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 & 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, (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. doi: 10.1512/iumj.2009.58.3714.

[3]

E. B. Davies, "Heat Kernel and Spectral Theory,", Cambridge Tracts in Mathematics, 92 (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. 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. 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.

[7]

A. Grigor'yan, Heat kernels on weighted manifolds and applications,, in, 398 (2006), 93.

[8]

S. Kamin and R. Kersner, Disappearance of interfaces in finite time,, Meccanica, 28 (1993), 117. 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.

[10]

S. Kamin and P. Rosenau, Nonlinear diffusion in a finite mass medium,, Comm. Pure Appl. Math., 35 (1982), 113.

[11]

S. Kumaresan and J. Prajapat, Serrin's result for hyperbolic space and sphere,, Duke Math. J., 91 (1998), 17. 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. 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. 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., ().

[15]

G. Reyes and J. L. Vazquez, The Cauchy problem for the inhomogeneous porous medium equation,, Netw. Heterog. Media, 1 (2006), 337. doi: 10.3934/nhm.2006.1.337.

show all references

References:
[1]

R. Benedetti and C. Petronio, "Lectures on Hyperbolic Geometry,", Universitext, (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. doi: 10.1512/iumj.2009.58.3714.

[3]

E. B. Davies, "Heat Kernel and Spectral Theory,", Cambridge Tracts in Mathematics, 92 (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. 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. 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.

[7]

A. Grigor'yan, Heat kernels on weighted manifolds and applications,, in, 398 (2006), 93.

[8]

S. Kamin and R. Kersner, Disappearance of interfaces in finite time,, Meccanica, 28 (1993), 117. 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.

[10]

S. Kamin and P. Rosenau, Nonlinear diffusion in a finite mass medium,, Comm. Pure Appl. Math., 35 (1982), 113.

[11]

S. Kumaresan and J. Prajapat, Serrin's result for hyperbolic space and sphere,, Duke Math. J., 91 (1998), 17. 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. 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. 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., ().

[15]

G. Reyes and J. L. Vazquez, The Cauchy problem for the inhomogeneous porous medium equation,, Netw. Heterog. Media, 1 (2006), 337. doi: 10.3934/nhm.2006.1.337.

[1]

A. M. Micheletti, Angela Pistoia. Multiple eigenvalues of the Laplace-Beltrami operator and deformation of the Riemannian metric. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 709-720. doi: 10.3934/dcds.1998.4.709

[2]

Micol Amar, Roberto Gianni. Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1739-1756. doi: 10.3934/dcdsb.2018078

[3]

Andrea Bonito, Roland Glowinski. On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2115-2126. doi: 10.3934/cpaa.2014.13.2115

[4]

Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761

[5]

Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics & Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005

[6]

Guoqing Zhang, Jia-yu Shao, Sanyang Liu. Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights. Communications on Pure & Applied Analysis, 2011, 10 (2) : 571-581. doi: 10.3934/cpaa.2011.10.571

[7]

Giuseppe Riey. Regularity and weak comparison principles for double phase quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4863-4873. doi: 10.3934/dcds.2019198

[8]

Yen-Lin Wu, Zhi-You Chen, Jann-Long Chern, Y. Kabeya. Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 949-960. doi: 10.3934/cpaa.2014.13.949

[9]

Alessandro Selvitella. Qualitative properties of stationary solutions of the NLS on the Hyperbolic space without and with external potentials. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2663-2677. doi: 10.3934/cpaa.2019118

[10]

Ruyun Ma, Man Xu. Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2701-2718. doi: 10.3934/dcdsb.2018271

[11]

Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715

[12]

Shingo Takeuchi. Partial flat core properties associated to the $p$-laplace operator. Conference Publications, 2007, 2007 (Special) : 965-973. doi: 10.3934/proc.2007.2007.965

[13]

Tohru Nakamura, Shinya Nishibata, Naoto Usami. Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space. Kinetic & Related Models, 2018, 11 (4) : 757-793. doi: 10.3934/krm.2018031

[14]

Zhigang Wang, Lei Wang, Yachun Li. Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1163-1182. doi: 10.3934/cpaa.2013.12.1163

[15]

Ana Maria Bertone, J.V. Goncalves. Discontinuous elliptic problems in $R^N$: Lower and upper solutions and variational principles. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 315-328. doi: 10.3934/dcds.2000.6.315

[16]

Yutian Lei, Congming Li, Chao Ma. Decay estimation for positive solutions of a $\gamma$-Laplace equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 547-558. doi: 10.3934/dcds.2011.30.547

[17]

Yucheng Bu, Yujun Dong. Existence of solutions for nonlinear operator equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4429-4441. doi: 10.3934/dcds.2019180

[18]

Anna Geyer. A note on uniqueness and compact support of solutions in a recent model for tsunami background flows. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1431-1438. doi: 10.3934/cpaa.2012.11.1431

[19]

Simona Fornaro, Stefano Lisini, Giuseppe Savaré, Giuseppe Toscani. Measure valued solutions of sub-linear diffusion equations with a drift term. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1675-1707. doi: 10.3934/dcds.2012.32.1675

[20]

Pedro Isaza, Jorge Mejía. On the support of solutions to the Kadomtsev-Petviashvili (KP-II) equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1239-1255. doi: 10.3934/cpaa.2011.10.1239

2017 Impact Factor: 0.561

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]