# zbMATH — the first resource for mathematics

Existence of solutions of the very fast diffusion equation in bounded and unbounded domain. (English) Zbl 1145.35075
The paper is devoted to proving the existence of a unique solution of the problem \begin{aligned} u_t & = (\varphi_m(u))_{xx} \quad \text{in } (a,b)\times (0,T),\\ u(x,0) & = u_0(x) \geq 0\quad \text{in }\;(a,b),\\ (\varphi_m(u))_x(a,t) & = g(t), (\varphi_m(u))_x(b,t) = - f(t), \quad 0 < t < T, \end{aligned} where $$\varphi_m(u) = u^m/m$$ if $$m < 0$$, $$\varphi_m = \log u$$ if $$m = 0$$, and
$T = \sup \left\{t' > 0:\int_{-a}^bu_0(x)\,dx > \int_0^{t'}(f + g)\,dt \right\}$ for the case $$m \leq 0$$, $$0 \leq f,g \in L_{\text{loc}}^{\infty}([0,\infty))$$, $$0 \leq u_0 \in L^{\infty}$$ and the case $$-1 < m \leq 0$$, $$f$$,$$g \in L^{\infty}_{\text{loc}}(a,b)$$ for some $$p > 1 - m$$. This result is extended for the case of higher dimension. The exact decay rate of the solution obtained at infinity is given. Moreover, the author gives a new proof of a result obtained by Rodriguez and Vazquez about the existence of many finite mass solutions of the above equation in $$\mathbb R \times (0,T)$$ for $$-1 < m \leq 0$$.

##### MSC:
 35K65 Degenerate parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35B25 Singular perturbations in context of PDEs 35K55 Nonlinear parabolic equations
##### Keywords:
Neumann boundary problem; exact decay rate
Full Text:
##### References:
  Anderson J.R. (1991). Local existence and uniqueness of solutions of degenerate parabolic equation. Comm. P.D.E. 16: 105–143 · Zbl 0738.35033 · doi:10.1080/03605309108820753  Aronson D.G. (1986). The porous medium equation, CIME Lectures, in Some problems in Nonlinear Diffusion, Lecture Notes in Mathematics 1224. Springer, New York  Aronson D.G., Bénilan P. (1979). Régularité des solutions de l’équation des milieux poreux dans R N . C. R. Acad. Sc. Paris, Série A, t. 288: 103–105 · Zbl 0397.35034  Dahlberg B.E.J., Kenig C. (1986). Non-negative solutions of generalized porous medium equations. Revista Matemática Iberoamericana 2: 267–305 · Zbl 0644.35057  Dahlberg B.E.J., Kenig C. (1988). Non-negative solutions to fast diffusions. Revista Matemática Iberoamericana 4: 11–29 · Zbl 0709.35054  Davis S.H., Dibenedetto E., Diller D.J. (1996). Some a priori estimates for a singular evolution equation arising in thin-film dynamics. SIAM J. Math. Anal. 27(3): 638–660 · Zbl 0854.35014 · doi:10.1137/0527035  Dibenedetto, E., Diller, D.J.: About a singular parabolic equation arising in thin film dynamics and in the Ricci flow for complete $$\mathbb{R}^2$$ . In: Marcellini, P., Giorgio, G. Talenti, Vesentini, E. (eds.) Partial differential equations and applications, Lecture Notes in Pure and Applied Mathematics, vol. 177, pp. 103–119, Dekker, New York (1996) · Zbl 0846.35070  Daskalopoulos P., Del Pino M.A. (1995). On a singular diffusion equation. Comm. Anal. Geometry 3(3): 523–542 · Zbl 0851.35072  Daskalopoulos P., Del Pino M.A. (1997). On nonlinear parabolic equations of very fast diffusion. Arch Rational Mech. Anal. 137: 363–380 · Zbl 0886.35081 · doi:10.1007/s002050050033  Esteban J.R., Rodriguez A., Vazquez J.L. (1988). A nonlinear heat equation with singular diffusivity. Commun. P.D.E. 13(8): 985–1039 · Zbl 0686.35066 · doi:10.1080/03605308808820566  Evans L.C., Spruck J. (1991). Motion of level sets by mean curvature I. J. Differ. Geometry 33(3): 635–681 · Zbl 