Existence results of renormalized solutions for nonlinear \(p(\cdot)\)-parabolic equations with possibly singular measure data. (English) Zbl 1492.35157
Summary: We study the existence of renormalized solutions to a nonlinear parabolic boundary value problem with a general and possibly singular measure data, whose model is
\[
(\mathcal{P}) \begin{cases} \frac{\partial b(u)}{\partial t} -\Delta_{p(x)}u =\mu \quad &\text{in } Q:=\Omega \times (0,T), \\ b(u)(t=0)=b(u_0)(x) \quad &\text{in } \Omega,\\ u(x,t)=0 \quad &\text{on } \partial \Omega \times (0,T),\\
\end{cases}
\]
where \(\Omega\) is an open bounded subset of \(\mathbb{R}^N\) (\(N\geq 2\)), \(T>0\), \(b\) is an increasing \(C^1\)-function, \( \Delta_{p(x)}u:=\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u\) (\(1<p_-\leq p(x)\leq p_+<N\)) is the \(p(x)\)-Laplacian operator which, roughly speaking, behaves as \(|\nabla u|^{p(x)-1}\), \(\mu\) is a bounded Radon measure with bounded total variation on \(Q\) and \(b(u_0)\) is an integrable function. We provide the assumptions, the notions of solution we are adopting and the statements of the existence result in the “generalized” Sobolev spaces with variable exponent using some “specific” decompositions on the data.
MSC:
| 35K92 | Quasilinear parabolic equations with \(p\)-Laplacian |
| 35D30 | Weak solutions to PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 35R06 | PDEs with measure |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
general measure data; \(p(\cdot)\)-parabolic capacity; renormalized solution; singular equations; Sobolev spaces with variable exponentReferences:
| [1] | Abdellaoui, M.; Azroul, E.; Ouaro, S.; Traoré, U., Nonlinear parabolic capacity and renormalized solutions for PDEs with diffuse measure data and variable exponent, Ann. Univ. Craiova Math. Comput. Sci. Ser., 46, 2, 269-297 (2019) · Zbl 1488.35167 |
| [2] | Aberqi, A.; Bennouna, J.; Redwane, H., Nonlinear parabolic problems with lower order terms and measure data, Thai J. Math., 14, 1, 115-130 (2016) · Zbl 1354.35061 |
| [3] | Aronson, DG, Removable singularities for linear parabolic equations, Arch. Ration. Mech. Anal., 17, 79-84 (1964) · Zbl 0128.09402 · doi:10.1007/BF00283868 |
| [4] | Aronson, DG, Removable singularities for linear parabolic equations, Arch. Ration. Mech. Anal., 17, 79-84 (1964) · Zbl 0128.09402 · doi:10.1007/BF00283868 |
| [5] | Aydin, I., Weighted variable Sobolev spaces and capacity, J. Funct. Spaces Appl., 132690, 17 (2012) · Zbl 1241.46019 |
| [6] | Azroul, E.; Hjiaj, H.; Lahmi, B., Existence of entropy solutions for some strongly nonlinear \(p(x)\)-parabolic problems with \(L^1\)-data, Ann. Univ. Craiova Ser. Math. Inform., 42, 2, 273-299 (2015) · Zbl 1374.35192 |
| [7] | Bendahmane, M.; Wittbold, P.; Zimmermann, A., Renormalized solutions for a nonlinear parabolic equation with variable exponents and \(L^1\)-data, J. Differ. Equ., 249, 483-515 (2010) · Zbl 1231.35106 · doi:10.1016/j.jde.2010.05.011 |
| [8] | Bénilan, P.; Boccardo, L.; Gallouët, T.; Gariepy, R.; Pierre, M.; Vázquez, JL, An \(L^1\)-theory of existence and uniqueness of nonlinear elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 22, 241-273 (1995) · Zbl 0866.35037 |
