# American Institute of Mathematical Sciences

March  2014, 13(2): 789-809. doi: 10.3934/cpaa.2014.13.789

## A strongly singular parabolic problem on an unbounded domain

 1 Department of Mathematics, National Technical University of Athens, 15780, Athens, Greece 2 Department of Mathematics & Engineering Sciences, Hellenic Army Academy, 16673, Athens

Received  April 2013 Revised  September 2013 Published  October 2013

We study the well-posedness and describe the asymptotic behavior of solutions of a strongly singular equation for the Cauchy problem on $R^N$. The strong singularity is exactly the critical case of the Caffarelli-Kohn-Nirenberg inequality. Moreover, we show the stabilization towards a radially symmetric solution in self-similar variables with a polynomial decay rate. This equation is closely related to a heat equation with inverse-square potential, posed on $R^N$. In this case we have the appearance of the Hardy singularity energy.
Citation: G. P. Trachanas, Nikolaos B. Zographopoulos. A strongly singular parabolic problem on an unbounded domain. Communications on Pure & Applied Analysis, 2014, 13 (2) : 789-809. doi: 10.3934/cpaa.2014.13.789
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##### References:
 [1] M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849 [2] Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 [3] Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055 [4] Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $\Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109 [5] Jean Dolbeault, Maria J. Esteban, Michał Kowalczyk, Michael Loss. Improved interpolation inequalities on the sphere. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 695-724. doi: 10.3934/dcdss.2014.7.695 [6] V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153 [7] Simone Calogero, Juan Calvo, Óscar Sánchez, Juan Soler. Dispersive behavior in galactic dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 1-16. doi: 10.3934/dcdsb.2010.14.1 [8] Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109 [9] Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363 [10] Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053 [11] Gioconda Moscariello, Antonia Passarelli di Napoli, Carlo Sbordone. Planar ACL-homeomorphisms : Critical points of their components. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1391-1397. doi: 10.3934/cpaa.2010.9.1391 [12] Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912 [13] Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 [14] Guillermo Reyes, Juan-Luis Vázquez. Long time behavior for the inhomogeneous PME in a medium with slowly decaying density. Communications on Pure & Applied Analysis, 2009, 8 (2) : 493-508. doi: 10.3934/cpaa.2009.8.493 [15] Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 [16] Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 [17] Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021035 [18] Vassili Gelfreich, Carles Simó. High-precision computations of divergent asymptotic series and homoclinic phenomena. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 511-536. doi: 10.3934/dcdsb.2008.10.511 [19] Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009 [20] Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

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