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Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension
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), 67-78. |
[2] |
S. Kamm, Source-type solution for equation of nonstationary filtration, J. Math. Anal. Appl., 64 (1978), 263-276.
doi: 10.1016/0022-247X(78)90036-7. |
[3] |
H. Brezis and A. Friedman, Nonlinear parabolic equations involving measure as initial conditions, J. Math. Pure. Appl., 62 (1983), 73-97. |
[4] |
S. Kamin and L. A. Peletier, Source-type solution of generate diffusive equation with absorption, Israel. J. Math., 50 (1985), 219-230.
doi: 10.1007/BF02761403. |
[5] |
J. Zhao, Source-type solutions of degenrate quasilinear parabolic equations, J. of Dff. Eq., 92 (1991), 179-198.
doi: 10.1016/0022-0396(91)90046-C. |
[6] |
T.-P. Liu and M. Pierre, Source-solution and asymptotic behavior in conservation laws, J. of Diff. Eq., 51 (1984), 419-441.
doi: 10.1016/0022-0396(84)90096-2. |
[7] |
M. Escobedo, J. L. Vazquez and E. Zuazua, Asymptotic behavior and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal., 124 (1993), 43-65.
doi: 10.1007/BF00392203. |
[8] |
G. Lu, Source-Type Solutions of Diffusion Equations with Nonlinear Convection, China J. of Contemporary Math., 28 (2000), 185-188. |
[9] |
G. Lu, Explicit and similarity solutions for certain nonlinear parabolic diffusion-convection equations, J. Sys. Sci and Math. Scis., 22 (2002), 210-222. |
[10] |
G. Lu and H. Yin, Source-type solutions of heat equation with convection in several variables spaces, Science in China, Series A, 54 (2011), 1145-1173.
doi: 10.1007/s11425-011-4219-4. |
[11] |
G. Lu, Source-type solutions of nonlinear fokker-planck equation of one-dimension, Science China Mathemathics, 56 (2013), 1845-1868.
doi: 10.1007/s11425-013-4612-2. |
[12] |
J. L. Vazquez, Perspectives in nonlinear diffusion: Between analysis, physics and geometry, International Congress of Mathematicians, Eur. Math. Soc., 1 (2007), 609-634.
doi: 10.4171/022-1/23. |
[13] |
Y. Chen, Hölder estimates for solutions of uniformly degenerate parabolic equations, Chin Ann. of Math., 5B (1984), 661-678. |
[14] |
G. Lu, A remark on $C^k$-regularity of free boundary for porous medium equation with gravity term in one-dimension, Appl. Math. A Journal of Chinese University, 7 (1992), 579-593. (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), 228-255. |
[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 second-order accurate godunov-type schemes, SIAM Journal on Numerical Analysis, 26 (1989), 830-853.
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), 67-78. |
[2] |
S. Kamm, Source-type solution for equation of nonstationary filtration, J. Math. Anal. Appl., 64 (1978), 263-276.
doi: 10.1016/0022-247X(78)90036-7. |
[3] |
H. Brezis and A. Friedman, Nonlinear parabolic equations involving measure as initial conditions, J. Math. Pure. Appl., 62 (1983), 73-97. |
[4] |
S. Kamin and L. A. Peletier, Source-type solution of generate diffusive equation with absorption, Israel. J. Math., 50 (1985), 219-230.
doi: 10.1007/BF02761403. |
[5] |
J. Zhao, Source-type solutions of degenrate quasilinear parabolic equations, J. of Dff. Eq., 92 (1991), 179-198.
doi: 10.1016/0022-0396(91)90046-C. |
[6] |
T.-P. Liu and M. Pierre, Source-solution and asymptotic behavior in conservation laws, J. of Diff. Eq., 51 (1984), 419-441.
doi: 10.1016/0022-0396(84)90096-2. |
[7] |
M. Escobedo, J. L. Vazquez and E. Zuazua, Asymptotic behavior and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal., 124 (1993), 43-65.
doi: 10.1007/BF00392203. |
[8] |
G. Lu, Source-Type Solutions of Diffusion Equations with Nonlinear Convection, China J. of Contemporary Math., 28 (2000), 185-188. |
[9] |
G. Lu, Explicit and similarity solutions for certain nonlinear parabolic diffusion-convection equations, J. Sys. Sci and Math. Scis., 22 (2002), 210-222. |
[10] |
G. Lu and H. Yin, Source-type solutions of heat equation with convection in several variables spaces, Science in China, Series A, 54 (2011), 1145-1173.
doi: 10.1007/s11425-011-4219-4. |
[11] |
G. Lu, Source-type solutions of nonlinear fokker-planck equation of one-dimension, Science China Mathemathics, 56 (2013), 1845-1868.
doi: 10.1007/s11425-013-4612-2. |
[12] |
J. L. Vazquez, Perspectives in nonlinear diffusion: Between analysis, physics and geometry, International Congress of Mathematicians, Eur. Math. Soc., 1 (2007), 609-634.
doi: 10.4171/022-1/23. |
[13] |
Y. Chen, Hölder estimates for solutions of uniformly degenerate parabolic equations, Chin Ann. of Math., 5B (1984), 661-678. |
[14] |
G. Lu, A remark on $C^k$-regularity of free boundary for porous medium equation with gravity term in one-dimension, Appl. Math. A Journal of Chinese University, 7 (1992), 579-593. (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), 228-255. |
[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 second-order accurate godunov-type schemes, SIAM Journal on Numerical Analysis, 26 (1989), 830-853.
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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