zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena. (English) Zbl 1072.35098
The paper deals with the Cauchy--Dirichlet problem for one-dimensional semilinear parabolic equations with a gradient nonlinearity of the form $$ \left\{\eqalign{& u_t=u_{xx}+|u_x|^p,\quad t>0,\ 0<x<1,\cr & u(t,0)=0,\quad u(t,1)=M,\qquad t>0,\cr & u(0,x)=u_0(x),\qquad 0<x<1,\cr }\right. $$ where $p>2,$ $M\geq0$ and $u_0\in X:=\{v\in C^1[0,1]\colon\ v(0)=0,\ v(1)=M\}.$ The authors provide a complete classification of large time behaviour of the classical solutions $u.$ Precisely, either the space derivative $u_x$ blows up in a finite time with $u$ itself remaining bounded, or $u$ is global and converges in $C^1$-norm to the unique steady state. The main difficulty concerns the proof of $C^1$-boundedness of all global solutions, and in avoinding it the authors compute a nontrivial Lyapunov functional by carrying out a method introduced by T. Zelenyak. After deriving precise estimates on the solutions and on the Lyapunov functional, they argue by contradiction by showing that any $C^1$ unbounded global solution should converge to a singular stationary solution, which does not exist.

MSC:
35K60Nonlinear initial value problems for linear parabolic equations
35K65Parabolic equations of degenerate type
35B45A priori estimates for solutions of PDE
WorldCat.org
Full Text: EuDML
References:
[1] N. Alaa, Solutions faibles d’équations paraboliques quasi-linéaires avec données initiales mesures, Ann. Math. Blaise-Pascal 3 (1996) 1-15. Zbl0882.35027 MR1435312 · Zbl 0882.35027 · doi:10.5802/ambp.64 · numdam:AMBP_1996__3_2_1_0 · eudml:79165
[2] N. Alikakos - P. Bates - C. Grant, Blow up for a diffusion-advection equation, Proc. Roy. Soc. Edinburgh A 113 (1989), 181-190. Zbl0707.35018 MR1037724 · Zbl 0707.35018 · doi:10.1017/S0308210500024057
[3] S. Angenent - M. Fila, Interior gradient blow-up in a semilinear parabolic equation, Differential Integral Equations 9 (1996), 865-877. Zbl0864.35052 MR1392084 · Zbl 0864.35052
[4] G. Barles - F. Da Lio, On the generalized Dirichlet for viscous Hamilton-Jacobi equations, J. Math. Pures et Appl., to appear. Zbl1056.35071 · Zbl 1056.35071 · doi:10.1016/S0021-7824(03)00070-9
[5] M. Benachour - S. Dabuleanu, The mixed Cauchy-Dirichlet for a viscous Hamilton-Jacobi equation, Adv. Differential Equations, to appear. Zbl1101.35043 MR2125405 · Zbl 1101.35043
[6] M. Ben-Artzi - Ph. Souplet - F. B. Weissler, The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces, J. Math. Pures et Appl. 81 (2002) 343-378. Zbl1046.35046 MR1967353 · Zbl 1046.35046 · doi:10.1016/S0021-7824(01)01243-0
[7] T. Cazenave - P.-L. Lions, Solutions globales d’equations de la chaleur semilinéaires, Commun. Partial Differential Equations 9 (1984), 955-978. Zbl0555.35067 MR755928 · Zbl 0555.35067 · doi:10.1080/03605308408820353
[8] C.-N. Chen, Infinite time blow-up of solutions to a nonlinear parabolic problem, J. Differential Equations 139 (1997), 409-427. Zbl0887.35079 MR1472354 · Zbl 0887.35079 · doi:10.1006/jdeq.1997.3289
[9] G. Conner - C. Grant, Asymptotics of blowup for a convection-diffusion equation with conservation, Differential Integral Equations 9 (1996), 719-728. Zbl0856.35011 MR1401433 · Zbl 0856.35011
[10] S. Dabuleanu, “Problèmes aux limites pour les équations de Hamilton-Jacobi avec viscosité et données initiales peu regulières”, PhD thesis, Université Nancy 1, 2003.