| [9] | Bennouna, J.; Mekkour, M.; Redwane, H.; Aberqi, A., Existence results for a nonlinear parabolic problems with lower order terms, Int. J. Math. Anal., 7, 27, 1323-1340 (2013) · Zbl 1284.35216 |
| [10] | Blanchard, D.; Murat, F., Renormalized solutions of nonlinear parabolic problems with \(L^1\) data, existence and uniqueness, Proc. R. Soc. Edinb. Sect. A, 127, 1137-1152 (1997) · Zbl 0895.35050 · doi:10.1017/S0308210500026986 |
| [11] | Blanchard, D.; Porretta, A., Nonlinear parabolic equations with natural growth terms and measure initial data, Ann. Sc. Norm. Super. Pisa Cl. Sci., 4, 583-622 (2001) · Zbl 1072.35089 |
| [12] | Blanchard, D.; Porretta, A., Stefan problems with nonlinear diffusion and convection, J. Differ. Equ., 210, 383-428 (2005) · Zbl 1075.35112 · doi:10.1016/j.jde.2004.06.012 |
| [13] | Blanchard, D.; Murat, F.; Redwane, H., Existence et unicité de la solution renormalisée d’un problème parabolique assez général, C. R. Acad. Sci. Paris Sér., I, 329, 575-580 (1999) · Zbl 0935.35073 · doi:10.1016/S0764-4442(00)80004-X |
| [14] | Blanchard, D.; Murat, F.; Redwane, H., Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, J. Differ. Equ., 177, 331-374 (2001) · Zbl 0998.35030 · doi:10.1006/jdeq.2000.4013 |
| [15] | Blanchard, D.; Redwane, H.; Petitta, F., Renormalized solutions of nonlinear parabolic equations with diffuse measure data, Manuscr. Math., 141, 601-635 (2013) · Zbl 1270.35274 · doi:10.1007/s00229-012-0585-7 |
| [16] | Boccardo, L.; Gallouët, T., Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87, 149-169 (1989) · Zbl 0707.35060 · doi:10.1016/0022-1236(89)90005-0 |
| [17] | Boccardo, L.; Gallouët, T., Unicité de la solution de certaines équations elliptiques non linéaires, C. R. Acad. Sci. Paris, 315, 1159-1164 (1992) · Zbl 0789.35056 |
| [18] | Boccardo, L.; Gallouët, T., Nonlinear elliptic equations with right-hand side measures, Commun. Partial Differ. Equ., 17, 3-3, 641-655 (1992) · Zbl 0812.35043 |
| [19] | Boccardo, L.; Giachetti, D.; Diaz, JI; Murat, F., Existence and regularity of renormalized solutions for some elliptic problems involving derivations of nonlinear terms, J. Differ. Equ., 106, 215-237 (1993) · Zbl 0803.35046 · doi:10.1006/jdeq.1993.1106 |
| [20] | Boccardo, L.; Giachetti, D.; Diaz, J-I; Murat, F., Existence and regularity of renormalized solutions of some elliptic problems involving derivatives of nonlinear terms, J. Differ. Equ., 106, 215-237 (1993) · Zbl 0803.35046 · doi:10.1006/jdeq.1993.1106 |
| [21] | Boccardo, L.; Gallouët, T.; Orsina, L., Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire Anal., 13, 5, 539-551 (1996) · Zbl 0857.35126 · doi:10.1016/S0294-1449(16)30113-5 |
| [22] | Boccardo, L.; Gallouët, T.; Orsina, L., Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire Anal., 13, 539-551 (1996) · Zbl 0857.35126 · doi:10.1016/S0294-1449(16)30113-5 |
| [23] | Boccardo, L.; Dall’Aglio, A.; Gallouët, T.; Orsina, L., Nonlinear parabolic equations with measure data, J. Funct. Anal., 147, 1, 237-258 (1997) · Zbl 0887.35082 · doi:10.1006/jfan.1996.3040 |
