Blow-Up Solution of Modified-Logistic-Diffusion Equation

Abstract

Modified-Logistic-Diffusion Equation $u_t=D{u}_{xx}+u|\mathrm{1}-u|$ with Neumann boundary condition has a global solution, if the given initial condition $\psi$ satisfies $\psi (x)\le \mathrm{1}$, for all $x\in [\mathrm{0,1}]$. Other initial conditions can lead to another type of solutions; i.e., an initial condition that satifies ${\int }_{\mathrm{0}}^{\mathrm{1}}\psi (x)dx>\mathrm{1}$ will cause the solution to blow up in a finite time. Another initial condition will result in another kind of solution, which depends on the diffusion coefficient $D$. In this paper, we obtained the lower bound of $D$, so that the solution of Modified-Logistic-Diffusion Equation with a given initial condition will have a global solution.