A Semilinear Heat Equation with Singular Nonlinearity

We are interested in the parabolic equation $u_t-\Delta u=f(x,u)$ in a bounded domain of $\R^N$ with Dirichlet boundary condition and $f:\Omega \por [0,\infty) \to [0,\infty)$ a Carath\'eodory function. We study the existence of solution, life span and analyze the behavior of the global(when the time $t\to \infty$) solution with respect to the solution of the ellipticcorresponding problem $-\Delta u=f(x,u)$ with the Dirichlet boundary condition. A typical example where the results areapplied is when $f(x,s)=a(x)s^q+b(x)s^p$ with $0