Existence of singular solutions of a degenerate equation in R2

6Let a(1), a(2),..., a(m) epsilon R-2, 2 <= f is an element of C([0,infinity)), g(i). is an element of C([0,infinity)) be such that 0 <= g(i) (t) <= 2 on [0,infinity). for all i = 1,..., m. For any u(0) is an element of L-1(R-2 \ U-i=1(m){a(i)}) boolean AND L-loc(p)(R-2 \ U-i=1(m){ai}), p > 1, we prove the existence and uniqueness of solutions of the equation u(t) = Delta(log u), u > 0, in (R-2\. U-i=1(m){ai}) x(0,T), u(x,0) = u(0)(x) in R-2\ U-i=1(m){ai}, satisfying integral(R2)\ U-i=1(m){ai} u(x,t) dx = integral(R2)\U-i=1(m){ai} u(0)dx - 2 pi integral(0)(t) fds + 2 pi Sigma(i=1)(m) integral(0)(t) for all 0 <= t < T and log u(x, t)/ log vertical bar x vertical bar -> - f (t) as vertical bar x vertical bar -> infinity, log u(x, t)/log vertical bar x - a(i)vertical bar -> g(i) (t) as vertical bar x - a(i)vertical bar -> 0, uniformly on every compact subset of (0, T) for any i = 1,..., m under a mild assumption on u0 where T = sup{t > 0 : integral(R2)\U-i=1({ai})m u(0)dx > 2 pi integral(0)(t) fds - 2 pi Sigma(i=1)(m) integral(0)(t) g(i)ds}. We also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of R-2 with prescribed singularities at a finite number of points in the domain.