Generalized Reaction-Diffusion Equation With Density Dependent Diffusion

Existence theorem for the traveling wave solutions of the “generalized” nonlinear reaction-diffusion equation $$u_{t}=(u^{m})_{xx}+f(u),$$ is established for any $$m>1.$$ This form generalizes the classical linear form with $$m=1.$$ A critical value of the minimum wave speed $$c^{*}$$ is obtained as $$c^{*}=2m\sqrt{f^{\prime }(0)}$$, which extends the results obtained by Fisher, Kolmogorov, and others.