35 Partial differential equations -- 35A General topics 35A18 Wave front sets
35 Partial differential equations -- 35K Parabolic equations and systems 35K91 Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian
The authors study the existence of monotone traveling fronts for a family of reaction-diffusion equations with delay, of the form $$ u_t(t,x)=u_{xx}(t,x)+g\left(u(t,x), u(t-\tau,x)\right), $$ where $(t,x)\in {\Bbb R}^2$, $\tau\geq 0$, and the reaction term $g$ is assumed to satisfy the following bistability condition: $g$ is a smooth function such that the equation $g(u,u)=0$ has exactly three solutions: $e_1MR3174733] concerning the real zeros of a specific transcendental function. \par The proofs are based on the homotopy method and on an alternative form of the Hale-Lin functional-analytic approach to heteroclinic solutions of functional differential equations (introduced in [J.~K. Hale and X.-B. Lin, J. Differential Equations {\bf 65} (1986), no.~2, 175--202; MR0861515]) by means of an appropriate Lyapunov-Schmidt reduction requiring a rigorous analysis of variational equations along with the monotone traveling wave. \par In addition, the article is complemented with two meaningful biological applications: a Mackey-Glass type bistable model and a version of virus infection spreading in tissues.