| [24] | Boccardo, L.; Dall’Aglio, A.; Gallouët, T.; Orsina, L., Nonlinear parabolic equations with measure data, J. Funct. Anal., 147, 237-258 (1997) · Zbl 0887.35082 · doi:10.1006/jfan.1996.3040 |
| [25] | Chen, Y.; Levine, S.; Rao, M., Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66, 1383-1406 (2006) · Zbl 1102.49010 · doi:10.1137/050624522 |
| [26] | Dal Maso, G.; Murat, F.; Orsina, L.; Prignet, A., Renormalized solutions for elliptic equations with general measure data, C. R. Acad. Sci. Ser. I, 325, 481-486 (1997) · Zbl 0887.35057 |
| [27] | Di Perna, R-J; Lions, P-L, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. Math., 130, 321-366 (1989) · Zbl 0698.45010 · doi:10.2307/1971423 |
| [28] | Diening, L., Harjulehto, P., Hästö, P., \(Ru{\ddot{\text{z}}}{\check{\text{i}}}{\text{c}}{\check{\text{k}}}{\text{a}} \), M.: Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017. Springer, Berlin (2011) · Zbl 1222.46002 |
| [29] | Diening, L., Hästö, P., Nekvinda, A.: Open problems in variable exponent Lebesgue and Sobolev spaces. In: Drabek, R. (ed.) FSDONA 2004 Proceedings, pp. 38-58. Milovy, Czech Republic (2004) |
| [30] | DiPerna, R-J; Lions, P-L, On the Fokker-Plank-Boltzmann equations, Commun. Math. Phys., 120, 1-23 (1988) · Zbl 0671.35068 · doi:10.1007/BF01223204 |
| [31] | DiPerna, R-J; Lions, P-L, On the Cauchy problem for Boltzmann equations, global existence and weak stability, Ann. Math., 130, 321-366 (1989) · Zbl 0698.45010 · doi:10.2307/1971423 |
| [32] | Droniou, J.; Prignet, A., Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data, No DEA, 14, 1-2, 181-205 (2007) · Zbl 1151.35045 |
| [33] | Droniou, J.; Porretta, A.; Prignet, A., Parabolic capacity and soft measures for nonlinear equations, Potential Anal., 19, 2, 99-161 (2003) · Zbl 1017.35040 · doi:10.1023/A:1023248531928 |
| [34] | Edmunds, DE; Peletier, LA, Removable singularities of solutions of quasi-linear parabolic equations, J. Lond. Math. Soc., 2, 2, 273-283 (1970) · Zbl 0195.39303 · doi:10.1112/jlms/s2-2.2.273 |
| [35] | Evans, CL; Gariepy, R., Measure Theory and Fine Properties of Functions. Advanced Mathematical Studies (1992), Boca Raton: CRC Press, Boca Raton · Zbl 0804.28001 |
| [36] | Fan, X-L; Zhao, D., On the spaces \(L^{p(x)}(\Omega )\) and \(W^{m, p(x)}(\Omega )\), J. Math. Anal. Appl., 263, 424-446 (2001) · Zbl 1028.46041 · doi:10.1006/jmaa.2000.7617 |
| [37] | Fukushima, M.; Sato, K.; Taniguchi, S., On the closable part of pre-Dirichlet forms and the finitene supports of underlying measures, Osaka J. Math., 28, 517-535 (1991) · Zbl 0756.60071 |
| [38] | Giaquinta, M.; Modica, G.; Soucěk, J., Cartesian Currents in the Calculus of Variations, vol. I (1998), Berlin: Springer, Berlin · Zbl 0914.49001 · doi:10.1007/978-3-662-06218-0 |
| [39] | Hamdaoui, BEL; Bennouna, J., On some nonlinear parabolic equations with variable exponents and measure data, Moroc. J. Pure Appl. Anal. (MJPAA), 6, 1, 93-117 (2020) · Zbl 07836853 · doi:10.2478/mjpaa-2020-0008 |
| [40] | Harjulehto, P.; Hästö, P.; Koskenoja, M.; Varonen, S., Sobolev capacity on the space \(W^{1, p(\cdot )}({\mathbb{R}}^n)\), J. Funct. Spaces Appl., 1, 17-33 (2003) · Zbl 1078.46021 · doi:10.1155/2003/895261 |
