Publication: Asymptotic behaviour of the solutions of a strongly nonlinear parabolic problem
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1981
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Université Toulouse III
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The authors consider the problem ut−div(|∇u|p−2∇u)=0 in (0,∞)×RN, u(x,0)=u0(x). They show that if N≥2 and 1<p<2N/(N+1) the solution has a finite extinction time for each u0∈Lm, m=N(2/p−1), and if N=1, p>1 or N≥2, p≥2N/(N+1) then conservation of total mass holds, i.e., ∫u(t,x)dx=∫u0(x)dx. Moreover the regularizing and decay estimate for ∥u(t)∥m (1<m≤∞) is proved for u0∈Lm0 with m0≥1, which is the extension of the corresponding result for bounded domains by L. Véron [same journal (5) 1 (1979), no. 2, 171–200] to the case of whole space. Finally the finite extinction time problem is discussed for the problem in a bounded domain, extending the result by A. Bamberger [J. Funct. Anal. 24 (1977), no. 2, 148–155].
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A. Bamberger. «Etude d'une équation doublement non linéaire». Journal of Functional Analysis 24 (1977), p. 148-155.
P. Benilan - M.G. Crandall. «The continuous dependence on φ of solutions of ut - Δφ(u) = 0».MRC report MC 578-01245.
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J.I. Diaz - M.A. Herrero. «Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems». Proc. Royal Soc. Edinburgh, to appear.
L.C. Evans. «Application of nonlinear semigroup theory to certain partial differential equations». in Nonlinear evolution equations, M.G. Grandall ed. (1979).
A. Friedman. «Partial Differential Equations». Holt, Rinehart and Winston (1973).
M.H. Protter - H.F. Weinberger. «Maximum principles in differential equations». Prentice Hall (1967).
E.S. Sabinina. «A class of nonlinear degenerating parabolic equations». Soviet Math. Dokl 3 (1962), p. 495-498 (in Russian).
J.L. Vazquez. «An a priori interior estimate for the solutions of a nonlinear problem representing weak diffusion». Nonlinear Analysis, 5 (1981), p. 95-103.
J.L. Vazquez - L. Veron. «Removable singularities of some strongly nonlinear elliptic equations». Manuscr. Math., 33 (1980), p. 129-144.
L. Veron. «Effets régularisants de semi-groupes non linéaires dans des espaces de Banach». Annales Fac. Sci. Toulouse, 1 (1979), p. 171-200.