Journal of the
Korean Mathematical Society JKMS

We here investigate an existence and uniqueness of the nontrivial,
nonnegative solution of a nonlinear ordinary differential
equation:
$$[|(w^{m})'|^{p-2}(w^{m})']'+\beta rw'+\alpha w+(w^{q})'=0$$
satisfying a specific decay rate: $\lim_{r\rightarrow\infty}r^{\alpha/\beta}w(r)=0$
with $\alpha:=(p-1)/[pq-(m+1)(p-1)]$ and $\beta:=[q-m(p-1)]/[pq-(m+1)(p-1)].$ Here $m(p-1)>1$
and $m(p-1) a doubly degenerate equation with nonlinear convection:
$$u_{t}=[|(u^{m})_{x}|^{p-2}(u^{m})_{x}]_{x}+(u^{q})_{x}$$
defined on the half line.

Keywords: very singular solution, existence, uniqueness, asymptotic behavior

MSC numbers: 60G18, 35K65, 35B40, 34A12