| [41] | Harjulehto, P.; Hästö, P.; Koskenoja, M., Properties of capacities in variable exponent Sobolev spaces, J. Anal. Appl., 5, 71-92 (2007) · Zbl 1143.31003 |
| [42] | Harjulehto, P.; Hästö, P.; Lê, UV; Nuortio, M., Overview of differential equations with non-standard growth, Nonlinear Anal., 72, 12, 4551-4574 (2010) · Zbl 1188.35072 · doi:10.1016/j.na.2010.02.033 |
| [43] | Harvey, R.; Polking, JC, A notion of capacity which characterizes removable singularities, Trans. Am. Math. Soc., 169, 183-195 (1972) · Zbl 0249.35012 · doi:10.1090/S0002-9947-1972-0306740-4 |
| [44] | Harvey, R.; Polkingn, J., A notion of capacity which characterizes removable singularities, Trans. Am. Math. Soc., 1669, 183-195 (1972) · Zbl 0249.35012 · doi:10.1090/S0002-9947-1972-0306740-4 |
| [45] | Heinonen, J.; Kilpeläinen, T.; Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations (1993), Oxford: Oxford University Press, Oxford · Zbl 0780.31001 |
| [46] | Kilpeläinen, T., Weighted Sobolev spaces and capacity, Ann. Acad. Sci. Fenn. Math., 19, 95-113 (1994) · Zbl 0801.46037 |
| [47] | Kovácǐk, O.; Rákosník, J., On spaces \(L^p(x)\) and \(W^{k,p(x)}\), Czechoslov. Math. J., 41, 116, 592-618 (1991) · Zbl 0784.46029 · doi:10.21136/CMJ.1991.102493 |
| [48] | Lanconelli, E., Sul problema di Dirichlet per l’equazione del calore, Ann. Mat. Pura Appl., 97, 1, 83-114 (1973) · Zbl 0277.35058 · doi:10.1007/BF02414910 |
| [49] | Lions, P.-L., Murat, F.: Solutions renormalisées d’équations elliptiques (In preparation) |
| [50] | Marah, A.; Redwane, H., Nonlinear parabolic equation with diffuse measure data, J. Nonlinear Evol. Equ. Appl., 3, 27-48 (2017) · Zbl 07069528 |
| [51] | Maz’ja, VG, Sobolev Spaces (1985), Berlin: Springer, Berlin · Zbl 0692.46023 · doi:10.1007/978-3-662-09922-3 |
| [52] | Nyanquini, I.; Ouaro, S.; Soma, S., Entropy solution to nonlinear multivalued elliptic problem with variable exponents and measure data, Ann. Univ. Craiova Math. Comput. Sci. Ser., 40, 174-198 (2013) · Zbl 1299.35106 |
| [53] | Ouaro, S.; Traoré, U., \(p(\cdot )\)-parabolic capacity and decomposition of measures, Ann. Univ. Craiova, 44, 1, 1-34 (2017) · Zbl 1413.35429 |
| [54] | Petitta, F.: Nonlinear parabolic equations with general measure data. Università di Roma, Italy (Ph.D. Thesis) (2006) · Zbl 1150.35060 |
| [55] | Petitta, F., Renormalized solutions of nonlinear parabolic equations with general measure data, Ann. Mat. Pura Appl., 187, 4, 563-604 (2008) · Zbl 1150.35060 · doi:10.1007/s10231-007-0057-y |
| [56] | Petitta, F.; Porretta, A., On the notion of renormalized solution to nonlinear parabolic equations with general measure data, J. Elliptic Parabol. Equ., 1, 201-214 (2015) · Zbl 1386.35190 · doi:10.1007/BF03377376 |
| [57] | Petitta, F.; Ponce, AC; Porretta, A., Approximation of diffuse measures for parabolic capacities, C. R. Acad. Sci. Paris Ser. I, 346, 161-166 (2008) · Zbl 1141.35313 · doi:10.1016/j.crma.2007.12.002 |
| [58] | Petitta, F.; Ponce, AC; Porretta, A., Diffuse measures and nonlinear parabolic equations, J. Evol. Equ., 11, 4, 861-905 (2011) · Zbl 1243.35108 · doi:10.1007/s00028-011-0115-1 |
