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

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.
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.   Google Scholar

[2]

S. Kamm, Source-type solution for equation of nonstationary filtration,, J. Math. Anal. Appl., 64 (1978), 263.  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.   Google Scholar

[4]

S. Kamin and L. A. Peletier, Source-type solution of generate diffusive equation with absorption,, Israel. J. Math., 50 (1985), 219.  doi: 10.1007/BF02761403.  Google Scholar

[5]

J. Zhao, Source-type solutions of degenrate quasilinear parabolic equations,, J. of Dff. Eq., 92 (1991), 179.  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.  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.  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.   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.   Google Scholar

[10]

G. Lu and H. Yin, Source-type solutions of heat equation with convection in several variables spaces,, Science in China, 54 (2011), 1145.  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.  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, 1 (2007), 609.  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.   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.   Google Scholar

[15]

O. A. Ladyzhenskaja, N. A. Solonnikov and N. N. Uralezeva, Linear and Quasilinear Equations of Parabolic Type,, Trans. Math. Mono., (1968).   Google Scholar

[16]

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

[17]

V. S. Varadarajan, Measure on topological spaces,, Amer. Math. Soci. Trans., 2 (1965).   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.  doi: 10.1137/0726046.  Google Scholar

[20]

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

[2]

S. Kamm, Source-type solution for equation of nonstationary filtration,, J. Math. Anal. Appl., 64 (1978), 263.  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.   Google Scholar

[4]

S. Kamin and L. A. Peletier, Source-type solution of generate diffusive equation with absorption,, Israel. J. Math., 50 (1985), 219.  doi: 10.1007/BF02761403.  Google Scholar

[5]

J. Zhao, Source-type solutions of degenrate quasilinear parabolic equations,, J. of Dff. Eq., 92 (1991), 179.  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.  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.  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.   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.   Google Scholar

[10]

G. Lu and H. Yin, Source-type solutions of heat equation with convection in several variables spaces,, Science in China, 54 (2011), 1145.  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.  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, 1 (2007), 609.  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.   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.   Google Scholar

[15]

O. A. Ladyzhenskaja, N. A. Solonnikov and N. N. Uralezeva, Linear and Quasilinear Equations of Parabolic Type,, Trans. Math. Mono., (1968).   Google Scholar

[16]

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

[17]

V. S. Varadarajan, Measure on topological spaces,, Amer. Math. Soci. Trans., 2 (1965).   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.  doi: 10.1137/0726046.  Google Scholar

[20]

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

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