# American Institute of Mathematical Sciences

June  2012, 5(3): 657-670. doi: 10.3934/dcdss.2012.5.657

## Support properties of solutions to nonlinear parabolic equations with variable density in the hyperbolic space

 1 Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, P.le A. Moro 5, I-00185 Roma

Received  June 2010 Revised  August 2010 Published  October 2011

We consider the Cauchy problem for a class of nonlinear parabolic equations with variable density in the hyperbolic space, assuming that the initial datum has compact support. We provide simple conditions, involving the behaviour of the density at infinity, so that the support of every nonnegative solution is not compact at some positive time, or it remains compact for any positive time. These results extend to the case of the hyperbolic space those given in [8] for the Cauchy problem in $\mathbb{R}^n$.
Citation: Fabio Punzo. Support properties of solutions to nonlinear parabolic equations with variable density in the hyperbolic space. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 657-670. doi: 10.3934/dcdss.2012.5.657
##### References:
 [1] R. Benedetti and C. Petronio, "Lectures on Hyperbolic Geometry," Universitext, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58158-8. [2] I. Birindelli and R. Mazzeo, Symmetry for solutions of two phase semilinear elliptic equations on hyperbolic space, Indiana Univ. Math. J., 58 (2009), 2347-2368. doi: 10.1512/iumj.2009.58.3714. [3] E. B. Davies, "Heat Kernel and Spectral Theory," Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1989. [4] D. Eidus and S. Kamin, The filtration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc., 120 (1994), 825-830. doi: 10.1090/S0002-9939-1994-1169025-2. [5] A. Grigor'yan and M. Noguchi, The heat kernel on hyperbolic space, Bull. Lond. Math. Soc., 30 (1998), 643-650. doi: 10.1112/S0024609398004780. [6] A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 135-249. [7] A. Grigor'yan, Heat kernels on weighted manifolds and applications, in "The Ubiquitous Heat Kernel," Cont. Math., 398, Amer. Math. Soc., Providence, RI, (2006), 93-191. [8] S. Kamin and R. Kersner, Disappearance of interfaces in finite time, Meccanica, 28 (1993), 117-120. doi: 10.1007/BF01020323. [9] S. Kamin, R. Kersner and A. Tesei, On the Cauchy problem for a class of parabolic equations with variable density, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 9 (1998), 279-298. [10] S. Kamin and P. Rosenau, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math., 35 (1982), 113-127. [11] S. Kumaresan and J. Prajapat, Serrin's result for hyperbolic space and sphere, Duke Math. J., 91 (1998), 17-28. doi: 10.1215/S0012-7094-98-09102-5. [12] M. A. Pozio and A. Tesei, On the uniqueness of bounded soutions to singular parabolic problems, Discr. Cont. Dyn. Syst., 13 (2005), 117-137. doi: 10.3934/dcds.2005.13.117. [13] F. Punzo, On the Cauchy problem for nonlinear parabolic equations with variable density, J. Evol. Equations, 9 (2009), 429-447. doi: 10.1007/s00028-009-0018-6. [14] F. Punzo, Well-posedness of the Cauchy problem for nonlinear parabolic equations with variable density in the hyperbolic space, Nonlin. Diff. Eq. Appl., to appear. [15] G. Reyes and J. L. Vazquez, The Cauchy problem for the inhomogeneous porous medium equation, Netw. Heterog. Media, 1 (2006), 337-351. doi: 10.3934/nhm.2006.1.337.

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##### References:
 [1] R. Benedetti and C. Petronio, "Lectures on Hyperbolic Geometry," Universitext, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58158-8. [2] I. Birindelli and R. Mazzeo, Symmetry for solutions of two phase semilinear elliptic equations on hyperbolic space, Indiana Univ. Math. J., 58 (2009), 2347-2368. doi: 10.1512/iumj.2009.58.3714. [3] E. B. Davies, "Heat Kernel and Spectral Theory," Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1989. [4] D. Eidus and S. Kamin, The filtration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc., 120 (1994), 825-830. doi: 10.1090/S0002-9939-1994-1169025-2. [5] A. Grigor'yan and M. Noguchi, The heat kernel on hyperbolic space, Bull. Lond. Math. Soc., 30 (1998), 643-650. doi: 10.1112/S0024609398004780. [6] A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 135-249. [7] A. Grigor'yan, Heat kernels on weighted manifolds and applications, in "The Ubiquitous Heat Kernel," Cont. Math., 398, Amer. Math. Soc., Providence, RI, (2006), 93-191. [8] S. Kamin and R. Kersner, Disappearance of interfaces in finite time, Meccanica, 28 (1993), 117-120. doi: 10.1007/BF01020323. [9] S. Kamin, R. Kersner and A. Tesei, On the Cauchy problem for a class of parabolic equations with variable density, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 9 (1998), 279-298. [10] S. Kamin and P. Rosenau, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math., 35 (1982), 113-127. [11] S. Kumaresan and J. Prajapat, Serrin's result for hyperbolic space and sphere, Duke Math. J., 91 (1998), 17-28. doi: 10.1215/S0012-7094-98-09102-5. [12] M. A. Pozio and A. Tesei, On the uniqueness of bounded soutions to singular parabolic problems, Discr. Cont. Dyn. Syst., 13 (2005), 117-137. doi: 10.3934/dcds.2005.13.117. [13] F. Punzo, On the Cauchy problem for nonlinear parabolic equations with variable density, J. Evol. Equations, 9 (2009), 429-447. doi: 10.1007/s00028-009-0018-6. [14] F. Punzo, Well-posedness of the Cauchy problem for nonlinear parabolic equations with variable density in the hyperbolic space, Nonlin. Diff. Eq. Appl., to appear. [15] G. Reyes and J. L. Vazquez, The Cauchy problem for the inhomogeneous porous medium equation, Netw. Heterog. Media, 1 (2006), 337-351. doi: 10.3934/nhm.2006.1.337.
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