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ISSN 1088-6850(e) ISSN 0002-9947(p)

From fast to very fast diffusion in the nonlinear heat equation

Author(s): Noureddine Igbida
Journal: Trans. Amer. Math. Soc. 361 (2009), 5089-5109.
MSC (2000): Primary 35K60, 35K65, 35B40
Posted: May 6, 2009
MathSciNet review: 2515804
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Abstract: We study the asymptotic behavior of the sign-changing solution of the equation $\displaystyle u_t=\nabla\cdot(\vert u\vert^{{-\alpha}} \nabla u)+f ,$ when the diffusion becomes very fast, i.e. as $\displaystyle \alpha\uparrow 1.$ We prove that a solution $u_\alpha(t)$ converges in $\displaystyle L^1(\Omega ),$ uniformly for in subsets with compact support in to a solution of $\displaystyle u_t=\nabla\cdot(\vert u\vert^{-1} \nabla u)+f .$ In contrast with the case of $\alpha<1,$ we prove that the singularity 0 created in the limiting problem, i.e. $\alpha=1,$ is an obstruction to the existence of sign-changing solutions. More precisely, we prove that, for each $t\geq 0,$ the limiting solutions are either positive or negative or identically equal to 0 in all $\Omega.$ This causes the limit to be singular, in the sense that a boundary layer appears at when one lets $\alpha\uparrow 1.$

References:

