The sub-supersolution method for the Fitzhugh-Nagumo type reaction-diffusion system with heterogeneity (Q1661071)

The main result of the paper concerns the existence of a solution \((u,v)\), defined on the half-line \(\mathbb{R}^+\), to the system \[ \begin{split} &-du''=\mu(x)f(u)-v, \\ &-v''+\gamma v=u \end{split} \] with boundary conditions that express the connection between two equilibria \[ (u,v)(0)=(0,0),\quad (u,v)(+\infty)=(a_\gamma,a_\gamma/\gamma), \] where \(d>0\), \(f(s)=s-s^3\), \(\gamma>1\), \(a_\gamma\) is the largest root of \(\gamma f(s)=\gamma\); \(\mu\) is an increasing \(C^1\) function on the half-line that takes a constant value \(\mu_0>0\) in some interval \([0,l_0]\) and \(0<1-\mu\in L^1(\mathbb{R}_+)\). According to the main result, there exists a solution to this problem provided that \(d<A\) and \(\gamma >B\), where \(A\) and \(B\) are positive constants that can be described explicitly in terms of the data. Moreover, the solution \((u,v)\) satisfies \[ W(x)\leq u(x)\leq T(x),\quad 0\leq v(x)\leq 1/_\gamma \;\,\forall x\in\mathbb{R}_+, \] where \(W\), \(T\) are a subsolution and a supersolution, respectively, such that for \(x\in\mathbb{R}_+\) \[ \begin{cases} -dT''=\mu(x)f(T(x)), \\ 0\leq T(x)\leq 1,\\ T'(x)\geq 0,\\ T(0)=0,\;\,T(+\infty)=1, \end{cases} \] \[ \begin{cases} -dW''=\mu_0f(W)-1/_\gamma, \\ 0\leq W(x)\leq u_\gamma,\\ W'(x)\geq 0, \\ W(0)=0,\;\,W(+\infty)=u_\gamma, \end{cases} \eqno{(a)} \] (here \(u_\gamma \) is the largest root of \(\mu_0f(s)=1/\gamma\)). The proof involves variational arguments (minimization), use of the subsolution-supersolution method and approximate solutions in finite subintervals. In particular, a central role is played by the minimum problem \[ \sigma (d,\gamma)=\inf\left\{\frac{J_d(\xi)}{\int_0^\infty|\xi(x)|^2}:\, \xi\in H_0^1(\mathbb{R}_+),\;\xi\neq 0\right\}, \] where the functional \(J_d\) is given by \(J_d(\xi)=\int_0^\infty\left[\frac{d}{2}\xi'(x)^2-\mu(x)(1-3W_{d,\gamma}(x)^2)\xi(x)^2\right]\,dx\). Here, \(W_{d,\gamma}\) means the solution of \((a)\). It is shown that for convenient values of \(\gamma\) and \(d\), \(\sigma (d,\gamma)\) is positive and in fact bounded away from zero. Then the key point for the existence of a solution is the condition \(\gamma\sigma (d,\gamma)>1\). The method is applicable also to the problem, where the righthandside of the first equation is modified to \(\mu(x)(f(u)-v(x))\).