Monotone wave fronts for $(p,q)$-Laplacian driven reaction-diffusion equations

Abstract

We study the existence of monotone heteroclinic traveling waves for the $1$-dimensional reaction-diffusion equation $$u_t=(|u_x{|}^{p-2}{u}_x+|u_x{|}^{q-2}{u}_x{)}_x+f(u),$$ where the non-homogeneous operator appearing on the right-hand side is known as $(p,q)$-Laplacian. Here we assume that $2\le q<p$ and $f$ is a nonlinearity of Fisher type, namely it is always positive out of its zeros. We give an estimate of the critical speed and we comment on the roles of $p$ and $q$ in the dynamics, providing some numerical simulations.