The Heat Equation with Singular Nonlinearity and Singular Initial Data

We study the existence, uniqueness and regularity of solutions of the parabolic equation $u_t -\Delta u = a(x)u^q + b(x)u^p$ in a bounded domain and with Dirichlet’s condition on the boundary. We consider here $a\in L^\alpha(\Omega)$; $b \in L^\beta(\Omega)$ and $0 < q \le 1 < p. The initial data $u(0) = u_0$ is considered in the space $L^r(\Omega)$, $r\ge 1$. In the main result $(0 < q < 1)$, we assumethat $a$, $b\ge 0$ a.e in $\Omega$ and we assume that $u_0\ge \gamma d_\Omega$ for some $\gamma > 0$. We find a unique solution $C([0; T];L^r(\Omega)) \cap L^\infty_{loc} ((0; T);L^\infty(\Omega))$.