation,
global attractor,
upper-semicontinuity
Received by editor(s):
February 19, 2013
Received by editor(s) in revised form:
April 2, 2013
Published electronically:
January 29, 2015
Article copyright:
© Copyright 2015
Brown University