Existence and uniqueness of entropy solutions to nonlinear parabolic problem with homogeneous Dirichlet boundary conditions involving variable exponent. (English) Zbl 1420.35123
Summary: In this paper, we prove the existence and uniqueness of entropy solution to nonlinear parabolic problem with homogeneous Dirichlet boundary conditions with \(L^1\)-data. The functional setting involves Lebesgue and Sobolev spaces with variable exponent. The general assumptions and the nonlinear semigroup theory are considered to prove the existence and uniqueness of mild solution satisfying the \(L^1\)-comparison principle. Moreover, under the same general assumptions and some a-priori estimations of the sequence of mild solutions, we obtain the existence and uniqueness of weak solution. Finally, we prove the existence and uniquess of the renormalized solution which is equivalent to the existence and uniqueness of entropy solution.
MSC:
| 35K55 | Nonlinear parabolic equations |
| 35D30 | Weak solutions to PDEs |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
Keywords:
parabolic equation; variable exponent; mild solution; maximal monotone graph; renormalized solution; entropy solution; \(L^1\)-dataReferences:
| [1] | Alt, H.W., Luckhaus, S.: Quasi-linear elliptic-parabolic differential equations. Math. Z. 183, 311-341 (1983) · Zbl 0497.35049 · doi:10.1007/BF01176474 |
| [2] | Antontsev, S.N., Rodrigues, J.F.: On stationary thermo-rheological viscous flows. Ann. del Univ. de Ferrara 52, 19-36 (2006) · Zbl 1117.76004 · doi:10.1007/s11565-006-0002-9 |
| [3] | Bendahmane, M., Wittbold, P., Zimmermann, A.: Renormalized solutions for a nonlinear parabolic equation with variable exponents and \[L^1-\] L1-data. J. Diff. Equ. 249, 1483-1515 (2010) · Zbl 1231.35106 · doi:10.1016/j.jde.2010.05.011 |
| [4] | Bénilan, Ph.: Equations d’évolution dans un espace de Banach quelconque et applications. Thèse d’état, Univ. Paris XI, Orsay (1972) |
| [5] | Bénilan, Ph, Boccardo, L., Gallouèt, T., Gariepy, R., Pierre, M., Vazquez, J.L.: An \[L^1\] L1 theory of existence and uniqueness of nonlinear elliptic equations. Ann. Sc. Norm. Sup. Pisa 22(2), 240-273 (1995) · Zbl 0866.35037 |
| [6] | Bénilan, Ph., Crandal, M.G., Pazy, A.: Evolution equations governed by accretive operators (forthcoming book) |
| [7] | Bénilan, Ph, Wittbold, P.: On mild and weak solutions of elliptic-parabolic problems. Ad. Differ. Equ. 1(6), 1053-1073 (1996) · Zbl 0858.35064 |
| [8] | Blanchard, D., Murat, F.: Renormalized solutions of nonlinear parabolic problems with \[L^1\] L1 data: existence and uniqueness. Proc. R. Soc. Edinb Sect. A 127(6), 1137-1152 (1997) · Zbl 0895.35050 · doi:10.1017/S0308210500026986 |
| [9] | Bonzi, B.K., Ouaro, S.: Entropy solutions for doubly nonlinear elliptic problems with variable exponent. J. Math. Anal. Appl. 370(2), 392-405 (2010) · Zbl 1200.35109 · doi:10.1016/j.jmaa.2010.05.022 |
| [10] | 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 |
| [11] | Crandall, M.C., Liggett, T.M.: Generation of semigroups of nonlinear transformations on general Banach spaces. Am. J. Math. 93(2), 265-298 (1971) · Zbl 0226.47038 · doi:10.2307/2373376 |
| [12] | Diening, L.: Theoritical and numerical results for electrorheological fluids. Ph. D. thesis, University of Freiburg, Germany, (2002) · Zbl 1022.76001 |
