This paper deals with the appearance of monotone bounded travelling wave solutions for a parabolic reaction-diffusion equation which frequently meets both in chemical and biological systems. In particular, we prove the existence of monotone front type solutions for any wave speed c greater than or equal to c* and give an estimate for the threshold value c*. Our model takes into account both of a density dependent diffusion term and of a non-linear convection effect. Moreover, we do not require the main non-linearity g to be a regular C1 function; in particular we are able to treat both the case when g'(0) = 0, giving rise to a degenerate equilibrium point in the phase plane, and the singular case when g'(0) = +∞. Our results generalize previous ones due to ARONSON and WEINBERGER [Adv. Math. 30 (1978), pp. 33 - 76], GIBBS and MURRAY (see MURRAY [Mathematical Biology, Springer-Verlag, Berlin, 1993]) and MCCABE, LEACH and NEEDHAM [SIAM J. Appl. Math. 59 (1998), pp. 870-899]. Finally, we obtain our conclusions by means of a comparison-type technique which was introduced and developed in this framework in a recent paper by the same authors.

Travelling wavefronts in reaction-diffusion equations with convection effects and non-regular terms / Malaguti, Luisa; C., Marcelli. - In: MATHEMATISCHE NACHRICHTEN. - ISSN 0025-584X. - STAMPA. - 242:(2002), pp. 148-164. [10.1002/1522-2616(200207)242:1<148::AID-MANA148>3.0.CO;2-J]

Travelling wavefronts in reaction-diffusion equations with convection effects and non-regular terms

MALAGUTI, Luisa;
2002

Abstract

This paper deals with the appearance of monotone bounded travelling wave solutions for a parabolic reaction-diffusion equation which frequently meets both in chemical and biological systems. In particular, we prove the existence of monotone front type solutions for any wave speed c greater than or equal to c* and give an estimate for the threshold value c*. Our model takes into account both of a density dependent diffusion term and of a non-linear convection effect. Moreover, we do not require the main non-linearity g to be a regular C1 function; in particular we are able to treat both the case when g'(0) = 0, giving rise to a degenerate equilibrium point in the phase plane, and the singular case when g'(0) = +∞. Our results generalize previous ones due to ARONSON and WEINBERGER [Adv. Math. 30 (1978), pp. 33 - 76], GIBBS and MURRAY (see MURRAY [Mathematical Biology, Springer-Verlag, Berlin, 1993]) and MCCABE, LEACH and NEEDHAM [SIAM J. Appl. Math. 59 (1998), pp. 870-899]. Finally, we obtain our conclusions by means of a comparison-type technique which was introduced and developed in this framework in a recent paper by the same authors.
2002
242
148
164
Travelling wavefronts in reaction-diffusion equations with convection effects and non-regular terms / Malaguti, Luisa; C., Marcelli. - In: MATHEMATISCHE NACHRICHTEN. - ISSN 0025-584X. - STAMPA. - 242:(2002), pp. 148-164. [10.1002/1522-2616(200207)242:1<148::AID-MANA148>3.0.CO;2-J]
Malaguti, Luisa; C., Marcelli
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/306808
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