A class of singular first order differential equations with applications in reaction-diffusion
The authors investigate the following first-order differential equation \[ y'=q(cy^{1/p}-f(u)), \] where \(c\) is a positive parameter, \(p, q\) greater than unity, are conjugate numbers and \(f\) is a continuous function such that \(f(0)=f(1)=0\). Solutions satisfying \(y(0)=y(1)=0\) are of particular interest from the physical point of view. Indeed, the equation stems from the Fisher reaction-diffusion equation when one looks for traveling wave solutions. According to the nonlinearity, three cases are discussed separately, including the stable and the bi-stable cases. Existence of positive solutions is then investigated together with their behavior near the origin and \(1\); the discussion takes into account the admissible values of the speed parameter \(c\) and the behavior of the ratio \(f(u)/u^{q-1}\) in the vicinity of the origin or on the interval \((0,1]\). The authors use shooting techniques.
scientific article; zbMATH DE number 6146681