New applications of monotonicity methods to a class of non-monotone parabolic quasilinear sub-homogeneous problems
| dc.contributor.author | Díaz Díaz, Jesús Ildefonso | |
| dc.date.accessioned | 2023-06-17T12:44:05Z | |
| dc.date.available | 2023-06-17T12:44:05Z | |
| dc.date.issued | 2019 | |
| dc.description.abstract | The main goal of this survey is to show how some monotonicity methods related with the subdifferential of suitable convex functions lead to new and unexpected results showing the continuous and monotone dependence of solutions with the respect to the data (and coefficients) of the problem. In this way, this paper offers ‘a common roof’ to several methods and results concerning monotone and non-monotone frameworks.Besides to present here some new results, this paper offers also a peculiar review to some topics which attracted the attention of many specialists in elliptic and parabolic nonlinear partial differential equations in the last years under the important influence of Häım Brezis .To be more precise,the model problem under consideration concerns to positive solutions of a class of doubly nonlinear diffusion parabolic equations with some sub-homogeneous non-monotone forcing terms. | |
| dc.description.department | Depto. de Análisis Matemático y Matemática Aplicada | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | FALSE | |
| dc.description.sponsorship | Ministerio de Economía, Industria y Competitividad (MINECO) | |
| dc.description.sponsorship | Universidad Complutense de Madrid | |
| dc.description.status | unpub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/73215 | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/12863 | |
| dc.language.iso | eng | |
| dc.relation.projectID | MTM2017-85449-P | |
| dc.relation.projectID | UCM (910480) | |
| dc.rights.accessRights | open access | |
| dc.subject.cdu | 517.95 | |
| dc.subject.keyword | Subhomogenous parabolic equations | |
| dc.subject.keyword | Monotone and continuous dependence | |
| dc.subject.keyword | Eigenvalue type problems | |
| dc.subject.ucm | Análisis funcional y teoría de operadores | |
| dc.subject.ucm | Ecuaciones diferenciales | |
| dc.subject.unesco | 1202.07 Ecuaciones en Diferencias | |
| dc.title | New applications of monotonicity methods to a class of non-monotone parabolic quasilinear sub-homogeneous problems | |
| dc.type | journal article | |
| dcterms.references | [1] N. Alikakos and R. Rostamian, Lower bound estimates and separable solutions for homogeneous equations of evolution in Banach space, J. Differential Equations 43 (1982), 323-344. [2] W. Allegretto and Y. X. Huang: A Picone’s identity for the p−Laplacian and applications, Nonlin. Anal. 32 (1998), 819–830. [3] H.W. Alt, An abstract existence theorem for parabolic systems, Commun. Pure Appl. Anal. 11 (2012), 2079–2123. [4] H.W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z. 183 (1983), 311- 341. [5] L. Alvarez and J.I. Díaz, On the retention of the interfaces in some elliptic and parabolic nonlinear problems, Discrete and Continuum Dynamical Systems 25 (2009), 1-17. [6] H. Amann, Existence and multiplicity theorems far semi-linear elliptic boundary value problems, Math. Z. 150 (1976), 281-295. [7] A. Anane, Simplicité et isolation de la premiere valeur propre du p−Laplacien avec poids, Comptes Rendus Acad. Sc. Paris Série I 305 (1987), 725–728. [8] F. Andreu, V. Caselles, J. I. Díaz and J. M. Mazón, Some Qualitative properties for the Total Variation Flow, Journal of Functional Analysis 188 (2002), 516-547. [9] D. Andreucci and A. F. Tedeev, A Fujita Type Result for Degenerate Neumann Problem in Domains with Noncompact Boundary, J. Math. Anal. Appl. 231 (1999), 543–567. [10] S. Antontsev, J. I. Díaz and S. Shmarev, Energy methods for free boundary problems, Birkäuser, Boston, 2002. [11] M. Arias, J. Campos, M. Cuesta and J. P. Gossez, Asymmetric elliptic problems with indefinite weights, Ann. Inst. H. Poincaré 19 (2002), 581–616. [12] M. Artola, Sur une classe de problemes paraboliques quasi-linéaires avec second membre Höldérien, Richerche di Matematica 52 (2003), 115-144. [13] A. Audrito and J.L. Vázquez, The Fisher–KPP problem with doubly nonlinear “fast” diffusion, Nonlinear Analysis, 157 (2017), 212-248. [14] A. Audrito and J.L. Vázquez, The Fisher–KPP problem with doubly nonlinear diffusion, J. Differential equations 263 (2017), 7617-7708. [15] M. Badii, J. I. Díaz and A. Tesei, Existence and attractivity results for a class of degenerate functional parabolic problems, Rend. Sem. Mat. Univ. Padova 78 (1987), 109-124. [16] M. Badra, K. Bal and J. Giacomoni, A singular parabolic equation: existence, stabilization, J. Differential Equations 252 (2012), 5042–5075. [17] C. Bandle, M. A. Pozio, and A. Tesei, The asymptotic behavior of the solutions of degenerate parabolic equations, Trans. Amer. Math. Soc. 303 (1987), 487–501. [18] V. Barbu, Non-linear semi-groups and differential equations in Banach spaces, Nordhoff Inf. Publ. Co., Leyden 1976. [19] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. [20] Y. Belaud, J.I. Díaz, Abstract results on the finite extinction time property: application to a singular parabolic equation, Journal of Convex Analysis 17 (2010), 827-860. [21] M. Belloni and B. Kawohl, A direct uniqueness proof for equations involving the p-Laplace operator, Manuscripta Math. 109 (2002), 229–231. [22] R. Benguria, The von Weizsacker and exchange corrections in the Thomas-Fermi theory, dissertation, Princeton University, 1979. [23] R. Benguria, H. Brezis and E. Lieb, The Thomas-Fermi-von Weizslicker theory of atoms and molecules, Communs Math. Phys. 79 (1981), 167-180. [24] Ph. Bénilan, Equations d’evolution dans un espace de Banach quelconque et applications, Thése, Orsay, 1972. [25] Ph. Bénilan, Semi-Groupes Invariants par un Cone de Fonctions Convexes. In: Analyse Convexe et Ses Applications, J.P. Aubin (ed.), Lecture Notes in Economics and Mathematical Systems 102, Springer, Berlin, 1974, 49-65. [26] Ph. Bénilan, Opérateurs accrétifs et semi-groupes dans les espaces Lp (1≤ p ≤ ∞), In: Functional analysis and numerical analysis: Japan-France Seminar, Tokyo and Kyoto 1976, H. Fujita (ed.), Japan Society for the Promotion of Science, Tokyo, 1978, 15-53. [27] Ph. Bénilan, Sur un probleme d’évolution non monotone dans L 2 (Ω), Publ. Math. Fac. Sc. Besancon, No. 2 (1976). [28] Ph. Bénilan, A strong regularity L p for solutions of the porous media equation, In: Contributions to Nonlinear Partial Differential Equations, C.Bardos, A. Damlamian, J.I. Díaz and J. Hernández (eds.), Pitman, London, 1983, 39-58. [29] Ph. Bénilan and M.G. Crandall, The continuous dependence on φ of solutions of ut − ∆φ(u) = 0, Indiana Univ Math J. 30 (1981), 162–177. [30] Ph. Bénilan and M.G. Crandall, Regularizing effects of homogeneous evolution equations, In: Contributions to analysis and geometry (Baltimore, Md., 1980), Johns Hopkins Univ. Press, Baltimore, Md., 1981, 23–39. [31] Ph. Bénilan, M.G. Crandall and A. Pazy, Nonlinear evolution equations governed by accretive operators, Unpublished book. [32] Ph. Bénilan, M.G. Crandall and P. Sacks, Some L1 existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions, Appl Math Optim 17 (1988), 203-224. [33] Ph. Bénilan and J.I. Díaz, Comparison of solutions of nonlinear evolutions equations with different nonlinear terms, Israel Journal of Mathematics 42 (1982), 241-257. [34] Ph. Bénilan and J. I. Díaz, Pointwise gradient estimates of solutions of onedimensional nonlinear parabolic problems, J. Evolution Equations 3 (2004), 557-602. [35] Ph.Bénilan and C. Picard, Quelques aspects non lineaires du principe du maximum. In: Séminaire de Théorie du Potentiel Paris, No. 4, F. Hirsch and G. Mokobodzki (eds.), Lecture Notes in Mathematics, 713, Springer, Berlin, 1979, 1-37. [36] Ph. Bénilan and P. Wittbold, Absorptions non linéaires, J. Funct. Anal. 114 (1993), 59–96. [37] Ph. Bénilan and P. Wittbold, Nonlinear evolution equations in Banach spaces: basic results and open problems. In: Functional analysis (Essen, 1991), K. D. Bierstedt et al. (eds.), Dekker, New York, 1994, 1–32. [38] S. Bensid and J.I. Díaz, On the exact number of monotone solutions of a simplified Budyko climate model and their different stability, Discrete and Continuous Dynamical Systems, Series B 24 (2019), 1033-1047. [39] A. Bensoussan, H. Brezis and A. Friedman, Estimates on