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

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  • 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.
    Mathematics Subject Classification: Primary: 35K65, 35K67; Secondary: 35M10, 35Q84.


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