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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 $x\in\partial\Omega$.

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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.

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