[11] K. Deng, Stabilization of solutions of a nonlinear parabolic equation with a gradient term, Math. Z., 216 (1994), 147-155. Zbl0798.35077 MR1273470 · Zbl 0798.35077 · doi:10.1007/BF02572313 · eudml:174642
[12] M. Fila - B. Kawohl, Is quenching in infinite time possible ?, Quart. Appl. Math. 8 (1990), 531-534. Zbl0737.35005 MR1074968 · Zbl 0737.35005
[13] M. Fila - G. Lieberman, Derivative blow-up and beyond for quasilinear parabolic equations, Differential Integral Equations 7 (1994), 811-821. Zbl0811.35059 MR1270105 · Zbl 0811.35059
[14] M. Fila - P. Sacks, The transition from decay to blow-up in some reaction-diffusion-convection equations, World Congress of Nonlinear Analysts ’92, Vol. I-IV (Tampa, FL, 1992), 455-463, de Gruyter, Berlin, 1996. Zbl0849.35057 MR1389096 · Zbl 0849.35057
[15] M. Fila - Ph. Souplet - F. B. Weissler, Linear and nonlinear heat equations in $L^q_{\delta }$ spaces and universal bounds for global solutions, Math. Ann. 320 (2001), 87-113. Zbl0993.35023 MR1835063 · Zbl 0993.35023 · doi:10.1007/PL00004471
[16] V. Galaktionov - J.-L. Vázquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997), 1-67. Zbl0874.35057 · Zbl 0874.35057 · doi:10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.0.CO;2-H
[17] Y. Giga, A bound for global solutions of semi-linear heat equations, Comm. Math. Phys. 103 (1986), 415-421. Zbl0595.35057 MR832917 · Zbl 0595.35057 · doi:10.1007/BF01211756
[18] M. Kardar - G. Parisi - Y. C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889-892. Zbl1101.82329 · Zbl 1101.82329 · doi:10.1103/PhysRevLett.56.889
[19] O. Ladyzenskaya - V. A. Solonnikov - N. N. Uralceva, “Linear and Quasilinear Equations of Parabolic Type”, Amer. Math. Soc. Translations, Providence, RI, 1967. Zbl0174.15403 MR241822 · Zbl 0174.15403
[20] G. Lieberman, The first initial-boundary value problem for quasilinear second order parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 13 (1986), 347-387. Zbl0655.35047 MR881097 · Zbl 0655.35047 · numdam:ASNSP_1986_4_13_3_347_0 · eudml:83983
[21] P. L. Lions, “Generalized solutions of Hamilton-Jacobi Equations”, Pitman Research Notes in Math. 62, 1982. MR667669 · Zbl 0497.35001
[22] W.-M. Ni - P. E. Sacks - J. Tavantzis, On the asymptotic behavior of solutions of certain quasi-linear equations of parabolic type, J. Differential Equations 54 (1984), 97-120. Zbl0565.35053 MR756548 · Zbl 0565.35053 · doi:10.1016/0022-0396(84)90145-1
[23] P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem, Acta Math. Univ. Comenianae 68 (1999), 195-203. Zbl0940.35112 MR1757788 · Zbl 0940.35112 · emis:journals/AMUC/_vol-68/_no_2/_quittne/quittner.html · eudml:120402
[24] P. Quittner, Universal bound for global positive solutions of a superlinear parabolic problem, Math. Ann. 320 (2001), 299-305. Zbl0981.35010 MR1839765 · Zbl 0981.35010 · doi:10.1007/PL00004475
[25] P. Quittner, Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, Houston J. Math. 29 (2003), 757-799. Zbl1034.35013 MR1998164 · Zbl 1034.35013
[26] P. Quittner - Ph. Souplet - M. Winkler, Initial blow-up rates and universal bounds for nonlinear heat equations, J. Differential Equations, to appear. Zbl1044.35027 MR2028111 · Zbl 1044.35027 · doi:10.1016/j.jde.2003.10.007
[27] Ph. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Electronic J. Differential Equations 20 (2001), 1-19. Zbl0982.35054 MR1824790 · Zbl 0982.35054 · emis:journals/EJDE/Volumes/2001/20/abstr.html · eudml:121711
[28] Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Differential Integral Equations 15 (2002), 237-256. Zbl1015.35016 MR1870471 · Zbl 1015.35016
[29] Ph. Souplet - F. B. Weissler, Poincaré’s inequality and global solutions of a nonlinear parabolic equation, Ann. Inst. H. Poincaré, Anal. Non linéaire 16 (1999), 337-373. Zbl0924.35065 MR1687278 · Zbl 0924.35065 · doi:10.1016/S0294-1449(99)80017-1 · numdam:AIHPC_1999__16_3_335_0 · eudml:78468
[30] T. I. Zelenyak, Stabilisation of solutions of boundary value problems for a second-order equation with one space variable, Differential Equations 4 (1968), 17-22. Zbl0232.35053 · Zbl 0232.35053