1.: F. ANDREU, N. IGBIDA, J. MAZON and J. TOLEDO.
Existence and Uniqueness Results for Quasi-linear Elliptic Equations with Nonlinear Boundary Conditions. Ann. Inst. Henri Poincaré Anal. Non Linéaire 24: 61-89, 2007. MR 2286559 (2007j:35056)
2.: D.G. ARONSON.
The porous medium equation, CIME Lectures, In some problems in nonlinear diffusion.
In K. Kirchgassner H. Amann, N. Bazely, editors, Lecture Notes in Mathematics 1224, Springer-Verlag, New York, 1986. MR 877986 (88a:35130)
3.: Ph. B´ENILAN, L. BOCCARDO, and M. HERRERO.
On the limit of solution of $u_t={\Delta} u^m$ as $m\rightarrow\infty$ .
In M. Bertch et al., editor, Proceedings Int. Conf., Torino, 1989. Some Topics in Nonlinaer PDE's.
4.: Ph. B´ENILAN, M.G. CRANDALL, and A. PAZY.
Evolution Equation Governed by Accretive Operators
(book to appear).
5.: Ph. B´ENILAN, M.G. CRANDALL, and P. SACKS.
Some ${L}^1$ existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions.
Appl. Math. Optim., 17:203-224, 1988. MR 922980 (89d:35055)
6.: Ph. B´ENILAN and N. IGBIDA.
Singular limit for perturbed nonlinear semigroup.
Comm. Applied Nonlinear Anal., 3(4):23-42, 1996. MR 1420283 (97k:34088)
7.: Ph. B´ENILAN and N. IGBIDA.
The Mesa problem for Neumann boundary value problem.
accepted in Jour. Func. Anal.
8.: J. G. BERRYMAN and C. J. HOLLAND.
Asymptotic behavior of the nonlinear diffusion equation $n_t =(n^{-1}n_x)_x$ .
J. Math. Phys., 23, 983-987, 1982. MR 659997 (83m:35070)
9.: H. BRÉZIS and A. PAZY.
Convergence and Approximation of Semigroups of Nonlinear Operators in Banach Spaces.
J. Funct. Anal., 9:63-74, 1972. MR 0293452 (45:2529)
10.: H. BRÉZIS.
Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations.
In E. Zarantonello, editor, Contribution to Nonlinear Functional Analysis. Academic Press, 1971, pp. 101-156. MR 0394323 (52:15126)
11.: L. A. CAFFARELLI and A. FRIEDMAN.
Continuity of the density of a gas flow in a porous medium.
Trans. Amer. Math. Soc., 252:99-113, 1979. MR 0534112 (80i:35090)
12.: J.R. ESTEBAN, A. RODRIGUEZ, and J. L. V´AZQUEZ.
A nonlinear heat equation with singular diffusivity.
Comm. Partial Diff. Eqs., 13:985-1039, 1988. MR 944437 (89h:35167)
13.: J.R. ESTEBAN, A. RODRIGUEZ, and J. L. V´AZQUEZ.
The maximal solution of the logarithmic fast diffusion equation in two space dimensions.
Adv. Differential Equations, 2(6):867-894, 1997. MR 1606339 (98k:35094)
14.: L. C. EVANS.
Appplication of nonlinear semigroup theory to certain partial differential equations.
In M.G. CRANDALL, editor, Nonlinear Evolution Equations, Academic Press, New York, 1978. MR 513818 (81b:47078)
15.: R.S. HAMILTON.
The Ricci flow on surfaces.
Contemporary Math., 71:237-262, Amer. Math. Soc., Providence, RI, 1988. MR 954419 (89i:53029)
16.: K.M. HUI.
Existence of solutions of the equation $\displaystyle u_t=\Delta \log u$ .
Nonlinear Anal. TMA, 37:875-914, 1999. MR 1695083 (2000c:35131)
17.: K.M. HUI.
Singular limit of solutions of the equation $u_t=\Delta\left(\tfrac{u^m}{m}\right)$ as $m\to\infty$ .
Pacific Jour. Math., 187:297-316, 1999. MR 1675029 (99j:35096)
18.: N. IGBIDA.
Limite singulière de problèmes d'évolution non linéaires.
Thèse de doctorat, Université de Franche-Comté, 1997.
19.: N. IGBIDA.
The mesa-limit of the porous medium equation and the Hele-Shaw problem.
Diff. Integral Equations, 15(2):129-146, 2002. MR 1870466 (2002m:35119)
20.: N. IGBIDA.
The Hele-Shaw problem with dynamical boundary conditions.
preprint.
21.: H. P. MCKEAN.
The central limit theorem for Carleman's equation.
Israel J. Math, 21:54-92, 1975. MR 0423553 (54:11529)
22.: No. KENMOCHI.
Neumann problems for a class of nonlinear degenerate parabolic equations.
Diff. Integral Equations, 3(2):253-273, 1990. MR 1025177 (91d:35120)
23.: S. N. KRUZHKOV.
Result Concerning the Nature of the Continuity of Solutions of Parabolic Equations and Some of Their Applications.
Mat. Zametki, 6(1):97-108, 1969.
24.: T.G. KURTZ.
Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics.
Trans. A.M.S., 186:259-272, 1974. MR 0336482 (49:1256)
25.: P.L. LIONS, P.E. SOUGANIDIS, and J.L. V´AZQUEZ.
The relation between the porous medium and the eikonal equations in several space dimensions.
Revista Mat. Iberoamericana, 3(3):275-310, 1987. MR 996819 (90g:35075)
26.: L.A. PELETIER.
The porous medium equation.
In K. Kirchgassner H. Amann, N. Bazely, editors, In application of nonlinear analysis in the physical sciences, Boston, 1981. Pitnam.
27.: Ph. ROSENAU.
Fast and superfast diffusion processes.
Physical Rev. Let, 74(7), 1056-1059, 1995.
28.: A. RODRIGUEZ and J.L. V´AZQUEZ.
Obstructions to existence in fast-diffusion equations.
preprint.
29.: L.F. WU.
A new result for the porous medium equation derived from the Ricci flow.
Bull. Amer. Math. Soc., 28:90-94, 1993. MR 1164949 (93f:58245)
30.: L.F. WU.
The Ricci flow on complete $\mathbb{R}^2$ .
Comm. in Analysis and Geometry, 1:439-472, 1993. MR 1266475 (95d:53043)

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