the free boundary for quasivariational inequalities, Comm. PDEs 2 (1977), 297–321. [40] N. El Berdan, J.I. Díaz and J.M. Rakotoson, The uniform Hopf inequality for discontinuous coefficients and optimal regularity in BMO for singular problems, J. Math. Anal. Appl. 437 (2016), 350–379. [41] H. Berestycki, Le nombre de solutions de certains probltmes semi-lineaires elliptiques, J. Functional Analysis 40 (1981), l-29. [42] F. Bernis, Existence results for doubly nonlinear higher order parabolic equations on unbounded domains, Math. Ann. 279 (1988), 373–394. [43] L. Boccardo and L. Orsina, Sublinear elliptic equations in L s with s small, Houston J. of Math. 20 (1994), 99-114. [44] V. Bögelein , F. Duzaar , P. Marcellini and Ch. Scheven, Doubly Nonlinear Equations of Porous Medium Type, Arch. Rational Mech. Anal. 229 (2018), 503–545. [45] B. Bougherara and J. Giacomoni, Existence of mild solutions for a singular parabolic equation and stabilization, Adv. Nonlinear Anal. 4 (2015), 123–134. [46] L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J. 37 (2014), 769–799. [47] H. Brezis, Problèmes unilatéraux, J. Math. Pures Appl. 51 (1972), 1-168. [48] H. Brezis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, In: Contributions to Nonlinear Funct. Analysis, Madison, 1971, E. Zarantonello (ed.), Acad. Press, NY, 1971, 101-156. [49] H.Brezis, Operateurs maximaux monotones et semigroupes de contraction dans les espaces de Hilbert. North Holland, Amsterdam, 1973. [50] H. Brezis, New results concerning monotone operators and nonlinear semigroups, in Proc. Symp. on Analysis of Nonlinear problems, RIMS, Kyoto, 1974, Surikaisekikenkyusho Kókyuroku No. 258, 1975, 2-27. [51] H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Analyse Math. 68 (1996), 277-304. [52] H. Brezis and T. Cazenave, Nonlinear evolution equations. Unpublished book. Part of it appeared under the title Linear semigroups of contractions; the Hille-Yosida theory and some applications, Publications du Laboratoire D’ Analyse Numerique, no 92004, Université Pierre et Marie Curie, 1993. [53] H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for ut − ∆u = g(u) revisited, Adv. Differential Equations 1 (1996), 73–90. [54] H. Brezis and S. Kamin, Sublinear elliptic equations in R N , Manuscripta Math. 74 (1992), 87-106. [55] H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Analysis 10 (1986), 55-64. [56] H. Brezis and G. Stampacchia, Sur la régularité de la solution d’inéquations elliptiques, Bull. Soc. Math. Fr. 96 (1968), 153-180. [57] H. Brezis and W. Strauss, Semilinear second order elliptic equations in L 1 , J. Math. Soc. Japan 25 (1974), 831-844. [58] J. Carmona, T. Leonori, S. López-Martínez and P.J. Martínez-Aparicio, Quasilinear elliptic problems with singular and homogeneous lower order terms, Nonlinear Anal. 179 (2019), 105–130. [59] T. Cazenave, T. Dickstein and M. Escobedo, A semilinear heat equation with concave-convex nonlinearity, Rendiconti di Matematica, Serie VII, 19 (1999), 211-242. [60] K. Chaib, Extension of Díaz-Saá’s inequality in R N and application to a system of p-Laplacian, Publ. Mat. 46 (2002), 473–488. [61] F. Charro and I. Peral, Zero order perturbations to fully nonlinear equations: comparison, existence and uniqueness, Communications in Contemporary Mathematics 11 (2009), 131–164. [62] M. Cuesta and P. Takac, A strong comparison principle for positive solutions of degenerate elliptic equations, Differential Integral Equations 13 (2000), 721–746. [63] A. N. Dao and J.I. Díaz, 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. Differential Equations 263 (2017), 6764–6804. [64] C. T. Dat and I. E. Verbitsky, Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms, Calc. Var. (2015), 529-546. [65] M. Del Pino, J. Dolbeault and I. Gentil, Nonlinear diffusions, hypercontractivity and the optimal Lp - Euclidean logarithmic Sobolev inequality, J. Math. Anal. Appl. 293 (2004), 375–388. [66] G. Díaz and J. I. Díaz, Uniqueness of the boundary behavior for large solutions to a degenerate elliptic equation involving the ∞−Laplacian, Rev. R. Acad. Cien. Serie A Matem. 