Well-Posedness of the Fisher–KPP Equation with Neumann, Dirichlet, and Robin Boundary Conditions on the Real Half Line

We consider the Fisher–KPP equation with Neumann boundary conditions on the real half line. We claim that the Fisher-KPP equation with Neumann boundary conditions is well-posed only for odd positive stationary solutions. We begin by proving that the Fisher-KPP equation with a Dirichlet boundary condition is stable, and with a Robin condition is stable only for odd positive stationary solutions. Then we inferred and proved that the Fisher-KPP equation with Neumann boundary conditions is stable only for odd positive stationary solutions. We solved the Fisher–KPP equation with Neumann boundary conditions to demonstrate the existence of the solution. In addition, we proved the uniqueness of the solution. Moreover, we proved the the solution of Fisher-Kpp equation with Dirichlet condition is stable. We also showed that the Fisher-KPP equation with Robin boundary conditions is stable only for odd positive stationary solutions. The uniqueness and existence proof of the Fisher-KPP equation with Robin condition are similar to the Neumann condition. Hence, we conclude that the Fisher-KPP equation on the real line is well-posed for the Dirichlet condition, and well-posed only for odd positive stationary solutions for both the Neumann condition and the Robin condition.