Journal of the
Korean Mathematical Society JKMS

We study the existence and uniqueness of nonnegative singular solutions $u(\x, t)$ of the semilinear parabolic equation $$u_t=\Delta u - \bold a \cdot \nabla (u^q)-u^p,$$ defined in the whole space $\bold R^N$ for $t>0$, with initial data $M\delta(\x)$, a Dirac mass, with $M >0$. The exponents $p, q$ are larger than $1$ and the direction vector $\bold a$ is assumed to be constant. We here show that a unique singular solution exists for every $M>0$ if and only if $1< q< (N+1)/(N-1)$ and $1< p< 1+(2q^*)/(N+1)$, where $q^*=\max \{q, (N+1)/N \}$. This result agrees with the earlier one for $N=1$. In the proof of this result, we also show that a unique singular solution of a diffusion-convection equation without absorption, $$u_t=\Delta u-\bold a \cdot \nabla (u^q),$$ exists if and only if $1< q < (N+1)/(N-1)$.

Keywords: existence and uniqueness, singular solution, semilinear heat equation

MSC numbers: 35K15