$ d, r, K, l, L $ with $ l<L $ are positive constants.
Question 1: Please solve this PDE for function $ I $: $$ \frac{\partial I}{\partial t}=d\frac{\partial^{2}I}{\partial x^{2}}+rI(1-\frac{I}{K}) $$ $$I(x,1)=L-x, x\in[l,L] $$ $$\frac{\partial I}{\partial x}(l,t)=\frac{\partial I}{\partial x}(L,t)=0, t\ge 1 $$ where $I(x,t): {\mathbb{R}}^{+^{2}} \to \mathbb{R}^{+}$ is with $x\in [l,L], t\ge 1$.
Question 2: If we change $L-x$ into general positive differentiable function $\phi(x)$, is there any general method to solve this equation?
p.s. This appears in Diffusive Logistic Model of social network service.
The Leibniz notation / choice of parmaeter letters is a headache to look at.
We have $$u_{t}-u_{xx}=ru-\alpha u^{2}.$$ This is a reaction-diffusion equation with a second order non-linearity. You cannot in general expect to obtain an exact soluion to the associated BVP for such a PDE. It is not even a trivial matter to verify that the problem is well-posed.