| [13] | Diening, L., Harjulehto, P., Hästö, P., Ruzicka, M.: Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics, vol. 2017. Springer-Verlag, Heidelberg (2011) · Zbl 1222.46002 |
| [14] | DiPerna, R.J., Lions, P.L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. 2(130), 321-366 (1989) · Zbl 0698.45010 · doi:10.2307/1971423 |
| [15] | Droniou, J., Prignet, A.: Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data. Nonlinear Diff. Equ. Appl. 14(1-2), 181-205 (2007) · Zbl 1151.35045 · doi:10.1007/s00030-007-5018-z |
| [16] | Fan, X., Zhao, D.: On the spaces \[L^{p(x)}(\varOmega )\] Lp(x)(Ω) and \[W^{m, p(x)}(\varOmega )\] Wm,p(x)(Ω). J. Math. Anal. Appl. 263, 424-446 (2001) · Zbl 1028.46041 · doi:10.1006/jmaa.2000.7617 |
| [17] | Giannetti, F., Passarelli di Napoli, A., Ragusa, M.A., Tachikawa, A., Takabayashi, H.: Partial regularity for minimizers of a class of non autonomous functionals with nonstandard growth. Calc. Var. Partial Differ. Equ. 56(6), 153 (2017) · Zbl 1383.49047 · doi:10.1007/s00526-017-1248-z |
| [18] | Kovacik, O., Rakosnik, J.: On spaces \[L^{p(x)}\] Lp(x) and \[W^{1, p(x)}\] W1,p(x). Czech. Math. J. 41, 592-618 (1991) · Zbl 0784.46029 |
| [19] | Landes, R.: On the existence of weak solutions for quasilinear parabolic initial-boundary value problems. Proc. R. Soc. Edinb. Sect. A 89(3-4), 217-237 (1981) · Zbl 0493.35054 · doi:10.1017/S0308210500020242 |
| [20] | Ouaro, S., Ouédraogo, A.: Nonlinear parabolic problems with variable exponent and \[L^1\] L1-data. Electron. J. Differ. Equ. 2017(32), 32 (2017) · Zbl 1357.35200 |
| [21] | Ouaro, S., Traoré, S.: Existence and uniqueness of entropy solutions to nonlinear elliptic problems with variable growth. Int. J. Evol. Equ. 4(4), 451-471 (2009) · Zbl 1243.35056 |
| [22] | Porretta, A.: Existence results for nonlinear parabolic equations via strong convergence of truncations. Ann. Mat. Pura Appl. 4(177), 143-172 (1999) · Zbl 0957.35066 · doi:10.1007/BF02505907 |
| [23] | Ragusa, M.A., Tachikawa, A.: On interior regularity of minimizers of \[p(x)\] p(x)-energy functionals. Nonlinear Anal. Theory Method. Appl. 93, 162-167 (2013) · Zbl 1281.49036 · doi:10.1016/j.na.2013.07.023 |
| [24] | Ragusa, M.A., Tachikawa, A.: Boundary regularity of minimizers of \[p(x)\] p(x)-energy functionals. Ann. Inst. Henri Poincaré Anal. Non Linéaire 33(2), 451-476 (2016) · Zbl 1333.49052 · doi:10.1016/j.anihpc.2014.11.003 |
| [25] | Rajagopal, K.R., Ruzicka, M.: Mathematical modeling of electrorheological materials. Contin. Mech. Thermodyn. 13, 59-78 (2001) · Zbl 0971.76100 · doi:10.1007/s001610100034 |
| [26] | Ruzicka, M.: Electrorheological fluids: modelling and mathematical theory. Lecture Notes in Mathematics 1748, Springer-Verlag, Berlin (2002) · Zbl 0962.76001 |
| [27] | Simondon, F.: Etude de l’équation \[\partial \_tb(u)- \text{div} a(b(u), Du) = 0\]∂_tb(u)-diva(b(u),Du)=0 par la méthode des semi-groupes dans \[L^1(\varOmega )\] L1(Ω). Publ. Math. Besançon, Anal. non linéaire 7, 1-18 (1983) · Zbl 0531.35043 |
| [28] | Wittbold, P., Zimmermann, A.: Existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponents and \[L^1\] L1-data. Nonlinear Anal. TMA 72, 2990-3008 (2010) · Zbl 1185.35088 · doi:10.1016/j.na.2009.11.041 |
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