= Solution
For a nonzero separated solution $u=X(x)T(t)$ of the <reaction-diffusion equation>, substitution and division by $XT$ gives
$$
\frac{T'}T+1=\frac{X''}X.
$$
The two sides depend on different variables, so both equal a separation constant. The homogeneous <Dirichlet boundary conditions> select a negative spatial constant, written $-k^2$:
$$
X''+k^2X=0,\qquad T'+(1+k^2)T=0.
$$
For $k>0$, $X=A\sin kx+B\cos kx$. The boundary value at zero makes $B=0$; the boundary value at $L$ then requires $kL=n\pi$, with $n=1,2,\ldots$. A zero spatial constant gives only a linear <function> vanishing at both endpoints, hence the zero <function>; a positive spatial constant similarly gives no nonzero admissible hyperbolic solution. Therefore
$$
\boxed{X_n(x)=A_n\sin\frac{n\pi x}{L},\qquad
T_n(t)=B_n\exp\left[-\left(1+\frac{n^2\pi^2}{L^2}\right)t\right]}.
$$
By <superposition>, the <general solution> with the stated boundary conditions is
$$
\boxed{u(x,t)=\sum_{n=1}^\infty c_n\sin\frac{n\pi x}{L}
\exp\left[-\left(1+\frac{n^2\pi^2}{L^2}\right)t\right]}.
$$
The coefficients are fixed only if initial data are supplied: for $u(x,0)=f(x)$, they are the <Fourier sine series> coefficients $c_n=(2/L)\int_0^L f(x)\sin(n\pi x/L)\,dx$. Here they remain arbitrary subject to convergence. Every mode has the ordinary diffusive decay rate plus the additional reaction decay rate one.
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