97 (2003), 425–430. [67] J.I. Díaz, Nonlinear partial differential equations and free boundaries, Research Notes in Mathematics 106, Pitman, Boston, MA, 1985. [68] J. I. Díaz, Qualitative Study of Nonlinear Parabolic Equations: an Introduction, Extracta Mathematicae 16 (2001), 303-341. [69] J. I. Díaz, On the Häım Brezis Pioneering Contributions on the Location of Free Boundaries, In: Proceedings of the Fifth European Conference on Elliptic and Parabolic Problems; A special tribute to the work of Häım Brezis, M. Chipot et al. (eds.), Birkhauser Verlag, Bassel, 2005, 217–234. [70] J. I. Díaz, On the ambiguous treatment of the Schrödinger equation for the infinite potential well and an alternative via flat solutions: the one-dimensional case, Interfaces and Free Boundaries 17 (2015), 333–351. [71] J.I. Díaz, On the ambiguous treatment of the Schrödinger equation for the infinite potential well and an alternative via singular potentials: the multi-dimensional case, SeMA-Journal 74 (2017), 225-278. [72] J.I. Díaz, Correction to: On the ambiguous treatment of the Schrödinger equation for the infinite potential well and an alternative via singular potentials: the multi-dimensional case, SeMA-Journal 75 (2018), 563– 568. [73] J.I. Díaz and J. Giacomoni, Uniquenees and monotone continuous dependence of solutions for a class of singular parabolic problems, To appear. [74] J. I. Díaz, D. Gómez-Castro, and J. M. Rakotoson, Existence and uniqueness of solutions of Schrödinger type stationary equations with very singular potentials without prescribing boundary conditions and some applications, Differential Equations and Applications 10 (2018), 47–74. [75] J. I. Díaz, D. Gómez-Castro, J. M. Rakotoson and R. Temam, Linear diffusion with singular absorption potential and/or unbounded convective flow: the weighted space approach, Discrete and Continuous Dynamical Systems 38 (2018), 509–546. [76] J. I. Díaz, D. Gómez-Castro and J. L. Vázquez, The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach, Nonlinear Analysis 177 (2018), 325-360. [77] J. I. Díaz, J. Hernández and Y. lI’yasov, On the existence of positive solutions and solutions with compact support for a spectral nonlinear elliptic problem with strong absorption, Nonlinear Analysis Series A: Theory, Mehods and Applications 119 (2015), 484-500. [78] J. I. Díaz, J. Hernández and Y. Il’yasov, Stability criteria on flat and compactly supported ground states of some non-Lipschitz autonomous semilinear equations, Chinese Ann. Math. 38 (2017), 345-378. [79] J.I. Díaz, J. Hernández and F. J. Mancebo, Nodal solutions bifurcating from infinity for some singular p-Laplace equations: flat and compact support solutions, Minimax Theory and its Applications 2 (2017), 27-40. [80] J. I. Díaz, J. Hernández and J. M. Rakotoson, On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spatial dependence terms, Milan J. Maths. 79 (2011), 233-245. [81] J. I. Díaz and M. A. Herrero, Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems, Proc. Royal Soc. Ed. 89A (1981), 249-258. [82] J.I. Díaz and S. Kamin, Convergence to travelling waves for quasilinear Fisher-KPP type equations, J. Math. Anal. Appl. 390 (2012), 74–85. [83] J. I. Díaz and O. A. Oleinik, Nonlinear elliptic boundary-value problems in unbounded domains and the asymptotic behaviour of its solutions, Comptes Rendus Acad. Sci. Paris, Série I 315 (1992), 787-792. [84] J.I. Díaz, J.M. Rakotoson, On very weak solutions of semilinear elliptic equations with right hand side data integrable with respect to the distance to the boundary, Discrete and Continuum Dynamical Systems 27 (2010), 1037-1058. [85] J. I. Díaz and J. E. Sáa, Uniqueness of nonnegative solutions for elliptic nonlinear diffusion equations with a general perturbation term. In: Actas del VIII Congreso de Ecuaciones Diferenciales y Aplicaciones (C. E. D. Y. A.), Santander September 1985. Unpublished book. [86] J. I. Díaz and J. E. Sáa, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, Comptes Rendus Acad. Sc. Paris, Série I, 305 (1987), 521–524. [87] J. I. Díaz and L. Tello, On a nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology, Collectanea Mathematica 50 (1999), 19-51. [88] J. I. Díaz and F. de Thelin, On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM J. Math. Anal. 25 (1995), 1085-1111. [89] J. I. Díaz and I.I. Vrabie, Existence for reaction-diffusion systems. A compactness method approach. Journal of Mathematical Analysis and Applications 188 (1994), 521-540. [90] F. Dickstein, On semilinear parabolic problems with non-Lipschitz nonlinearities, Mat. Contemp., 18 (2000), 111-121. [91] O. Doslý, The Picone identity for a class of partial differential equations, Mathematica Bohemica, 127 (2002), 581–589. [92] P. Dr´abek and J. Hernández, Quasilinear eigenvalue problems with singular weights for the p-Laplacian, Annali di Matematica Pura ed Applicata 198 (2019), 1069–1086. [93] L. C. Evans, Application of nonlinear semigroup theory to certain partial differential equations, In: Nonlinear Evolution Equations, M. G. Crandall (ed.), Academic Press, New York, 1979, 163-188. [94] E. Feireisl, A note on uniqueness for parabolic problems with discontinuous nonlinearities, Nonlinear Anal. 16 (1991), 1053-1056. [95] M. Fila, Ph. Souplet and F.B. Weissler, Linear and nonlinear heat equations in L p δ spaces and universal bounds for global solutions, Math. Ann. 320 (2001), 87–113. [96] J. Fleckinger, J. Hernández and F. de Thélin, On maximum principles and existence of positive solutions for some cooperative elliptic systems, Differential Integral Equations 8 (1995), 69–85. [97] J. Fleckinger, J. Hernández, P. Takac and F. de Thélin, Uniqueness and positivity for solutions of equations with the p-Laplacian, In: Reaction diffusion systems (Trieste, 1995), Lecture Notes in Pure and Appl. Math. 194, Dekker, New York, 1998, 141–155. [98] H. Fujita and S. Watanabe, On the uniqueness and non-uniqueness of solutions of initial value problems for some quasi-linear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 631-652. [99] M. Ghergu and V. Radulescu, The influence of the distance function in some singular elliptic problems, Potential theory and stochastics in Albac, Theta Ser. Adv. Math., 11, Theta, Bucharest, 2009, 125–137. [100] M. Ghergu and V. Radulescu, Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and Its Applications, 37, Oxford University Press, New York, 2008. [101] M. Ghergu and V. Radulescu, Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics, Springer, Berlin, 2012. [102] J. Giacomoni, P. Sauvy and S. Shmarev, Complete quenching for a quasilinear parabolic equation, J. Math. Anal. Appl. 410 (2014), 607–624. [103] B. H. Gilding and R. Kersner, A necessary and sufficient condition for finite speed of propagation in the theory of doubly nonlinear degenerate parabolic equations, Proceedings of the Royal Society of Edinburgh: Section A Mathematics 126 (1996), 739-767. [104] P. Girg and P. Takác, Bifurcations of positive and negative continua in quasilinear elliptic eigenvalue problems, Ann. Inst. H. Poincar´e 9 (2008), 275–327. [105] J.-S. Guo, Large time behaviour of solutions of a fast diffusion equation with source, Nonlinear Anoiysis, Theory, Methods & AppIications 23 (1994), 1559-1568. [106] J. Hernández and F. J. Mancebo, Singular elliptic and parabolic equations, In: Handbook of Differential Equations: Stationary Partial Differential Equations, M. Chipot and P. Quittner (eds.) 3, Elsevier BV, 2006, 317-400. [107] K. M. Hui, Nonnegative solutions of the fast diffusion equation with strong reaction term, Nonlinear Analysis, Theory, Methods & Applicationss, 19 (1992), 1155-1178. [108] A.S. Kalashnikov, Some problems of qualitative theory of nonlinear degenerate second-order parabolic equations, Uspekhi Mat. Nauk. 42 (1987), 135-176. [109] H. Keller and D. Cohen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech. 16 (1967), 1361-1367. [110] H. Keller and J. Keener, Positive solutions of convex nonlinear eigenvalue problems, J. Diff. Eq. 16 (1967), 103-125. [111] M. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, London, 1964. [112] H. Levine, The role of critical exponents in blow-up theorems, SIAM Rev. 32 (1990), 262–288. [113] H. Levine and P. Sacks, Some existence and nonexistence theorems for solutions of degenerate parabolic equations, J. Differential Equations 52 (1984), 135–161. [114] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. [115] M. Marcus and I. Shafrir, An eigenvalue problem related to Hardy’s Lp inequality, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, 29 (2000), 581-604. [116] M. Marcus and L. Véron, Nonlinear Second Order Elliptic Equations Involving Measures, De Gruyter Series in Nonlinear Analysis and Applications, Berlin, 2013. [117] T. Namba, Density-dependent dispersal and spatial distribution of a population, J.Theor. Biol. 86 (1980), 351-363. [118] L. Orsina and A. C. Ponce, Hopf potentials for Schroedinger operators, Analysis & PDE, 11 (2018), 2015– 2047. [119] A de Pablo and J.L. Vázquez, The balance between strong reaction and slow diffusion, Communs Partial differential Equations 15 (1990), 159-183. [120] M. C. Palmeri, Existence and regularity results for some sublinear parabolic equations, Communications in Applied Analysis, 6 (2002), 297-316. [121] M. Picone, Sui valori eccezionalle di un parametro da cui dipende une equatione differenzialle lineare del secondo ordine, Ann. Scuola Norm. Sup. Pisa 11 (1910), 1–141. [122] Y. Pinchover, A. Tertikas and K. Tintarev, A Liouville-type theorem for the p-Laplacian with potential term, Ann. I. H. Poincaré – AN 25 (2008), 357–368. [123] A. Ponce, Elliptic PDEs, Measures and Capacities, Europeam Mathematical Society, Zurich, 2016. [124] P. Pucci and J. Serrin, The strong maximum principle revisited, J. Differential Equations 196 (2004), 1–66. [125] P. Quittner and Ph. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, 2nd ed. Birkhäuser, Basel 2019. [126] J.M. Rakotoson, Regularity of a very weak solution for parabolic equations and applications, Advances in Differential Equations 16 (2011), 867–894. [127] J. E. Saá, Nonlinear Diffusion Elliptic and Parabolic Problems, Ph.D. thesis, Universidad Complutense de Madrid, 1988. [128] J. E. Saá, Large Time Behaviour of the Doubly Nonlinear Porous Medium Equation, Journal of Mathematical Analysis and Applications 155 (1991), 345-363. [129] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov, Blow-up in quasilinear parabolic equations, Nauka, Moscow, 1987; English translation: Walter de Gruyter, Berlin, 1995. [130] M. Schatzman, Stationary solutions and asymptotic behaviour of a quasilinear degenerate parabolic equation, Indiana Univ. Math. J. 33 (1984), 1-30. [131] P. Takac, Nonlinear spectral problems for degenerate elliptic operators, In: Handbook of Differential Equations, M. Chipot M. and P. Quittner (eds.), Elsevier, Amsterdam, 2004, 385–489. [132] P. Takac, L. Tello and M. Ulm, Variational problems with a p-homogeneous energy, Positivity 6 (2001), 75–94. [133] P. Takác and J. Giacomoni, A p(x)-Laplacian extension of the Díaz-Saá inequality and some applications, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1-28. doi:10.1017/prm.2018.91 [134] M. Tsutsumi, On solution of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. Appl., 132 (1988), 187-212. [135] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191–202. [136] J. L. Vázquez, The porous media equation. Mathematical theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. [137] J. L. Vázquez, The mathematical theories of diffusion: nonlinear and fractional diffusion, In: Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions, Lecture Notes in Math., vol. 2186, Springer, Cham, 2017, 205–278. [138] L. Véron, Coercivité et propriétés régularisantes des semi-groupes non-linéaires dans les espaces de Banach, Publ. Math. Fac. Se. Besan¸con 3 (1977). [139] L. Véron, Elliptic equations involving measures, In: Stationary Partial Differential Equations. Vol. I. Handb. Differ. Equ., North-Holland, Amsterdam, 2004, 593–712. [140] L. Veron, Local and Global Aspects of Quasilinear Degenerate Elliptic Equations: Quasilinear Elliptic Singular Problems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. [141] I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, Second Edition, Pitman Monographs and Surveys in Pure and Applied Mathematics 75, Addison-Wesley and Longman 1995. | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | 34ef57af-1f9d-4cf3-85a8-6a4171b23557 | |
| relation.isAuthorOfPublication.latestForDiscovery | 34ef57af-1f9d-4cf3-85a8-6a4171b23557 |
Download
Original bundle
1 - 1 of 1