0726.53029  Evans L.C., Spruck J. (1992). Motion of level sets by mean curvature II. Trans. Amer. Math. Soc. 330(1): 321–332 · Zbl 0776.53005 · doi:10.2307/2154167  Folland G.B. (1976). Introduction to partial differential equations, Mathematical notes series 17. Princeton University Press, Princeton · Zbl 0325.35001  Giga Y., Goto S. (1992). Motion of hypersurfaces and geometric equations. J. Math. Soc. Jpn. 44: 99–111 · Zbl 0739.53005 · doi:10.2969/jmsj/04410099  Giga Y., Yamaguchi K. (1993). On a lower bound for the extinction time of surfaces moved by mean curvature. Cal. Var. 1: 413–428 · Zbl 0854.53011  Hsu S.Y. (2003). Asymptotic behaviour of solution of the equation u t = $$\Delta$$log u near the extinction time. Adv. Differ. Eqn. 8(2): 161–187 · Zbl 1028.35079  Hsu S.Y. (2001). Global existence and uniqueness of solutions of the Ricci flow equation. Differ. Integral Eqn. 14(3): 305–320 · Zbl 1011.35085  Hsu S.Y. (2001). Large time behaviour of solutions of the Ricci flow equation on R 2. Pac. J. Math. 197(1): 25–41 · Zbl 1053.53045 · doi:10.2140/pjm.2001.197.25  Hsu S.Y. (2003). Uniqueness of solutions of a singular diffusion equation. Differ. Integral Eqn. 16(2): 181–200 · Zbl 1036.35103  Hsu S.Y. (2002). Dynamics near extinction time of a singular diffusion equation. Math. Annalen 323(2): 281–318 · Zbl 1007.35009 · doi:10.1007/s002080100304  Hui, K.M.: Singular limit of solutions of the equation $$u_t=\Delta (\frac{u^m}{m})$$ as m 0. Pac. J. Math. 187(2), 297–316 (1999) · Zbl 0931.35081 · doi:10.2140/pjm.1999.187.297  Hui K.M. (1999). Existence of solutions of the equation u t = $$\Delta$$log u. Nonlinear Anal. TMA 37: 875–914 · Zbl 0936.35098 · doi:10.1016/S0362-546X(98)00081-9  Hui K.M. (2001). Asymptotic behaviour of solutions of u t = $$\Delta$$log u in a bounded domain. Differ. Integral Eqn. 14(2): 175–188 · Zbl 1021.35055  Hui, K.M.: Singular limit of solutions of the very fast diffusion equation (preprint) · Zbl 1153.35351  Kato T. (1973). Schrödinger operators with singular potentials. Israel J. Math. 13: 135–148 · Zbl 0246.35025 · doi:10.1007/BF02760233  Kurtz T.G. (1973). Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics. Trans. Amer. Math. Soc. 186: 259–272 · Zbl 0275.47047 · doi:10.1090/S0002-9947-1973-0336482-1  Ladyzenskaya, O.A., Solonnikov, V.A., Uraltceva, N.N.: Linear and quasilinear equations of parabolic type Transl. Math. Mono. vol 23, Amer. Math. Soc. Providence, R.I. (1968)  Lions P.L., Toscani G. (1997). Diffusive limit for finite velocity Boltzmann kinetic models. Revista Matematica Iberoamericana 13(3): 473–513 · Zbl 0896.35109  Peletier, L.A.: The porous medium equation in applications of nonlinear analysis in the physical sciences. In: Amann, H., Bazley, N., Kirchgassner, K. (eds) Pitman, Boston (1981)  Rosen G. (1979). Nonlinear heat conduction in solid H 2. Phys. Rev. B 19: 2398–2399 · doi:10.1103/PhysRevB.19.2398  Rodriguez A., Vazquez J.L. (1990). A well posed problem in singular Fickian diffusion. Arch Rational Mech. Anal. 110: 141–163 · Zbl 0695.76043 · doi:10.1007/BF00873496  Soner H.M., Souganidis P.E. (1993). Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature. Comm. P.D.E. 18: 859–894 · Zbl 0804.53006 · doi:10.1080/03605309308820954  Vazquez, J.L.: Nonexistence of solutions for nonlinear heat equations of fast-diffusion type. J. Math. Pures Appl. 503–526 (1992) · Zbl 0694.35088
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.