Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 1, pp. 1-15.

We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions u: either the space derivative u x blows up in 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 is to prove C 1 boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out the method of Zelenyak. After deriving precise estimates on the solutions and on the Lyapunov functional, we proceed by contradiction by showing that any C 1 unbounded global solution should converge to a singular stationary solution, which does not exist. As a consequence of our results, we exhibit the following interesting situation: - the trajectories starting from some bounded set of initial data in C 1 describe an unbounded set, although each of them is individually bounded and converges to the same limit; - the existence time T * is not a continuous function of the initial data.

Classification: 35K60, 35K65, 35B45
Arrieta, José M. 1; Rodriguez-Bernal, Anibal 1; Souplet, Philippe 2

1 Departamento de Matemática Aplicada Universidad Complutense 28040 Madrid, Spain
2 Département de Mathématiques INSSET Université de Picardie 02109 St-Quentin, France and Laboratoire de Mathématiques Appliquées UMR CNRS 7641 Université de Versailles 45 avenue des Etats-Unis 78035 Versailles, France
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Arrieta, José M.; Rodriguez-Bernal, Anibal; Souplet, Philippe. Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 1, pp. 1-15. http://archive.numdam.org/item/ASNSP_2004_5_3_1_1_0/

[1] N. Alaa, Solutions faibles d'équations paraboliques quasi-linéaires avec données initiales mesures, Ann. Math. Blaise-Pascal 3 (1996) 1-15. | Numdam | MR | Zbl

[2] N. Alikakos - P. Bates - C. Grant, Blow up for a diffusion-advection equation, Proc. Roy. Soc. Edinburgh A 113 (1989), 181-190. | MR | Zbl

[3] S. Angenent - M. Fila, Interior gradient blow-up in a semilinear parabolic equation, Differential Integral Equations 9 (1996), 865-877. | MR | Zbl

[4] G. Barles - F. Da Lio, On the generalized Dirichlet for viscous Hamilton-Jacobi equations, J. Math. Pures et Appl., to appear. | Zbl

[5] M. Benachour - S. Dabuleanu, The mixed Cauchy-Dirichlet for a viscous Hamilton-Jacobi equation, Adv. Differential Equations, to appear. | MR | Zbl

[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. | MR | Zbl

[7] T. Cazenave - P.-L. Lions, Solutions globales d'equations de la chaleur semilinéaires, Commun. Partial Differential Equations 9 (1984), 955-978. | MR | Zbl

[8] C.-N. Chen, Infinite time blow-up of solutions to a nonlinear parabolic problem, J. Differential Equations 139 (1997), 409-427. | MR | Zbl

[9] G. Conner - C. Grant, Asymptotics of blowup for a convection-diffusion equation with conservation, Differential Integral Equations 9 (1996), 719-728. | MR | Zbl

[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. | MR | Zbl

[12] M. Fila - B. Kawohl, Is quenching in infinite time possible ?, Quart. Appl. Math. 8 (1990), 531-534. | MR | Zbl

[13] M. Fila - G. Lieberman, Derivative blow-up and beyond for quasilinear parabolic equations, Differential Integral Equations 7 (1994), 811-821. | MR | Zbl

[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. | MR | Zbl

[15] M. Fila - Ph. Souplet - F. B. Weissler, Linear and nonlinear heat equations in L δ q spaces and universal bounds for global solutions, Math. Ann. 320 (2001), 87-113. | MR | Zbl

[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. | Zbl

[17] Y. Giga, A bound for global solutions of semi-linear heat equations, Comm. Math. Phys. 103 (1986), 415-421. | MR | Zbl

[18] M. Kardar - G. Parisi - Y. C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889-892. | Zbl

[19] O. Ladyzenskaya - V. A. Solonnikov - N. N. Uralceva, “Linear and Quasilinear Equations of Parabolic Type”, Amer. Math. Soc. Translations, Providence, RI, 1967. | MR | Zbl

[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. | Numdam | MR | Zbl

[21] P. L. Lions, “Generalized solutions of Hamilton-Jacobi Equations”, Pitman Research Notes in Math. 62, 1982. | MR

[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. | MR | Zbl

[23] P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem, Acta Math. Univ. Comenianae 68 (1999), 195-203. | MR | Zbl

[24] P. Quittner, Universal bound for global positive solutions of a superlinear parabolic problem, Math. Ann. 320 (2001), 299-305. | MR | Zbl

[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. | MR | Zbl

[26] P. Quittner - Ph. Souplet - M. Winkler, Initial blow-up rates and universal bounds for nonlinear heat equations, J. Differential Equations, to appear. | MR | Zbl

[27] Ph. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Electronic J. Differential Equations 20 (2001), 1-19. | MR | Zbl

[28] Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Differential Integral Equations 15 (2002), 237-256. | MR | Zbl

[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. | Numdam | MR | Zbl

[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. | Zbl