zbMATH — the first resource for mathematics

On maximum and comparison principles for parabolic problems with the \(p\)-Laplacian. (English) Zbl 1416.35056
Summary: We investigate strong and weak versions of maximum and comparison principles for a class of quasilinear parabolic equations with the \(p\)-Laplacian \[ \partial _t u - \Delta _p u = \lambda |u|^{p-2} u + f(x,t) \] under zero boundary and nonnegative initial conditions on a bounded cylindrical domain \(\Omega \times (0, T)\), \(\lambda \in \mathbb{R}\), and \(f \in L^\infty (\Omega \times (0, T))\). Several related counterexamples are given.

35B50 Maximum principles in context of PDEs
35B51 Comparison principles in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K92 Quasilinear parabolic equations with \(p\)-Laplacian
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35K20 Initial-boundary value problems for second-order parabolic equations
Full Text: DOI
[1] Anane, A.: Etude des valeurs propres et de la résonance pour l’opérateur \(p\)-Laplacien, Thése de doctorat (Doctoral dissertation), Université libre de Bruxelles (1988)
[2] Arrieta, JM; Rodríguez-Bernal, A.; Valero, J., Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, Int. J. Bifurcation Chaos, 16, 2965-2984, (2006) · Zbl 1185.37161
[3] Barenblatt, GI, On self-similar motions of a compressible fluid in a porous medium, Prikladnaya Matematika i Mekhanika (Appl. Math. Mech.), 16, 679-699, (1952)
[4] Bensid, S., Díaz, J. I.: Stability results for discontinuous nonlinear elliptic and parabolic problems with a \(S\)-shaped bifurcation branch of stationary solutions. Discrete & Continuous Dynamical Systems-Series B, 22(5) (2017)
[5] Bobkov, VE; Takáč, P., A strong maximum principle for parabolic equations with the \(p\)-Laplacian, J. Math. Anal. Appl., 419, 218-230, (2014) · Zbl 1295.35291
[6] Cuesta, M., Takáč, P.: A strong comparison principle for the Dirichlet \(p\)-Laplacian. In G. Caristi & E. Mitidieri (Eds.), Lecture Notes in Pure and Applied Mathematics, Vol. 194 , (pp. 79-87). Marcel Dekker, Inc., New York, Basel (1998). https://books.google.com/books?id=8owrhdOK-G8C · Zbl 0910.35006
[7] Dao, NA; Díaz, JI, A gradient estimate to a degenerate parabolic equation with a singular absorption term: The global quenching phenomena, J. Math. Anal. Appl., 437, 445-473, (2016) · Zbl 1377.35287
[8] Dao, N. A., Díaz, J.: Existence and uniqueness of singular solutions of \(p\)-Laplacian with absorption for Dirichlet boundary condition. Proceedings of the American Mathematical Society (2017). https://doi.org/10.1090/proc/13647
[9] Dao, NA; Díaz, JI, The extinction versus the blow-up: global and non-global existence of solutions of source types of degenerate parabolic equations with a singular absorption, J. Differ. Equ., 263, 6764-6804, (2017) · Zbl 1433.35182
[10] Dao, A.N., Díaz, J.I., Sauvy, P.: Quenching phenomenon of singular parabolic problems with \(L^1\) initial data. Electronic Journal of Differential Equations, 201(136), 1-16 (2016). https://ejde.math.txstate.edu/Volumes/2016/136/dao.pdf
[11] Deguchi, H., Existence, uniqueness and non-uniqueness of weak solutions of parabolic initial-value problems with discontinuous nonlinearities, Proc R Soc Edinburgh Sect A: Math, 135, 1139-1167, (2005) · Zbl 1130.35076
[12] Derlet, A.; Takáč, P., A quasilinear parabolic model for population evolution, Differ. Equ. Appl., 4, 121-136, (2012) · Zbl 1243.35107
[13] Díaz, JI; Thélin, F., On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM J. Math. Anal., 25, 1085-1111, (1994) · Zbl 0808.35066
[14] Díaz, J.I., Saá, J.E.: Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. Comptes rendus de l’Académie des sciences. Série 1, Mathématique, 305(12), 521-524 (1987) http://gallica.bnf.fr/ark:/12148/bpt6k62167875/f537.item · Zbl 0656.35039
