We consider the following initial-boundary value problem: \begin{cases} \begin{align} u_t-u_{xx} &= -u^{q} & x \in \Omega, t > 0 \newline \dfrac{\partial u}{\partial\nu} &= u^{p} & x \in \partial \Omega, t > 0 \newline u(x,0) &= u_0(x) & x \in \Omega \end{align} \end{cases} where $0 < p, q < \infty$, $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth $\partial \Omega$, $\nu$ is the outward normal, $0 \le u_0(x) \le M$, and $\frac{\partial u_0}{\partial\nu}=u^{p}_0$ on $\partial \Omega$.
What's the method to solve $u(x,t)$?