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ISSN 1088-6826(e) ISSN 0002-9939(p)

Extinction and decay estimates for viscous Hamilton-Jacobi equations in ${\mathbb{R}}^N$

Author(s): Said Benachour; Philippe Laurençot; Didier Schmitt
Journal: Proc. Amer. Math. Soc. 130 (2002), 1103-1111.
MSC (1991): Primary 35B40, 35B05, 35K55
Posted: October 1, 2001
MathSciNet review: 1873785
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Abstract: We consider non-negative and integrable classical solutions to the Cauchy problem $u_t-\Delta u+\vert\nabla u\vert^p=0$ when $p\in (0,+\infty)$ . For $p\in (0,N/(N+1))$ we prove that any such solution vanishes identically after a finite time. For higher values of temporal decay estimates are obtained.

References:

1.: M. Ben-Artzi and H. Koch, Decay of mass for a semilinear parabolic equation, Comm. Partial Differential Equations 24 (1999), 869-881. MR 2000a:35098
2.: S. Benachour and Ph. Laurençot, Global solutions to viscous Hamilton-Jacobi equations with irregular initial data, Comm. Partial Differential Equations 24 (1999), 1999-2021. MR 2000g:35091
3.: S. Benachour and Ph. Laurençot, Very singular solutions to a nonlinear parabolic equation with absorption. I. Existence, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 27-44. CMP 2001:09
4.: S. Benachour, Ph. Laurençot and D. Schmitt, in preparation.
5.: S. Benachour, B. Roynette and P. Vallois, Asymptotic estimates of solutions of $u_t - \Delta u = -\vert\nabla u\vert$ in $\mathbb{R}_+\times\mathbb{R}^d$ , $d\ge 2$ , J. Funct. Anal. 144 (1997), 301-324. MR 97m:35118
6.: H. Brézis, Analyse fonctionnelle. Théorie et applications, Masson, Paris, 1983. MR 85a:46001
7.: L.C. Evans, Regularity properties for the heat equation subject to nonlinear boundary constraints, Nonlinear Anal. 1 (1977), 593-602. MR 58:29276
8.: L.C. Evans and B.F. Knerr, Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities, Illinois J. Math. 23 (1979), 153-166. MR 80d:35082
9.: V.A. Galaktionov and J.L. Vazquez, Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach, J. Funct. Anal. 100 (1991), 435-462. MR 92k:35128
10.: V.A. Galaktionov and J.L. Vazquez, Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997), 1-67. MR 97h:35085
11.: B.H. Gilding, M. Guedda and R. Kersner, The Cauchy problem for $u_t = \Delta u + \vert\nabla u\vert^q$ , prépublication LAMFA 28, Université de Picardie, 1998.
12.: M.A. Herrero and J.J.L. Velázquez, Approaching an extinction point in one-dimensional semilinear heat equations with strong absorption, J. Math. Anal. Appl. 170 (1992), 353-381. MR 93k:35043
13.: A.S. Kalashnikov, The propagations of disturbances in problems of non-linear heat conduction with absorption, Comput. Math. Math. Phys. 14 (1974), 70-85.
14.: O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural'ceva, Linear and quasilinear equations of parabolic type, Transl. Math. Monogr. 23, Amer. Math. Soc., Providence, 1968. MR 39:3159b

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Additional Information:

Said Benachour
Affiliation: Institut Elie Cartan - Nancy, Université de Nancy I, BP 239, F-54506 Vandouvre les Nancy cedex, France
Email: benachou@iecn.u-nancy.fr

Philippe Laurençot
Affiliation: Institut Elie Cartan - Nancy, Université de Nancy I, BP 239, F-54506 Vandouvre les Nancy cedex, France
Address at time of publication: Mathématiques pour l'Industrie et la Physique, UNR CNRS 5640, Université Paul Sabatier-Toulouse 3, 118, route de Narbonne, F-31062 Toulouse Cedex 4, France
Email: laurenco@iecn.u-nancy.fr, laurencot@mip.ups-tlse.fr

Didier Schmitt
Affiliation: Institut Elie Cartan - Nancy, Université de Nancy I, BP 239, F-54506 Vandouvre les Nancy cedex, France
Email: dschmitt@iecn.u-nancy.fr

DOI: 10.1090/S0002-9939-01-06140-8
PII: S 0002-9939(01)06140-8
Keywords: Extinction in finite time, temporal decay estimates, viscous Hamilton-Jacobi equations
Received by editor(s): March 23, 2000
Received by editor(s) in revised form: October 12, 2000
Posted: October 1, 2001
Communicated by: David S. Tartakoff
Copyright of article: Copyright 2001, American Mathematical Society

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