The authors study the existence of weak travelling wave solutions \(u(x,t)= \phi(x- at)\) for the equation \[ u_t= [D(u)u_x]_x+ g(u),\quad (x,t)\in\mathbb{R}\times \mathbb{R}_+,\tag{1} \] where, for a given \(\alpha\in(0, 1)\), the function \(D\) and \(g\), defined on the interval \([0,1]\), satisfy the conditions: 1) \(g(0)= g(\alpha)= g(1)= 0\), \(g(u)0\), for all \(u\in(\alpha, 1)\); 2) \(g\in C^2\), \(g'(0)0\), \(g'(1)0\); 3) \(D(0)= 0\), \(D(u)>0\) for all \(u\in(0, 1]\); 4) \(D\in C^2\), \(D'(u)>0\) for all \(u\in[0,1]\), \(D''(0)>0\). The solution \(u(x,t)\) must also satisfy the initial condition \(u(x,0)= u_0(x)\) with \(0\leq u_0(x)\leq 1\), where \(u_0\) is a piecewise differentiable function on \(\mathbb{R}\), and the function \(\phi\) must satisfy the inequality \(0\leq \phi(\xi)\leq 1\), for all \(\xi\in(-\infty, \infty)\). The equation (1) is a generalization of the Nagumo equation arising in nerve conduction theory.