Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T21:08:24.815Z Has data issue: false hasContentIssue false
  • English
  • Français

The Continuous Dependence on the Nonlinearities of Solutions of Fast Diffusion Equations

Published online by Cambridge University Press:  20 November 2018

Jiaqing Pan
Jiaqing Pan*
Affiliation:
Institute of Mathematics, Jimei University, Xiamen, 361021, P. R. Chinae-mail: jqp4300@Yahoo.com.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we consider the Cauchy problem

$$\left\{ \begin{align} & {{u}_{t}}=\Delta ({{u}^{m}}),\,\,\,\,\,x\in {{\mathbb{R}}^{N}},t>0,N\ge 3, \\ & u(x,0)={{u}_{0}}(x),\,\,\,\,\,x\in {{\mathbb{R}}^{N}}. \\ \end{align} \right.$$

We will prove that

(i) for ${{m}_{c}}\,<\,m,\,{{m}_{0}}\,<\,1,\,\left| u(x,\,t,m)-u(x,\,t,{{m}_{0}}) \right|\,\to \,0$ as $m\,\to \,{{m}_{0}}$ uniformly on every compact subset of ${{\mathbb{R}}^{N}}\,\times \,{{\mathbb{R}}^{+}}$, where ${{m}_{c}}\,=\,\frac{{{(N-2)}_{+}}}{N}$;

(ii) there is a ${{C}^{*}}$ that explicitly depends on $m$ such that

$${{\left\| u(\cdot ,\cdot ,m)-u(\cdot ,\cdot ,1) \right\|}_{{{L}^{2}}({{\mathbb{R}}^{N}}\times {{\mathbb{R}}^{+}})}}\le {{C}^{*}}\left| m-1 \right|.$$

Keywords

35K0535K1035K15fast diffusion equationsCauchy problemcontinuous dependence on nonlinearity
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 3 , 01 September 2012 , pp. 623 - 631
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Aronson, D. G. and Bénilan, P., Régularité des solutions de l’équation des milieux poreux dans ℝ n. C. R. Acad. Sci. Paris Sér. A-B 288(1979), no. 2, A103–A105.Google Scholar
[2] Bénilan, P. and Crandall, M. G., The continuous dependence on ϕ of solution of ut– Δϕ(u) = 0. J. Indiana Univ. Math. 30(1981), no. 2. 161–177. http://dx.doi.org/10.1512/iumj.1981.30.30014 Google Scholar
[3] Cockburn, B. and Gripenberg, G., Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations. J. Differential Equations 151(1999), no. 2, 231–251. http://dx.doi.org/10.1006/jdeq.1998.3499 Google Scholar
[4] Ebmeyer, C., Regularity in Sobolev spaces for the fast diffusion and the porus medium equation. J. Math. Anal. Appl. 307(2005), no. 1, 134–152. http://dx.doi.org/10.1016/j.jmaa.2005.01.009 Google Scholar
[5] Esteban, J. R., Rodriguez, A., and Vázquez, J. L., A nonlinear heat equation with singular diffusivity. Commun. Partial Differential Equations 13(1988), no. 8, 985–1039. http://dx.doi.org/10.1080/03605308808820566 Google Scholar
[6] Gilding, B. H. and Peletier, L. A., The Cauchy problem for an equation in the theory of infiltration. Arch. Rational Mech. Anal. 61(1976), no. 2, 127–140.Google Scholar
[7] Herrero, M. A. and Pierre, M., The Cauchy problem for ut = Δum when 0 < m < 1. Tran. Amer. Math. Soc. 291(1985), no. 1, 145–158.Google Scholar
[8] Hui, K. M., Existence of solutions of the very fast diffusion equation. Nonlinear Anal. 58(2004), no. 1–2, 75–101. http://dx.doi.org/10.1016/j.na.2004.05.001 Google Scholar
[9] Pan, J. and Gang, L., The linear approach for a nonlinear infiltration equation. European J. Appl. Math. 17(2006), no. 6, 665–675. http://dx.doi.org/10.1017/S095679250700681X Google Scholar
[10] Vázquez, J. L., Symmetrization and mass comparison for degenerate nonlinear parobolic and related elliptic equations. Adv. Nonlinear Stud. 5(2005), no. 1, 87–131.Google Scholar
[11] Vázquez, J. L., An introduction to the mathematical theory of the porous medium equation. In: Shape optimization and free boundaries (Montreal, PQ, 1990), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 1992, pp. 347–389.Google Scholar
[12] Vázquez, J. L., Smoothing and decay estimates for nonlinear diffusion equations. Equations of porous medium type. Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006.Google Scholar