Eymard, Nathalie; Volpert, Vitaly; Vougalter, Vitali Existence of pulses for local and nonlocal reaction-diffusion equations. (English) Zbl 1373.35169 J. Dyn. Differ. Equations 29, No. 3, 1145-1158 (2017). Summary: Reaction-diffusion equations with a space dependent nonlinearity are considered on the whole axis. Existence of pulses, stationary solutions which vanish at infinity, is studied by the Leray-Schauder method. It is based on the topological degree for Fredholm and proper operators with the zero index in some special weighted spaces and on a priori estimates of solutions in these spaces. Existence of solutions is related to the speed of travelling wave solutions for the corresponding autonomous equations with the limiting nonlinearity. Cited in 3 Documents MSC: 35K57 Reaction-diffusion equations 35A16 Topological and monotonicity methods applied to PDEs 35Q92 PDEs in connection with biology, chemistry and other natural sciences 92D15 Problems related to evolution Keywords:reaction-diffusion equation; existence of pulse solutions; Leray-Schauder method PDFBibTeX XMLCite \textit{N. Eymard} et al., J. Dyn. Differ. Equations 29, No. 3, 1145--1158 (2017; Zbl 1373.35169) Full Text: DOI HAL References: [1] Ambrosetti, A., Malchiodi, A.: Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge University Press, Cambridge (2007) · Zbl 1125.47052 · doi:10.1017/CBO9780511618260 [2] Berestycki, H., Lions, P.L., Peletier, L.A.: An ODE approach to the existence of positive solutions for semilinear problems in \[\mathbb{R}^N\] RN. Indiana Univ. Math. J. 30(1), 141-157 (1981) · Zbl 0522.35036 · doi:10.1512/iumj.1981.30.30012 [3] Bessonov, N., Reinberg, N., Volpert, V.: Mathematics of Darwin’s diagram. Math. Model. Nat. Phenom. 9(3), 5-25 (2014) · Zbl 1321.92060 · doi:10.1051/mmnp/20149302 [4] Caffarelli, L.A., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42(3), 271-297 (1989) · Zbl 0702.35085 · doi:10.1002/cpa.3160420304 [5] Chen, W.X., Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63(3), 615-622 (1991) · Zbl 0768.35025 · doi:10.1215/S0012-7094-91-06325-8 [6] Chen, W.X., Li, C.: Qualitative properties of solutions to some nonlinear elliptic equations in \[R^2\] R2. Duke Math. J. 71(2), 427-439 (1993) · Zbl 0923.35055 · doi:10.1215/S0012-7094-93-07117-7 [7] Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34(4), 525-598 (1981) · Zbl 0465.35003 · doi:10.1002/cpa.3160340406 [8] Gidas, B., Spruck, J.: A priori bounds for positive solutions of nonlinear elliptic equations. Commun. Partial Differ. Equ. 6(8), 883-901 (1981) · Zbl 0462.35041 · doi:10.1080/03605308108820196 [9] Kuzin, I., Pohozaev, S.: Entire Solutions of Semilinear Elliptic Equations. Birkhäuser, Basel (1997) · Zbl 0882.35003 [10] Volpert, A.I., Volpert, V., Volpert, V.A.: Traveling Wave Solutions of Parabolic Systems. Translation of Mathematical Monographs, vol. 140. AMS, Providence (1994) · Zbl 0805.35143 [11] Volpert, V.: Elliptic Partial Differential Equations. Volume 1. Fredholm Theory of Elliptic Problems in Unbounded Domains. Birkhäuser, Base (2011) · Zbl 1222.35002 [12] Volpert, V.: Elliptic Partial Differential Equations. Volume 2. Reaction-Diffusion Equations. Birkhäuser, Basel (2014) · Zbl 1307.35004 · doi:10.1007/978-3-0348-0813-2 [13] Volpert, V.: Pulses and waves for a bistable nonlocal reaction-diffusion equation. Appl. Math. Lett. 44, 21-25 (2015) · Zbl 1435.35200 · doi:10.1016/j.aml.2014.12.011 [14] Volpert, V., Reinberg, N., Benmir, M., Boujena, S.: On pulse solutions of a reactiondiffusion system in population dynamics. Nonlinear Anal. 120, 76-85 (2015) · Zbl 1327.35189 · doi:10.1016/j.na.2015.02.017 [15] Volpert, V., Vougalter, V.: Existence of stationary pulses for nonlocal reaction-diffusion equations. Doc. Math. 19, 1141-1153 (2014) · Zbl 1319.35093 [16] Vougalter, V., Volpert, V.: On the existence of stationary solutions for some integro-differential equations. Doc. Math. 16, 561-580 (2011) · Zbl 1235.35279 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.