| [59] | Pierre, M., Parabolic capacity and Sobolev spaces, SIAM J. Math. Anal., 14, 522-533 (1983) · Zbl 0529.35030 · doi:10.1137/0514044 |
| [60] | Porretta, A., Existence results for nonlinear parabolic equations via strong convergence of trauncations, Anna. Mat. Pura Appl., 177, 143-172 (1999) · Zbl 0957.35066 · doi:10.1007/BF02505907 |
| [61] | Prignet, A.: Problèmes elliptiques et paraboliques dans un cadre non variationnel, Ph.D. Thesis, UMPA-ENS Lyon, France (1997) |
| [62] | Prignet, A., Remarks on existence and uniqueness of solutions of elliptic problems with right hand side measures, Rend. Mat., 15, 321-337 (1995) · Zbl 0843.35127 |
| [63] | Prignet, A., Existence and uniqueness of entropy solutions of parabolic problems with \(L1\) data, Nonlinear Anal., 28, 1943-1954 (1997) · Zbl 0909.35075 · doi:10.1016/S0362-546X(96)00030-2 |
| [64] | Rakotoson, JM, Resolution of the critical cases for the problems with \(L^1\)-data, Asymptot. Anal., 6, 285-293 (1993) · Zbl 0793.35042 · doi:10.3233/ASY-1993-6305 |
| [65] | Redwane, H., Existence of a solution for a class of parabolic equations with three unbounded nonlinearities, Adv. Dyn. Syst. Appl., 2, 241-264 (2007) |
| [66] | Redwane, H., Nonlinear parabolic equation with variable exponents and diffuse measure data, J. Nonlinear Evol. Equ. Appl., 2019, 6, 95-114 (2020) · Zbl 07308384 |
| [67] | Ružic̆ka, M., Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics (2000), Berlin: Springer, Berlin · Zbl 0962.76001 · doi:10.1007/BFb0104029 |
| [68] | Samko, S., On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integral Transforms Spec. Funct., 16, 5-6, 461-482 (2005) · Zbl 1069.47056 · doi:10.1080/10652460412331320322 |
| [69] | Samko, S.; Cialdea, A.; Lanzara, F.; Ricci, PE, On some classical operators of variable order in variable exponent spaces, Analysis, Partial Differential Equations and Applications, The Vladimir Maz’ya Anniversary Volume in Operator Theory Advances and Applications, 281-301 (2009), Basel: Birkhäuser, Basel · Zbl 1189.46022 |
| [70] | Saraiva, LMR, Removable singularities of solutions of degenerate quasilinear equations, Ann. Mat. Pura Appl., 141, IV, 187-221 (1985) · Zbl 0592.35077 · doi:10.1007/BF01763174 |
| [71] | Serrin, J., Pathological solutions of elliptic differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18, 385-387 (1964) · Zbl 0142.37601 |
| [72] | Simon, J., Compact sets in \(L^p(0, T; B)\), Ann. Mat. Pura Appl., 146, 65-96 (1987) · Zbl 0629.46031 · doi:10.1007/BF01762360 |
| [73] | Watson, NA, Thermal capacity, Proc. Lond. Math. Soc., 37, 342-362 (1978) · Zbl 0395.35034 · doi:10.1112/plms/s3-37.2.342 |
| [74] | Zhang, C.; Zhou, S., Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and \(L^1\) data, J. Differ. Equ., 248, 1376-1400 (2010) · Zbl 1195.35097 · doi:10.1016/j.jde.2009.11.024 |
| [75] | Ziemer, WP, Behavior at the boundary of solutions of quasilinear parabolic equations, J. Differ. Equ., 35, 3, 291-305 (1980) · Zbl 0451.35037 · doi:10.1016/0022-0396(80)90030-3 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
