# How many steady state solutions does $u_t=d\Delta u+au-bu^2$ possess?

Consider the following evolution equation

$$u_t=d\Delta u+au-bu^2$$ in a bounded and regular open subset $$\Omega$$ of $$\mathbb{R}^N$$, with smooth initial conditions $$u_0\geq 0$$ and homogeneous Dirichlet boundary conditions.

How many steady state solutions does this equation possess?. I mean by steady state solution a function of space $$w:\Omega\to\mathbb{R}$$ such that $$d\Delta w+aw-bw^2=0$$ and $$w(x)=0$$ for $$x\in\partial\Omega$$.

Hint: In the case $$N=1$$ and $$b=0$$, we have $$d w'' + aw = 0$$, which solutions are $$w(x) = A \cos kx + B \sin kx , \qquad k = \sqrt{\tfrac{a}{d}} \, .$$ The boundary conditions $$w(x_\min) = 0 = w(x_\max)$$ yield the system $$\begin{pmatrix} \cos kx_\min & \sin kx_\min \\ \cos kx_\max & \sin kx_\max \end{pmatrix} \begin{pmatrix} A\\ B \end{pmatrix} = \begin{pmatrix} 0\\ 0 \end{pmatrix}$$ which has one unique solution (the zero function) if its determinant is nonzero. If the determinant is zero, then we have an infinity of solutions, which satisfy $$x_\max = x_\min + n\frac{\pi}{k} \, , \qquad n \in \Bbb Z$$ and $$B = -A\tan kx_\min$$. Hence, for particular cases where the domain $$\Omega$$ and the parameters $$a$$, $$b$$, $$d$$ satisfy a given relationship, we may obtain an infinity of steady states.