[15] Díaz, JI; Tello, L., A nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology, Collectanea Mathematica, 50, 19-51, (1999) · Zbl 0936.35095
[16] DiBenedetto, E.: Degenerate parabolic equations. Springer, New York (1993). https://doi.org/10.1007/978-1-4612-0895-2 · Zbl 0794.35090
[17] DiBenedetto, E., Gianazza, U.P., Vespri, V.: Harnack’s inequality for degenerate and singular parabolic equations. Springer, New York (2011). https://doi.org/10.1007/978-1-4614-1584-8 · Zbl 1237.35004
[18] Feireisl, E.; Norbury, J., Some existence, uniqueness and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities, Proc. R. Soc. Edinburgh Sect. A: Math., 119, 1-17, (1991) · Zbl 0784.35117
[19] Fleckinger-Pellé, J., Hernández, J., Takáč, P., de Thélin, F.: Uniqueness and positivity for solutions of equations with the \(p\)-Laplacian. In G. Caristi & E. Mitidieri (Eds.), Lecture Notes in Pure and Applied Mathematics, Vol. 194 , (pp. 141-156). Marcel Dekker, Inc., New York-Basel. (1998). https://books.google.com/books?id=8owrhdOK-G8C · Zbl 0912.35064
[20] Gianni, R.; Hulshof, J., The semilinear heat equation with a Heaviside source term, Eur. J. Appl. Math., 3, 367-379, (1992) · Zbl 0789.35088
[21] Il’yasov, Y.; Runst, T., Positive solutions of indefinite equations with \(p\)-Laplacian and supercritical nonlinearity, Complex Var. Elliptic Equ., 56, 945-954, (2011) · Zbl 1229.35052
[22] Kamin, S.; Vázquez, JL, Fundamental solutions and asymptotic behaviour for the \(p\)-Laplacian equation, Revista Matemática Iberoamericana, 4, 339-354, (1988) · Zbl 0699.35158
[23] Kilpeläinen, T.; Lindqvist, P., On the Dirichlet boundary value problem for a degenerate parabolic equation, SIAM J. Math. Anal., 27, 661-683, (1996) · Zbl 0857.35071
[24] Lieberman, GM, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. Theory, Methods Appl., 12, 1203-1219, (1988) · Zbl 0675.35042
[25] Lieberman, GM, Boundary and initial regularity for solutions of degenerate parabolic equations, Nonlinear Anal. Theory, Methods Appl., 20, 551-569, (1993) · Zbl 0782.35036
[26] Moser, J.: A Harnack inequality for parabolic differential equations. Communications on pure and applied mathematics, 17(1), 101-134. (1964). https://doi.org/10.1002/cpa.3160170106 Correction: Communications on pure and applied mathematics, 20 (1967), 231-236. https://doi.org/10.1002/cpa.3160200107 · Zbl 0149.06902
[27] Nazaret, B., Principe du maximum strict pour un opérateur quasi linéaire, Comptes Rendus de l’Académie des Sciences-Series I-Mathematics, 333, 97-102, (2001) · Zbl 1014.35011
[28] Nazarov, A.; Uraltseva, NN, The Harnack inequality and related properties for solutions of elliptic and parabolic equations with divergence-free lower-order coefficients, St. Petersburg Math. J., 23, 93-115, (2012) · Zbl 1235.35054
[29] Padial, J. F., Takáč, P., Tello, L.: An antimaximum principle for a degenerate parabolic problem. Adv. Differ. Equ. 15(7/8), 601-648. (2010). http://projecteuclid.org/euclid.ade/1355854621 · Zbl 1195.35237
[30] Protter, M.H., Weinberger, H.F.: Maximum principles in differential equations. Prentice-Hall, Englewood Cliffs (1967). https://doi.org/10.1007/978-1-4612-5282-5 · Zbl 0153.13602
[31] Pucci, P., Serrin, J.B.: The maximum principle, vol. 73. Springer, New York (2007). https://doi.org/10.1007/978-3-7643-8145-5 · Zbl 1134.35001
[32] Smoller, J.: Shock waves and reaction-diffusion equations, vol. 258. Springer, New York (1983). https://doi.org/10.1007/978-1-4684-0152-3 · Zbl 0508.35002
[33] Takeuchi, S., Stationary profiles of degenerate problems with inhomogeneous saturation values, Nonlinear Anal. Theory, Methods Appl., 63, e1009-e1016, (2005) · Zbl 1224.35153
[34] Vázquez, JL, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12, 191-202, (1984) · Zbl 0561.35003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.