# zbMATH — the first resource for mathematics

On the Cauchy problem for the fast diffusion equation. (English) Zbl 1140.35496
Summary: For $$u_0$$, $${1\over u_0}\in L^1_{\text{loc}}(\mathbb{R}^n)$$, the author studies the existence of a kind of weak solution to the Cauchy problem
\begin{aligned} u_t= \text{div}(u^{m-1} Du)&\quad\text{in }\mathbb{R}^n\times (0,T],\\ u(x,0)= u_0(x)\geq 0 &\quad\text{in }\mathbb{R}^N, \end{aligned} where $$m< 0$$ is a constant. The uniqueness and regularity of solutions are also discussed.
##### MSC:
 35K65 Degenerate parabolic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000) 35K15 Initial value problems for second-order parabolic equations 35K55 Nonlinear parabolic equations
##### Keywords:
existence; regularity
Full Text:
##### References:
 [1] Aronson, D.G., The porous medium equation, () · Zbl 0626.76097 [2] Aronson, D.G.; Caffarelli, L., The initial trace of a solution of the porous medium equation, Trans. amer. math. soc., 280, 351-366, (1983) · Zbl 0556.76084 [3] Andreucci, D.; Dibenedetto, E., On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources, Ann. sc. norm. super. Pisa ser. 4, 18, 3, 393-441, (1991) · Zbl 0762.35052 [4] Bénilan, P.; Crandall, M.G.; Pierre, M., Solutions of the porous medium equation in $$R^n$$ under optimal conditions on initial values, Indiana univ. math. J., 33, 51-87, (1984) · Zbl 0552.35045 [5] Bernard, G., Existence theorems for fast diffusion equations, Nonlinear anal., 43, 575-590, (2001) · Zbl 0963.35090 [6] Bertsch, M.; Dal Passo, R.; Ughi, M., Discontinuous “viscosity” solutions of a degenerate parabolic equation, Trans. amer. math. soc., 320, 2, 779-798, (1990) · Zbl 0714.35039 [7] Chayes, J.T.; Osher, S.J.; Ralston, J.V., On the singular diffusion equations with applications to self-organized criticality, Comm. pure appl. math., 46, 1363-1377, (1993) · Zbl 0832.35142 [8] Dibenedetto, E., Degenerate parabolic equations, (1993), Springer-Verlag New York · Zbl 0794.35090 [9] Dibenedetto, E.; Friedman, A., Hölder estimates for nonlinear degenerate parabolic systems, J. reine angew. math., 357, 1-22, (1985) · Zbl 0549.35061 [10] Dibenedetto, E.; Friedman, A., Regularity of solutions of nonlinear degenerate parabolic systems, J. reine angew. math., 349, 83-128, (1984) · Zbl 0527.35038 [11] Dahlberg, B.E.G.; Kenig, C.E., Nonnegative solutions of the porous medium equation, Comm. partial differential equations, 9, 409-437, (1984) · Zbl 0547.35057 [12] Daskalopoulos, P.; del Pino, M.A., On fast diffusion nonlinear heat equations and a related singular elliptic problem, Indiana univ. math. J., 43, 2, 703-728, (1994) · Zbl 0806.35086 [13] Daskalopoulos, P.; del Pino, M.A., On nonlinear parabolic equations of very fast diffusion, Arch. ration. mech. anal., 137, 363-380, (1997) · Zbl 0886.35081 [14] De Gennes, P.G., Wetting: statics and dynamics, Rev. modern phys., 57, 827-863, (1985) [15] Herrero, M.A.; Pierre, M., The Cauchy problem for $$u_t = \operatorname{\Delta} u^m$$ when $$0 < m < 1$$, Trans. amer. math. soc., 291, 145-158, (1985) · Zbl 0583.35052 [16] Ladyzhenskaya, O.A.; Solonikov, V.A.; Ural’ceva, N.N., Linear and quasilinear equations of parabolic type, (1968), Amer. Math. Soc. Providence, RI [17] Vázquez, J.L., Nonexistence of solutions for nonlinear heat equations of fast diffusion type, J. math. pures appl., 71, 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.