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July  2016, 21(5): 1567-1586. doi: 10.3934/dcdsb.2016011

Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension

Guofu Lu 1,

1. 

Institute of Applied Mathematics, Putian University, Putian 351100, China

Received  September 2013 Revised  March 2014 Published  April 2016

In this paper we consider the following equation $$ u_t=(u^m)_{xx}+(u^n)_x, \ \ (x, t)\in \mathbb{R}\times(0, \infty) $$ with a Dirac measure as initial data, i.e., $u(x, 0)=\delta(x)$. The solution of the Cauchy problem is well-known as source-type solution. In the recent work [11] the author studied the existence and uniqueness of such kind of singular solutions and proved that there exists a number $n_0=m+2$ such that there is a unique source-type solution to the equation when $0 \leq n < n_0$. Here our attention is focused on the nonexistence and asymptotic behavior near the origin for a short time. We prove that $n_0$ is also a critical number such that there exits no source-type solution when $n \geq n_0$ and describe the short time asymptotic behavior of the source-type solution to the equation when $0 \leq n < n_0$. Our result shows that in the case of existence and for a short time, the source-type solution of such equation behaves like the fundamental solution of the standard porous medium equation when $0 \leq n < m+1$, the unique self-similar source-type solution exists when $n = m+1$, and the solution does like the nonnegative fundamental entropy solution in the conservation law when $m+1 < n < n_0$, while in the case of nonexistence the singularity gradually disappears when $n \geq n_0$ that the mass cannot concentrate for a short time and no such a singular solutions exists. The results of previous work [11] and this paper give a perfect answer to such topical researches.
Keywords: nonexistence, porous medium equation, Source-type solution, entropy condition, convection, asymptotic behavior., Barenblatt-Pattle solution.
Mathematics Subject Classification: Primary: 35K65, 35K67; Secondary: 35M10, 35Q8.
Citation: Guofu Lu. Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1567-1586. doi: 10.3934/dcdsb.2016011
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.  Google Scholar

[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.  Google Scholar

[3]

H. Brezis and A. Friedman, Nonlinear parabolic equations involving measure as initial conditions, J. Math. Pure. Appl., 62 (1983), 73-97.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

G. Lu, Source-Type Solutions of Diffusion Equations with Nonlinear Convection, China J. of Contemporary Math., 28 (2000), 185-188.  Google Scholar

[9]

G. Lu, Explicit and similarity solutions for certain nonlinear parabolic diffusion-convection equations, J. Sys. Sci and Math. Scis., 22 (2002), 210-222.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

Y. Chen, Hölder estimates for solutions of uniformly degenerate parabolic equations, Chin Ann. of Math., 5B (1984), 661-678. Google Scholar

[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)  Google Scholar

[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.  Google Scholar

[16]

S. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR. Sb., 81 (1970), 228-255.  Google Scholar

[17]

V. S. Varadarajan, Measure on topological spaces, Amer. Math. Soci. Trans., Series. 2 (1965), p48. Google Scholar

[18]

R. J. LeVeque, Finite Volue Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[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.  Google Scholar

[20]

T. Ding and C. Li, Ordinary differential equations, China Hihgher Education Press, Beijing, 1991. (In Chinease). Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

H. Brezis and A. Friedman, Nonlinear parabolic equations involving measure as initial conditions, J. Math. Pure. Appl., 62 (1983), 73-97.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

G. Lu, Source-Type Solutions of Diffusion Equations with Nonlinear Convection, China J. of Contemporary Math., 28 (2000), 185-188.  Google Scholar

[9]

G. Lu, Explicit and similarity solutions for certain nonlinear parabolic diffusion-convection equations, J. Sys. Sci and Math. Scis., 22 (2002), 210-222.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

Y. Chen, Hölder estimates for solutions of uniformly degenerate parabolic equations, Chin Ann. of Math., 5B (1984), 661-678. Google Scholar

[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)  Google Scholar

[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.  Google Scholar

[16]

S. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR. Sb., 81 (1970), 228-255.  Google Scholar

[17]

V. S. Varadarajan, Measure on topological spaces, Amer. Math. Soci. Trans., Series. 2 (1965), p48. Google Scholar

[18]

R. J. LeVeque, Finite Volue Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[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.  Google Scholar

[20]

T. Ding and C. Li, Ordinary differential equations, China Hihgher Education Press, Beijing, 1991. (In Chinease). Google Scholar

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