A heteroclinic orbit connecting traveling waves pertaining to different nonlinearities

Abstract

In this paper we consider a semilinear parabolic equation in an infinite cylinder. The spatially varying nonlinearity is such that it connects two (spatially independent) bistable nonlinearities in a compact set in space. We prove that, given such a setting, a traveling wave obeying the equation with the one bistable nonlinearity and starting at the respective side of the cylinder, will converge to a traveling wave solution prescribed by the nonlinearity on the other side.

On the mean speed of bistable transition fronts in unbounded domains

“…Lastly, even in the homogeneous space R N , non-standard transition fronts which are not invariant in any moving frame were also constructed in [24] under assumptions (1.2)-(1.5). More generally speaking, there is now a large literature devoted to transition fronts for bistable reactions in homogeneous or heterogeneous settings [6,13,18,22,48,53,60], as well as for other types of homogeneous or space/time dependent reactions in dimension 1 [16,29,30,34,35,37,40,42,43,51,52,57,58] and in higher dimensions [1,9,38,39,49,50,59,61].…”

“…18) where r M δ +Rε > 0 is given in (1.8) with M := M δ + R ε . Lemma 4.5 then yields the existence of two real numbers T 1…”

Front blocking versus propagation in the presence of drift term varying in the direction of propagation

“…Now we have everything in place to proof Theorem 1.3. The proof resembles the respective one of Theorem 4.1 in [7] and will be slightly modified and added for the sake of completeness.…”

Front blocking in the presence of gradient drift

“…For the sake of completeness and to ensure the reader that we are not investigating the empty set of solutions or assume uniqueness of solutions of (1.1) without justification, let us mention that existence and uniqueness for solutions of (1.1) can be obtained almost literally copying the proof of Theorem 2.1 in [4] or Appendix A in [8].…”