Existence and regularity results for degenerate parabolic problems in the presence of strongly increasing regularizing lower-order terms and $L^{m}$-data/Dirac mass

  • Mohammed Abdellaoui LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, Atlas Fez, Morocco
  • Hicham Redwane Faculté dÉconomie et de Gestion, Université Hassan 1, Settat, Morocco
Keywords: Regularizing effect, Degenerate coercivity, Distributional/Entropy solutions, $L^{m}$/Dirac-data, Parabolic problems

Abstract

UDC 517.9

We study the existence and regularity results for degenerate parabolic problems in the presence of strongly increasing regularizing lower-order terms and $L^{m}$-data/Dirac mass.

References

M. Abdellaoui, Regularizing effect for some parabolic problems with perturbed terms and irregular data, J. Elliptic Parabol. Equat. (to appear).

M. Abdellaoui, Stability/nonstability properties of renormalized/entropy solutions for degenerate parabolic equations with $L^{1}$/measure data, SEMA, 77, 457–506 (2020). DOI: https://doi.org/10.1007/s40324-020-00222-1

M. Abdellaoui, E. Azroul, Non-stability result of entropy solutions for nonlinear parabolic problems with singular measures, E. J. Elliptic and Parabol. Equat., 5, № 1, 1–26 (2019). DOI: https://doi.org/10.1007/s41808-019-00036-x

A. Alvino, L. Boccardo, V. Ferone, L. Orsina, G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity, Ann. Mat. Pura ed Appl. (4), 182, № 1, 53–79 (2003). DOI: https://doi.org/10.1007/s10231-002-0056-y

A. Alvino, V. Ferone, G. Trombetti, A priori estimates for a class of nonuniformly elliptic equations, Atti Semin. Mat. Fis. Univ. Modena, 46, 381–391 (1998).

F. Andreu, J. M. Mazon, S. Segura de Leon, J. Toledo, Existence and uniquence for a degenerate parabolic equation with $L^{1}$ data, Trans. Amer. Math. Soc., 351, № 1, 285–306 (1999). DOI: https://doi.org/10.1090/S0002-9947-99-01981-9

Ph. Bénilan, H. Brezis, Nonlinear problems related to the Thomas–Fermi equation, J. Evol. Equat., 3, 673–770 (2004). DOI: https://doi.org/10.1007/978-3-0348-7924-8_35

P. Bénilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre, J. L. Vazquez, An $L^{1}$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 22, 241–273 (1995).

Ph. Bénilan, H. Brezis, M. G. Crandall, A semilinear equation in $L^{1}R^N)$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 2, 523–555 (1975).

H. Berestycki, S. Kamin, G. Sivashinsky, Metastability in a flame front evolution equation, Interfaces Free Bound., 3, № 4, 361–392 (2001). DOI: https://doi.org/10.4171/ifb/45

L. Boccardo, On the regularizing effect of strongly increasing lower-order terms, J. Evol. Equat., 3, № 2, 225–236 (2003). DOI: https://doi.org/10.1007/s00028-003-0089-8

L. Boccardo, Dirichlet problems with singular and gradient quadratic lower-order terms, ESAIM Control, Optim. and Calc. Var., 14, 411–426 (2008). DOI: https://doi.org/10.1051/cocv:2008031

L. Boccardo, Some elliptic problems with degenerate coercivity, Adv. Nonlinear Stud., 6, № 1, 1–12 (2006). DOI: https://doi.org/10.1515/ans-2006-0101

L. Boccardo, Quasilinear elliptic equations with natural growth terms: the regularizing effects of the lower-order terms, Nonlinear and Convex Anal., 7, № 3, 355–365 (2006).

L. Boccardo, H. Brezis, Some remarks on a class of elliptic equations with degenerate coercivity, Boll. Unione Mat. Ital. (8), 6-B, № 3, 521–530 (2003).

L. Boccardo, A. Dall'Aglio, L. Orsina, Existence and regularity results for some elliptic equations with degenerate coercivity, Atti Semin. Mat. Fis. Univ. Modena, 46, suppl., 51–81 (1998).

