Global continuation of monotone waves for a unimodal bistable reaction-diffusion equation with delay

Sergei Trofimchuk, Vitaly Volpert

Published 2017-06-11Version 1

We consider a delayed reaction-diffusion equation $u_t(t,x) = u_{xx}(t,x) + g(u(t,x), u(t-\tau,x)) \ (*)$ with bistable nonlinearity $g$. In difference with previous works, we do not assume the monotonicity of $g(u,v)$ with respect to $v$ that hinders application of the comparison techniques. For two different types of $v-$unimodal nonlinearity $g(u,v)$, we prove the existence of a maximal continuous family of bistable monotone wavefronts $u= \phi(x+c(\tau)t,\tau)$. Our proof is based on a variant of the Hale-Lin functional-analytic approach to heteroclinic solutions where Lyapunov-Schmidt reduction is realised in appropriate weighted spaces of $C^2$-smooth functions. This method requires a detailed analysis of associated linear differential Fredholm operators and their formal adjoints. Depending on type of unimodality (equivalently, on the sign of $c(\tau)$), two different scenarios can be observed for our wave solutions of $(*)$: i) independently on the size of delay $\tau$, each bistable wavefront is monotone; ii) wavefronts are monotone for moderate values of $\tau$ and can oscillate for large $\tau$. Our results are illustrated by two biological models and one `toy' example.