Standing pulse solutions to FitzHugh-Nagumo equations
Let \(\beta\) be a fixed constant with \(0<\beta<\frac{1}{2}\). This paper deals with existence and qualitative properties of a standing pulse solution to the FitzHugh-Nagumo system \[ (P)\left\{ \begin{aligned} u_t&=du_{xx}+f(u)-v,\\ \tau v_t&=v_{xx}+u-\gamma v, \end{aligned} \right. \] where \(f(u)=u(u-\beta)(1-u),d>0\) and \(\gamma>0\) are adjusted constants, \(\tau>0\) is a given constant. Let \( (P_s)\) be the stationary system associated to \((P)\). A standing pulse for \((P)\) is a weak solution \( (u,v)\) of \((P_s)\) that belongs to \((H^1(\mathbb R))^2\) and is even in \(x\). The relation \[ v(x)=\mathcal L u(x) \] is obtained where \(\mathcal L:L^2(0,\infty)\to L^2(0,\infty)\) is the self adjoint operator defined by \[ \mathcal Lw(x)\equiv\int_0^{\infty} G(x,s)w(s)ds. \] Here \(G\) is the Green's function for \((\gamma-\frac{d^2}{dx^2})\) associated with homogeneous Neumann condition at \(x=0\) and decay at large \(x\), that is symmetric in \(x\) and \(s\). Let \(f(\xi)=-\int_0^{\xi} f(\eta) d\eta.\) The functional \(\hat{J}:H^1(0,\infty)\to\mathbb R\), defined by \[ \hat{J}(w)=\int_0^{\infty}\biggl(\frac{d}{2}w_x^2+\frac{1}{2}w\mathcal L w+F(w)\biggl) dx, \] is introduced. It is stated that, for suitable constants \(\gamma,d,\) \(\hat{J}\) possesses a local minimizer \(u_o\in H^1(0,\infty)\) with \(\hat{J}(u_o)<0\). Next, \(u_o\) and \(v_o=\mathcal L u_o\) are both extended to be even functions on \((-\infty,+\infty)\). Then \(u_o\) and \(v_o\) are \(C^{\infty}\)-functions and \((u_o,v_o)\) is a standing pulse solution to \((P)\). Moreover, on \((0,\infty)\), \(u_o\) changes sign exactly once, while \(v_o>0\) and \(v^{\prime}<0\). Two possibilities for the profile of \(u_o\) on \((0,\infty)\) are described. A graph of \(u_o\) and \(v_o\) is presented for suitable values of \(\gamma,\beta, d\). The proof is elaborate. Its starting point is the restriction \(J\) of \(\hat {J}\) to a suitable class of functions \(\mathcal A\) in which \(J\) has a global minimizer.
scientific article; zbMATH DE number 6110419
scientific article