Global Travelling Waves in Reaction–Convection–Diffusion Equations

Travelling Waves in Nonlinear Diffusion-Convection Reaction

“…Many of the conclusions of this lemma and therewith Theorems 60 and 61 can be obtained from results we have already established and straightforward study of the integral equation ( The exposition above provides an upper bound on the magnitude of the critical value σ * in Theorem 60(c) when γ = 0 which is sharper than the corresponding bound in [208]. By further analysis of the integral equation (12.1) one can also obtain a lower bound for γ = 0 which is sharper than that previously obtained.…”

“…Recently de Pablo and Sánchez [208] have conducted a thorough analysis of semi-wavefront solutions decreasing to 0 for the full power-law reactionconvection-diffusion equation with b 0 = 0 and c 0 = 0. Terms which are used in that analysis include a local wave to distinguish a strict semi-wavefront solution from a global solution, a finite wave for a travelling-wave solution whose support is bounded above or below, and, a positive wave for a global travelling-wave solution whose support is unbounded above and below.…”

“…Terms which are used in that analysis include a local wave to distinguish a strict semi-wavefront solution from a global solution, a finite wave for a travelling-wave solution whose support is bounded above or below, and, a positive wave for a global travelling-wave solution whose support is unbounded above and below. Also for the reader referring to [208], at the risk of causing confusion where we intend to clarify, we mention that a bounded wave in that article is synonymous with a global travelling-wave solution, while an unbounded wave is an unbounded strict semi-wavefront solution in our terminology. The analysis of [208] is a characterization of unbounded monotonic travelling-wave solutions decreasing to 0 tantamount to the following theorems.…”

“…Also for the reader referring to [208], at the risk of causing confusion where we intend to clarify, we mention that a bounded wave in that article is synonymous with a global travelling-wave solution, while an unbounded wave is an unbounded strict semi-wavefront solution in our terminology. The analysis of [208] is a characterization of unbounded monotonic travelling-wave solutions decreasing to 0 tantamount to the following theorems. Table 9 for n < 1 and Table 10 for n > 1.…”

“…The characterization of unbounded monotonic travelling-wave solutions of equation (12.18) decreasing to 0 by de Pablo and Sánchez [208] is obtained using a sophisticated phase-plane analysis. This has the following interpretation for solutions of the corresponding integral equation (12.1).…”