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The quasiconvex envelope through firstorder partial differential equations which characterize quasiconvexity of nonsmooth functions
Selfsimilar focusing in porous media: An explicit calculation
1.  Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, United States 
References:
[1] 
S. B. Angenent and D. G. Aronson, The focussing problem for the radially symmetric porous medium equation,, Comm. PDE, 20 (1995), 1217. doi: 10.1080/03605309508821130. 
[2] 
S. B. Angenent and D. G. Aronson, Selfsimilarity in the postfocussing regime in porous medium flows,, Euro. J. Appl. Math., 7 (1996), 277. 
[3]  
[4] 
D. G. Aronson and J. Graveleau, A selfsimilar solution to the focusing problem for the porous medium equation,, Euro. J. Appl. Math., 4 (1993), 65. 
[5] 
G. I. Barenblatt, "Scaling, SelfSimilarity, and Intermediate Asymptotics,", With a foreword by Ya. B. Zeldovich, 14 (1996). 
[6] 
G. I. Barenblatt, On some unsteady motions of a liquid or gas in a porous medium,, Akad. Nauk SSSR Prikl. Mat. Meh., 16 (1952), 67. 
[7] 
Ph. Bénilan, M. G. Crandall and M. Pierre, Solutions of the porous medium equation in $\mathbfR^N$ under optimal conditions on initial values,, Indiana U. Math. J., 33 (1984), 51. doi: 10.1512/iumj.1984.33.33003. 
[8] 
B. E. J. Dahlberg and C. E. Kenig, Nonnegative solutions of the porous medium equation,, Comm. PDE, 9 (1984), 409. 
show all references
References:
[1] 
S. B. Angenent and D. G. Aronson, The focussing problem for the radially symmetric porous medium equation,, Comm. PDE, 20 (1995), 1217. doi: 10.1080/03605309508821130. 
[2] 
S. B. Angenent and D. G. Aronson, Selfsimilarity in the postfocussing regime in porous medium flows,, Euro. J. Appl. Math., 7 (1996), 277. 
[3]  
[4] 
D. G. Aronson and J. Graveleau, A selfsimilar solution to the focusing problem for the porous medium equation,, Euro. J. Appl. Math., 4 (1993), 65. 
[5] 
G. I. Barenblatt, "Scaling, SelfSimilarity, and Intermediate Asymptotics,", With a foreword by Ya. B. Zeldovich, 14 (1996). 
[6] 
G. I. Barenblatt, On some unsteady motions of a liquid or gas in a porous medium,, Akad. Nauk SSSR Prikl. Mat. Meh., 16 (1952), 67. 
[7] 
Ph. Bénilan, M. G. Crandall and M. Pierre, Solutions of the porous medium equation in $\mathbfR^N$ under optimal conditions on initial values,, Indiana U. Math. J., 33 (1984), 51. doi: 10.1512/iumj.1984.33.33003. 
[8] 
B. E. J. Dahlberg and C. E. Kenig, Nonnegative solutions of the porous medium equation,, Comm. PDE, 9 (1984), 409. 
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