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Complete quenching phenomenon and instantaneous shrinking of support of solutions of degenerate parabolic equations with nonlinear singular absorption

Part of: Parabolic equations and systems Partial differential equations

Published online by Cambridge University Press:  17 January 2019

Nguyen Anh Dao ,
Jesus Ildefonso Díaz  and
Huynh Van Kha
Nguyen Anh Dao
Affiliation:
Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam ([email protected])
Jesus Ildefonso Díaz
Affiliation:
Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, 28040 Madrid, Spain ([email protected])
Huynh Van Kha
Affiliation:
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam ([email protected])
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Abstract

This paper deals with nonnegative solutions of the one-dimensional degenerate parabolic equations with zero homogeneous Dirichlet boundary condition. To obtain an existence result, we prove a sharp estimate for |ux|. Besides, we investigate the qualitative behaviours of nonnegative solutions such as the quenching phenomenon, and the finite speed of propagation. Our results of the Dirichlet problem are also extended to the associated Cauchy problem on the whole domain ℝ. In addition, we also consider the instantaneous shrinking of compact support of nonnegative solutions.

Keywords

gradient estimatesquenching type of parabolic equationsirregular initial datumfree boundaryinstantaneous shrinking of support

MSC classification

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35K65: Degenerate parabolic equations 35B99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 5 , October 2019 , pp. 1323 - 1346
Copyright
Copyright © Royal Society of Edinburgh 2019 

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References

1Aris, R.. The mathematical theory of diffusion and reaction in permeable catalysts (Oxford University Press, 1975).Google Scholar
2Bandle, C. and Brauner, C.-M., Singular perturbation method in a parabolic problem with free boundary, BAIL IV (Novosibirsk, 1986) Boole Press Conf. Ser., vol. 8, Boole, Dún Laoghaire, 1986, pp. 7–14.Google Scholar
3Belaud, Y. and Díaz, J. I.. Abstract results on the finite extinction time property: application to a singular parabolic equation. J. Convex. Anal. 17 (2010), 827–860.Google Scholar
4Benilan, Ph. and Díaz, J. I.. Pointwise gradient estimates of solutions of one dimensional nonlinear parabolic problems. J. Evol. Equ. 3 (2004), 557–602.Google Scholar
5Boccardo, L. and Gallouet, T.. Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87 (1989), 149–169.Google Scholar
6Boccardo, L. and Murat, F.. Almost everywhere convergence of the gradients of solutions to Elliptic and Parabolic equations. Nonlinear Anal. Theory, Methods and Applications 19 (1992), 581–597.Google Scholar
7Borelli, M. and Ughi, M.. The fast diffusion equation with strong absorption: the instantaneous shrinking phenomenon. Rend. Istit. Mat. Univ. Trieste 26 (1994), 109–140.Google Scholar
8Brezis, H. and Friedman, A.. Estimates on the support of solutions of parabolic variational inequalities. Illinois J. Math. 20 (1976), 82–97.Google Scholar
9Dao, A. N. and Díaz, J. I.. A gradient estimate to a degenerate parabolic equation with a singular absorption term: global and local quenching phenomena. J. Math. Anal. Appl. 437 (2016), 445–473.Google Scholar
10Dao, A. N. and Díaz, J. I., The extinction versus the blow-up: Global and non-global existence of solutions of source types of degenerate parabolic equations with a singular absorption, Submitted.Google Scholar
11Dao, A. N., Díaz, J. I. and Sauvy, P.. Quenching phenomenon of singular parabolic problems with L 1 initial data. Electron. J. Diff. Equa. 2016 (2016), 1–16.Google Scholar
12Dávila, J. and Montenegro, M.. Existence and asymptotic behavior for a singular parabolic equation. Trans. AMS 357 (2004), 1801–1828.Google Scholar
13Díaz, J. I. and Herrero, M. Á.. Propriétés de support compact pour certaines équations elliptiques et paraboliques non linéaires. C.R. Acad. Sc. París,t. 286, Série I, (1978),815–817.Google Scholar
14Díaz, J. I.. Nonlinear partial differential equations and free boundaries, research notes in mathematics, vol. 106 (London: Pitman, 1985).Google Scholar
15Díaz, J. I.. On the free boundary for quenching type parabolic problems via local energy methods. Commun. Pure Appl. Math. 13 (2014), 1799–1814.Google Scholar
16DiBenedetto, E.. Degenerate parabolic equations (New York: Springer-Verlag, 1993).Google Scholar
17Evans, L. C. and Gariepy, R.. Measure theory and fine properties of functions (Boca Raton, Ann Arbor, and London: CRC Press, 1992).Google Scholar
18Evans, L. C. and Knerr, B. F.. Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities. Illinois J. Math. 23 (1979), 153–166.Google Scholar
19Fila, M. and Kawohl, B.. Is quenching in infinite time possible. Q. Appl. Math. 48 (1990), 531–534.Google Scholar
20Giacomoni, J., Sauvy, P. and Shmarev, S.. Complete quenching for a quasilinear parabolic equation. J. Math. Anal. Appl. 410 (2014), 607–624.Google Scholar
21Herrero, M. A.. On the behavior of the solutions of certain nonlinear parabolic problems (Spanish). Rev. Real Acad. Cienc. Exact. Fs. Natur. Madrid 75 (1981), 1165–1183.Google Scholar
22Herrero, M. A. and Vázquez, J. L.. On the propagation properties of a nonlinear degenerate parabolic equation. Comm. PDE 7 (1982), 1381–1402.Google Scholar
23Kawohl, B.. Remarks on quenching. Doc. Math. J. DMV 1 (1996), 199–208.Google Scholar
24Kawohl, B. and Kersner, R.. On degenerate diffusion with very strong absorption. Math. Method Appl. Sci. 15 (1992), 469–477.Google Scholar
25Ladyzenskaja, O. A., Solonnikov, V. A. and Uralceva, N. N.. Linear and quasi-linear equations of parabolic type vol. 23 (Providence: AMS, 1968).Google Scholar
26Levine, H. A., Quenching and beyond: a survey of recent results. GAKUTO Internat. Ser. Math. Sci. Appl. 2 (1993), Nonlinear mathematical problems in industry II, Gakkotosho, Tokyo, 501–512.Google Scholar
27Phillips, D.. Existence of solutions of quenching problems. Appl. Anal. 24 (1987), 253–264.Google Scholar
28Porretta, A.. Existence results for nonlinear parabolic equations via strong convergence of truncations. Ann. Mat. Pura Appl. (IV) 177 (1999), 143–172.Google Scholar
29Simon, J.. Compact sets in the space L p(0, T; B). Ann. Mat. Pura Appl. 196 (1987), 65–96.Google Scholar
30Strieder, W. and Aris, R.. Variational methods applied to problems of diffusion and reaction (Berlin: Springer-Verlag, 1973).Google Scholar
31Winkler, M.. Nonuniqueness in the quenching problem. Math. Ann. 339 (2007), 559–597.Google Scholar
32Nonlinear diffusion equations (Singapore: World Scientific, 2001).Google Scholar
33Zhao, J. N.. Existence and Nonexistence of Solutions for $u_{t}=div(\vert \nabla u \vert ^{p-2}\nabla u)+ f(\nabla u, u, x, t)$. J. Math. Anal. Appl. 172, (1993), 130–146.Google Scholar