Local and global solvability of a class of semilinear parabolic equations (Q1090847)

The authors study the problem \[ u_ t=\Delta u+q(x)u^ n,\quad u(x,0)=u_ 0(x),\quad x\in R^ N. \] Here \(n>1\) and the functions \(u_ 0\) and q are continuous and nonnegative on \(R^ N\). Necessary conditions, expressed in terms of \(u_ 0\) and q, for the existence of a local resp. global solution of the above problem are formulated. Also some rather involved sufficient conditions are stated.