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Schauder estimates for singular parabolic and elliptic equations of Keldysh type
Nonexistence and short time asymptotic behavior of sourcetype solution for porous medium equation with convection in onedimension
1.  Institute of Applied Mathematics, Putian University, Putian 351100, China 
References:
[1] 
G. I. Barenblatt, On some unsteady motions of a liquid and gas in a porous medium, Prikladnaja Mathematika Mechanika, 16 (1952), 6778. 
[2] 
S. Kamm, Sourcetype solution for equation of nonstationary filtration, J. Math. Anal. Appl., 64 (1978), 263276. doi: 10.1016/0022247X(78)900367. 
[3] 
H. Brezis and A. Friedman, Nonlinear parabolic equations involving measure as initial conditions, J. Math. Pure. Appl., 62 (1983), 7397. 
[4] 
S. Kamin and L. A. Peletier, Sourcetype solution of generate diffusive equation with absorption, Israel. J. Math., 50 (1985), 219230. doi: 10.1007/BF02761403. 
[5] 
J. Zhao, Sourcetype solutions of degenrate quasilinear parabolic equations, J. of Dff. Eq., 92 (1991), 179198. doi: 10.1016/00220396(91)90046C. 
[6] 
T.P. Liu and M. Pierre, Sourcesolution and asymptotic behavior in conservation laws, J. of Diff. Eq., 51 (1984), 419441. doi: 10.1016/00220396(84)900962. 
[7] 
M. Escobedo, J. L. Vazquez and E. Zuazua, Asymptotic behavior and sourcetype solutions for a diffusionconvection equation, Arch. Rational Mech. Anal., 124 (1993), 4365. doi: 10.1007/BF00392203. 
[8] 
G. Lu, SourceType Solutions of Diffusion Equations with Nonlinear Convection, China J. of Contemporary Math., 28 (2000), 185188. 
[9] 
G. Lu, Explicit and similarity solutions for certain nonlinear parabolic diffusionconvection equations, J. Sys. Sci and Math. Scis., 22 (2002), 210222. 
[10] 
G. Lu and H. Yin, Sourcetype solutions of heat equation with convection in several variables spaces, Science in China, Series A, 54 (2011), 11451173. doi: 10.1007/s1142501142194. 
[11] 
G. Lu, Sourcetype solutions of nonlinear fokkerplanck equation of onedimension, Science China Mathemathics, 56 (2013), 18451868. doi: 10.1007/s1142501346122. 
[12] 
J. L. Vazquez, Perspectives in nonlinear diffusion: Between analysis, physics and geometry, International Congress of Mathematicians, Eur. Math. Soc., 1 (2007), 609634. doi: 10.4171/0221/23. 
[13] 
Y. Chen, Hölder estimates for solutions of uniformly degenerate parabolic equations, Chin Ann. of Math., 5B (1984), 661678. 
[14] 
G. Lu, A remark on $C^k$regularity of free boundary for porous medium equation with gravity term in onedimension, Appl. Math. A Journal of Chinese University, 7 (1992), 579593. (In Chinese) 
[15] 
O. A. Ladyzhenskaja, N. A. Solonnikov and N. N. Uralezeva, Linear and Quasilinear Equations of Parabolic Type, Trans. Math. Mono., 23, AMS Providence, R. I., 1968. 
[16] 
S. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR. Sb., 81 (1970), 228255. 
[17] 
V. S. Varadarajan, Measure on topological spaces, Amer. Math. Soci. Trans., Series. 2 (1965), p48. 
[18] 
R. J. LeVeque, Finite Volue Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253. 
[19] 
P. J. Vila, An analysis of a class of secondorder accurate godunovtype schemes, SIAM Journal on Numerical Analysis, 26 (1989), 830853. doi: 10.1137/0726046. 
[20] 
T. Ding and C. Li, Ordinary differential equations, China Hihgher Education Press, Beijing, 1991. (In Chinease). 
show all references
References:
[1] 
G. I. Barenblatt, On some unsteady motions of a liquid and gas in a porous medium, Prikladnaja Mathematika Mechanika, 16 (1952), 6778. 
[2] 
S. Kamm, Sourcetype solution for equation of nonstationary filtration, J. Math. Anal. Appl., 64 (1978), 263276. doi: 10.1016/0022247X(78)900367. 
[3] 
H. Brezis and A. Friedman, Nonlinear parabolic equations involving measure as initial conditions, J. Math. Pure. Appl., 62 (1983), 7397. 
[4] 
S. Kamin and L. A. Peletier, Sourcetype solution of generate diffusive equation with absorption, Israel. J. Math., 50 (1985), 219230. doi: 10.1007/BF02761403. 
[5] 
J. Zhao, Sourcetype solutions of degenrate quasilinear parabolic equations, J. of Dff. Eq., 92 (1991), 179198. doi: 10.1016/00220396(91)90046C. 
[6] 
T.P. Liu and M. Pierre, Sourcesolution and asymptotic behavior in conservation laws, J. of Diff. Eq., 51 (1984), 419441. doi: 10.1016/00220396(84)900962. 
[7] 
M. Escobedo, J. L. Vazquez and E. Zuazua, Asymptotic behavior and sourcetype solutions for a diffusionconvection equation, Arch. Rational Mech. Anal., 124 (1993), 4365. doi: 10.1007/BF00392203. 
[8] 
G. Lu, SourceType Solutions of Diffusion Equations with Nonlinear Convection, China J. of Contemporary Math., 28 (2000), 185188. 
[9] 
G. Lu, Explicit and similarity solutions for certain nonlinear parabolic diffusionconvection equations, J. Sys. Sci and Math. Scis., 22 (2002), 210222. 
[10] 
G. Lu and H. Yin, Sourcetype solutions of heat equation with convection in several variables spaces, Science in China, Series A, 54 (2011), 11451173. doi: 10.1007/s1142501142194. 
[11] 
G. Lu, Sourcetype solutions of nonlinear fokkerplanck equation of onedimension, Science China Mathemathics, 56 (2013), 18451868. doi: 10.1007/s1142501346122. 
[12] 
J. L. Vazquez, Perspectives in nonlinear diffusion: Between analysis, physics and geometry, International Congress of Mathematicians, Eur. Math. Soc., 1 (2007), 609634. doi: 10.4171/0221/23. 
[13] 
Y. Chen, Hölder estimates for solutions of uniformly degenerate parabolic equations, Chin Ann. of Math., 5B (1984), 661678. 
[14] 
G. Lu, A remark on $C^k$regularity of free boundary for porous medium equation with gravity term in onedimension, Appl. Math. A Journal of Chinese University, 7 (1992), 579593. (In Chinese) 
[15] 
O. A. Ladyzhenskaja, N. A. Solonnikov and N. N. Uralezeva, Linear and Quasilinear Equations of Parabolic Type, Trans. Math. Mono., 23, AMS Providence, R. I., 1968. 
[16] 
S. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR. Sb., 81 (1970), 228255. 
[17] 
V. S. Varadarajan, Measure on topological spaces, Amer. Math. Soci. Trans., Series. 2 (1965), p48. 
[18] 
R. J. LeVeque, Finite Volue Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253. 
[19] 
P. J. Vila, An analysis of a class of secondorder accurate godunovtype schemes, SIAM Journal on Numerical Analysis, 26 (1989), 830853. doi: 10.1137/0726046. 
[20] 
T. Ding and C. Li, Ordinary differential equations, China Hihgher Education Press, Beijing, 1991. (In Chinease). 
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