Publication: Pointwise gradient estimates and stabilization for Fisher-KPP type equations with a concentration dependent diffusion
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We prove a pointwise gradient estimate for the bounded weak solution of the Cauchy problem associated to the quasilinear Fisher-KPP type equation ut ='(u)xx + (u) when ' satisÖes that '(0)=0; and (u) is vanishing only for levels u = 0 and u = 1. As a Örst consequence we prove that the bounded weak solution becomes instantaneously a continuous function even if the initial datum is merely a discontinuous bounded function. Moreover the obtained estimates also prove the stabilization of the gradient of bounded weak solutions as t ! +1 for suitable initial data.
D. G. Aronson. Regularity properties of flows through porous media, SIAM J. Appl. Math. 17 (1969), 461-467. Ph. Benilan. Evolution Equations and Accretive Operators, Lecture Notes, Univ. Kentucky, manuscript, 1981. Ph. Benilan and J.I. Díaz, Pointwise gradient estimates of solutions of onedimensional nonlinear parabolic problems, J. Evolution Equations, 3 (2004) 557-602. J.I. Díaz, On some onedimensional reaction-diffusion-convection equations. To appear. J.I. Díaz and S. Kamin, Convergence to Travelling Waves for Quasilinear FisherKPP Type Equations. To appear. E. DiBenedetto, Continuity of Weak Solutions to a General Porous Medium Equation, Indiana Univ. Math. J. 32 No. 1 (1983), 83-118. R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics 7 (1937), 355-369. M.A. Herrero, J.L. Vázquez, The one-dimensional nonlinear heat equation with absorption. Regularity of solutions and interfaces, SIAM J. Math. Anal. 18 (1987) 149-167. A.S. Kalashnikov, The propagation of disturbances in problems of non-linear heat conduction with absorption, USSR Comput. Math. and Math. Phys. 14 (1974), 70-85. S. Kamin and P. Rosenau, Convergence to the Travelling Wave Solution for a Non-linear Reaction-Diffusion Equation, Rendiconti Mat. Acc. Lincei Cl. Sci. Fis. Mat. Natur. 15 (2004), 271-280. R. Kersner, Degenerate parabolic equations with general nonlinearities, Nonlinear Anal. 4 (1984), 1043-1062. S.N. Kruzhkov, Results concerning the nature of the continuity of solutions of parabolic equations and some of their applications, Math. Zam. 6, 1 (1969) 97-98. English tr. in Math. Notes, V 6, (1969) 517-523. O.A. Ladyzenskaya, V.A. Solonnikov and N.N. Uralítseva, Linear and Quasilinear Equations of Parabolic Type. Transl. Math. Monographs, Vol. 23, Amer. Math. Soc., Providence, RI. 1968. A. Kolmogorov, I. Petrovsky and N. Piscunov, Etude de líequation de la di§usion avec croissance de la quantité de matiere et son application a un probleme biologique, Bulletin Univ. Moscow, Ser. Internationale, Math., Mec. 1 (1937),1-25. English translation in P. Pelce, (ed.), Dynamics of Curved Fronts, Academic Press, Boston, 1988, 105-130. O.A. Oleinik, A.S. Kalashnikov and Y.-L. Chzhou, The Cauchy problem and boundary problems for equations of the type of nonstationary Öltration, Izv. Akad. Nauk. SSSR Ser. Mat. 22 (1958), 667-704( Russian). Ph. Souplet, An optimal Liouville theorem for radial entire solutions of the porous medium equation with source, J. Differential Equations 246 (2009), 3980-4005. J.L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Univ. Press, 2007.