L. Boccardo, A. Dall'Aglio, T. Gallouët, L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147, 237–258 (1997). DOI: https://doi.org/10.1006/jfan.1996.3040

L. Boccardo, T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87, № 1, 149–169 (1989). DOI: https://doi.org/10.1016/0022-1236(89)90005-0

L. Boccardo, T. Gallouët, L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13, 539–551 (1996). DOI: https://doi.org/10.1016/s0294-1449(16)30113-5

L. Boccardo, T. Gallouët, J. L. Vázquez, Nonlinear elliptic equations in $R^{N}$ without growth restrictions on the data, J. Different. Equat., 105, № 2, 334–363 (1993). DOI: https://doi.org/10.1006/jdeq.1993.1092

L. Boccardo, H. Brezis, Some remarks on a class of elliptic equations with degenerate coercivity, Boll. Unione Mat. Ital., 6, 521–530 (2003).

L. Boccardo, F. Murat, Increasing powers leads to bilateral problems, Composite Media and Homogenization Theory, World Sci. (1995).

L. Boccardo, F. Murat, J. P. Puel, Existence results for some quasilinear parabolic equations, Nonlinear Anal., 13, 373–392 (1989). DOI: https://doi.org/10.1016/0362-546X(89)90045-X

H. Brezis, Nonlinear problems related to the Thomas–Fermi equation, Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Int. Symp., North Holland, Amsterdam (1978), p. 74–80. DOI: https://doi.org/10.1016/S0304-0208(08)70859-4

H. Brezis, Some variational problems of the Thomas–Fermi type, Variational Inequalities and Complementarity Problems, Proc. Int. School, Erice, 1978, Wiley, Chichester (1980), p. 53–73.

H. Brezis, Problèmes elliptiques et paraboliques non linéaires avec données mesures, Goulaouic–Meyer–Schwartz Seminar, 1981/1982, Ecole Polytech., Palaiseau (1982), p. X.1–X.12.

H. Brezis, Nonlinear elliptic equations involving measures, Contributions to Nonlinear Partial Differential Equations, Madrid (1981), Pitman, Boston, MA (1983), p. 82–89.

H. Brezis, F. E. Browder, Some properties of higher order Sobolev spaces, J. Math. Pures et Appl., 61, 245–259 (1982).

H. Brezis, M. Marcus, A. C. Ponce, Nonlinear elliptic equations with measures revisited, Mathematical Aspects of Nonlinear Dispersive Equations (AM-163), Princeton Univ. Press, Princeton (2009), p. 55–110. DOI: https://doi.org/10.1515/9781400827794.55

H. Brezis, W. A. Strauss, Semi-linear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan, 25, 565–590 (1973). DOI: https://doi.org/10.2969/jmsj/02540565

G. R. Cirmi, On the existence of solutions to nonlinear degenerate elliptic equations with measures data, Ric. Mat., 42, № 2, 315–329 (1993).

R. Dautray, J. L. Lions, Mathematical analysis and numerical methods for science and technology, vol. 5, Springer-Verlag (1992).

G. Dal Maso, F. Murat, L. Orsina, A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci., 28, 741–808 (1999).

A. Dall'Aglio, L. Orsina, Existence results for some nonlinear parabolic equations with nonregular data, Different. Integral Equat., 5, № 6, 1335–1354 (1992). DOI: https://doi.org/10.57262/die/1370875550

E. DiBenedetto, Partial differential equations, Birkhäuser, Boston (1995). DOI: https://doi.org/10.1007/978-1-4899-2840-5

J. Droniou, A. Porretta, A. Prignet, Parabolic capacity and soft measures for nonlinear equations, Potential Anal., 19, № 2, 99–161 (2003). DOI: https://doi.org/10.1023/A:1023248531928

J. Droniou, A. Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data, NoDEA, 14, № 1-2, 181–205 (2007). DOI: https://doi.org/10.1007/s00030-007-5018-z

L. Dupaigne, A. C. Ponce, A. Porretta, Elliptic equations with vertical asymptotes in the nonlinear term, J. Anal. Math., 98, № 1, 349–396 (2006). DOI: https://doi.org/10.1007/BF02790280

D. Giachetti, M. M. Porzio, Elliptic equations with degenerate coercivity: gradient regularity, Acta Math. Sin. (Engl. Ser.), 19, № 2, 349–370 (2003). DOI: https://doi.org/10.1007/s10114-002-0235-1

D. Giachetti, M. M. Porzio, Existence results for some nonuniformly elliptic equations with irregular data, J. Math. Anal. and Appl., 257, № 1, 100–130 (2001). DOI: https://doi.org/10.1006/jmaa.2000.7324

G. Croce, The regularizing effects of some lower-order terms on the solutions in an elliptic equation with degenerate coercivity, Rend. Mat. (7), 27, 299–314 (2007).

