A Simple Approach to Stability of Semi-wavefronts in Parabolic-Difference Systems

We consider the parabolic-difference system \( \Big ({\dot{u}}(t,x), v(t, x)\Big )=\Big (D\, u_{xx}(t, x)\hspace{-0.06cm}-\hspace{-0.06cm}f(u(t, x))+Hv(t-h, \cdot )(x), \,\, g(u(t, x))+B v(t-h, \cdot )(x)\Big )\), \( t>0, x\in {{\mathbb {R}}},\) which appears in a model for hematopoietic cells population. We prove the global stability of semi-wavefronts \((\phi _c, \varphi _c)\) for this system. More precisely, for an initial history \((u_0, v_0)\) we study the convergence to zero of the associated perturbation \(P(t)=(u(t)-\phi _c, v(t)-\varphi _c)\), as \(t\rightarrow +\infty \), in a suitable Banach space Y; we prove that if the initial perturbation satisfies \(P_0\in C([-h, 0], Y)\), then \(P(t)\rightarrow 0\) in two cases: (i) \(v_0=\varphi _c\), for all \(h\ge 0\) or (ii) \(v_0\not \equiv \varphi _c\) for all \(h\le h_*\) and some \(h_*=h_*(B)\). This result is obtained by analyzing an abstract integral equation with infinite delay. Also, our main result allow us to obtain a result about the uniqueness of these semi-wavefronts.