RT Journal Article T1 A note on spatial uniformation for Fisher-KPPtype equations with a concentration dependentdiffusion A1 Díaz Díaz, Jesús Ildefonso AB We prove a pointwise gradient estimate for the solution of the Cauchy problem associated to the quasilinear Fisher-KPP type equation with a diffusion coefficient ϕ(u) satisfying that ϕ(0) = 0, ϕ(1) = 1 and a source term ψ(u) which is vanishing only for levels u = 0 and u = 1. As consequence we prove that the bounded weak solution becomes instantaneously a continuous function even if the initial datum is merely a bounded function. PB Inderscience publishers SN 1752-3583 YR 2012 FD 2012 LK https://hdl.handle.net/20.500.14352/44514 UL https://hdl.handle.net/20.500.14352/44514 LA eng NO Benilan, Ph. and Díaz, J.I. (2004) ‘Pointwise gradient estimates of solutions of onedimensional nonlinear parabolic problems’, J. Evolution Equations, Vol. 3,pp.557–602.Diaz, J.I. (2012) ‘On some onedimensional parabolic reaction-diffusion-convection equations’, Journal of Mathematical Analysis and Applications, to appear.Díaz, J.I. and Kamin, S. (2012) ‘Convergence to travelling waves for quasilinear fisher-KPP type equations’, Journal of Mathematical Analysis and Applications, to appear.DiBenedetto, E. (1983) ‘Continuity of weak solutions to a general porous medium equation’,Indiana Univ. Math. J., Vol. 32, No. 1, pp.83–118.Fisher, R.A. (1937) ‘The wave of advance of advantageous genes’, Annals of Eugenics, Vol. 7, pp.355–369.Gilding, B.H. (1976) ‘Hölder continuity of solutions of parabolic equations’, J. London Math. Soc., Vol. s2-13, No. 1, pp.103-106.Kalashnikov, A.S. (1974) ‘The propagation of disturbances in problems of non-linear heat conduction with asorption’, USSR Comput. Math. and Math. Phys., Vol. 14, pp.70–85.Kamin, S. and Rosenau, P. (2004) ‘Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation’, Rendiconti Mat. Acc. Lincei Cl. Sci. Fis. Mat. Natur., Vol. 15, pp.271–280.Kersner, R. (1984) ‘Degenerate parabolic equations with general nonlinearities’, Nonlinear Anal., Vol. 4, pp.1043–1062.Kolmogorov, A., Petrovsky, I. and Piscunov, N. (1937) Etude de l’equation de la diffusion avec croissance de la quantité de matiere et son application a un probleme biologique,Bulletin Univ. Moscow, Ser. Internationale, Math., Mec., in Pelce, P. (Ed.), Vol. 1, pp.1–25. English translation, Dynamics of Curved Fronts, Academic Press, Boston, 1988,pp.105–130.Kruzhkov, S.N. (1969) ‘Results concerning the nature of the continuity of solutions of parabolic equations and some of their applications’, Math. Zam. English tr. in Math. Notes V, Vol. 6, No. 1, pp.97–98, pp.517–523.Ladyzenskaya, O.A., Solonnikov, V.A. and Ural’tseva, N.N. (1968) Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monographs, Vol. 23,Amer. Math. Soc, Providence, RI.Oleinik, O.A., Kalashnikov, A.S. and Chzhou, Y-L. (1958) ‘The Cauchy problem and boundary problems for equations of the type of nonstationary filtration’, Izv. Akad.Nauk. SSSR Ser. Mat, Vol. 22, pp.667–704 (Russian). NO Unión Europea. FP7 NO DGISPI (Spain) NO UCM DS Docta Complutense RD 17 may 2024