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

We consider the parabolic-difference system \( \Big ({\dot{u}}(t,x), v(t, x)\Big )=\Big (D\, u_{xx}(t, x)\hspace{-0.06cm}-\hspace{-0.06cm}f(u(t, x))+Hv(t-h, \cdot )(x), \,\, g(u(t, x))+B v(t-h, \cdot )(x)\Big )\) $(\dot{u}(t,x),v(t,x))=(D{u}_{\mathit{xx}}(t,x)-f(u(t,x))+Hv(t-h,·)(x),g(u(t,x))+Bv(t-h,·)(x))$ , \( t>0, x\in {{\mathbb {R}}},\) $t>0,x\in \mathbb{R},$ which appears in a model for hematopoietic cells population. We prove the global stability of semi-wavefronts \((\phi _c, \varphi _c)\) $({\varphi }_c,{\phi }_c)$ for this system. More precisely, for an initial history \((u_0, v_0)\) $(u_0,v_0)$ we study the convergence to zero of the associated perturbation \(P(t)=(u(t)-\phi _c, v(t)-\varphi _c)\) $P(t)=(u(t)-{\varphi }_c,v(t)-{\phi }_c)$ , as \(t\rightarrow +\infty \) $t\to +\infty$ , in a suitable Banach space Y; we prove that if the initial perturbation satisfies \(P_0\in C([-h, 0], Y)\) $P_0\in C([-h,0],Y)$ , then \(P(t)\rightarrow 0\) $P(t)\to 0$ in two cases: (i) \(v_0=\varphi _c\) $v_0={\phi }_c$ , for all \(h\ge 0\) $h\ge 0$ or (ii) \(v_0\not \equiv \varphi _c\) $v_0\not\equiv {\phi }_c$ for all \(h\le h_*\) $h\le h_{\ast }^{}$ and some \(h_*=h_*(B)\) $h_{\ast }^{}=h_{\ast }^{}(B)$ . 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.