Z. Guo, J. Wei, Hausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity, Manuscripta Math., 120, 193–209 (2006). DOI: https://doi.org/10.1007/s00229-006-0001-2

J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod et Gauthier-Villars (1969).

O. A. Ladyzenskaya, V. A. Solonnikov, N. N. Ural'tseva, Linear and quasi-linear equations of parabolic type, Translated from the Russian by S. Smith, Amer. Math. Soc. (1968). DOI: https://doi.org/10.1090/mmono/023

J. Leray, J.-L. Lions, Quelques résultats de Visik sur les problèmes elliptiques semi-linéaires par les méthodes de Minty et Browder, Bull. Soc. Math. France, 93, 97–107 (1965). DOI: https://doi.org/10.24033/bsmf.1617

F. Li, Existence of entropy solutions to some parabolic problems with $L^{1}$ data, Acta Math. Sinica (Chin. Ser.), 18, № 1, 119–128 (2002). DOI: https://doi.org/10.1007/s101140100119

F. Li, Existence and regularity results for some parabolic equations with degenerate coercivity, Ann. Acad. Sci. Fenn. Math., 37, 605–633 (2012). DOI: https://doi.org/10.5186/aasfm.2012.3738

Pedro J. Martinez-Aparicio, F. Petitta Francesco, Parabolic equations with nonlinear singularities, Nonlinear Anal., 74, № 1, 114–131 (2011). DOI: https://doi.org/10.1016/j.na.2010.08.023

L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, 13, 116–162 (1959).

F. Petitta, Renormalized solutions of nonlinear parabolic equations with general measure data, Ann. Mat. Pure ed Appl., 563–808 (2008). DOI: https://doi.org/10.1007/s10231-007-0057-y

F. Petitta, A non-existence result for nonlinear parabolic equations with singular measure data, Proc. Roy. Soc. Edinburgh Sect. A, 139, 381–392 (2009). DOI: https://doi.org/10.1017/S0308210507001163

A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura ed Appl. (4), 177, 143–172 (1999). DOI: https://doi.org/10.1007/BF02505907

A. Porretta, Uniqueness and homogenization for a class of noncoercive operators in divergence form, Atti Sem. Mat. Fis. Univ. Modena, 46, suppl., 915–936 (1998).

M. M. Porzio, M. A. Pozio, Parabolic equations with nonlinear, degenerate and space-time dependent operators, J. Evol. Equat., 8, 31–70 (2008). DOI: https://doi.org/10.1007/s00028-007-0317-8

A. Prignet, Existence and uniqueness of entropy solutions of parabolic problems with $L^{1}$ data, Nonlinear Anal., 28, 1943–1954 (1997). DOI: https://doi.org/10.1016/S0362-546X(96)00030-2

S. Segura de León, J. Toledo, Regularity for entropy solutions of parabolic $P$-Laplacian equations, Publ. Mat., 43, 665–683 (1999). DOI: https://doi.org/10.5565/PUBLMAT_43299_08

J. Simon, Compact sets in the space $L^{p}(0,T;B)$, Ann. Mat. Pura ed Appl., 146, 65–96 (1987). DOI: https://doi.org/10.1007/BF01762360

Published
24.10.2023
How to Cite
AbdellaouiM., and RedwaneH. “Existence and Regularity Results for Degenerate Parabolic Problems in the Presence of Strongly Increasing Regularizing Lower-Order Terms and $L^{m}$-data/Dirac Mass”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 10, Oct. 2023, pp. 1317 -46, doi:10.3842/umzh.v75i10.7260.
Section
Research articles