Solvability of doubly nonlinear parabolic equation with <i>p</i>-laplacian

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June 2022, 11(3): 975-1000. doi: 10.3934/eect.2021033

Solvability of doubly nonlinear parabolic equation with p-laplacian

Faculty of Science and Technology, Oita University, 700 Dannoharu, Oita, Japan

Received November 2020 Published June 2022 Early access July 2021

Fund Project: The author is supported by the Fund for the Promotion of Joint International Research (Fostering Joint International Research (B)) #18KK0073, JSPS Japan

In this paper, we consider a doubly nonlinear parabolic equation $ \partial _t \beta (u) - \nabla \cdot \alpha (x , \nabla u) \ni f $ with the homogeneous Dirichlet boundary condition in a bounded domain, where $ \beta : \mathbb{R} \to 2 ^{ \mathbb{R} } $ is a maximal monotone graph satisfying $ 0 \in \beta (0) $ and $ \nabla \cdot \alpha (x , \nabla u ) $ stands for a generalized $ p $-Laplacian. Existence of solution to the initial boundary value problem of this equation has been studied in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on $ \beta $. However, there are a few results for the case where such assumptions are removed and it is difficult to construct an abstract theory which covers the case for $ 1 < p < 2 $. Main purpose of this paper is to show the solvability of the initial boundary value problem for any $ p \in (1, \infty ) $ without any conditions for $ \beta $ except $ 0 \in \beta (0) $. We also discuss the uniqueness of solution by using properties of entropy solution.

Keywords: Doubly nonlinear equation, parabolic equation, p-Laplacian, initial boundary value problem, well-posedness, entropy solution.

Mathematics Subject Classification: Primary 35K92; Secondary 35K61, 47J35, 34G25.

Citation: Shun Uchida. Solvability of doubly nonlinear parabolic equation with p-laplacian. Evolution Equations and Control Theory, 2022, 11 (3) : 975-1000. doi: 10.3934/eect.2021033

References:

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[2]	G. Akagi and U. Stefanelli, Doubly nonlinear equations as convex minimization, SIAM J. Math. Anal., 46 (2014), 1922-1945. doi: 10.1137/13091909X.
[3]	H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. doi: 10.1007/BF01176474.
[4]	L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000.
[5]	K. Ammar, Renormalized entropy solutions for degenerate nonlinear evolution problems, Electron. J. Differential Equations, 147 (2009), 32 pp.
[6]	K. Ammar and P. Wittbold, Existence of renormalized solutions of degenerate elliptic-parabolic problems, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 477-496. doi: 10.1017/S0308210500002493.
[7]	F. Andreu, J. M. Mazon, J. Toledo and N. Igbida, A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions, Interfaces Free Bound., 8 (2006), 447-479. doi: 10.4171/IFB/151.
[8]	H. Attouch, Variational Convergence for Functions and Operators, Pitman, London, 1984.
[9]	A. Bamberger, Étude d'une équation doublement non linéaire, J. Functional Analysis, 24 (1977), 148-155. doi: 10.1016/0022-1236(77)90051-9.
[10]	V. Barbu, Existence for nonlinear Volterra equations in Hilbert spaces, SIAM J. Math. Anal., 10 (1979), 552-569. doi: 10.1137/0510052.
[11]	V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.
[12]	V. Barbu and A. Favini, Existence for an implicit nonlinear differential equation, Nonlinear Anal., 32 (1998), 33-40. doi: 10.1016/S0362-546X(97)00450-1.
[13]	P. Bénilan, Equations d'évolution dans un Espace de Banach Quelconque et Applications, Thèse d'état, Orsay, 1972.
[14]	P. Bénilan and P. Wittbold, On mild and weak solutions of elliptic-parabolic problems, Adv. Differential Equations, 1 (1996), 1053-1073.
[15]	F. Bernis, Existence results for doubly nonlinear higher order parabolic equations on unbounded domains, Math. Ann., 279 (1988), 373-394. doi: 10.1007/BF01456275.
[16]	H. Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in Contributions to Nonlinear Functional Analysis (ed. E. H. Zarantonello), Academic Press, New York, (1971), 101–156. doi: 10.1016/B978-0-12-775850-3.50009-1.
[17]	H. Brézis, Intégrales convexes dans les espaces de Sobolev, Israel J. Math., 13 (1972), 9-23. doi: 10.1007/BF02760227.
[18]	H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.
[19]	H. Brézis, M. G. Crandall and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl. Math., 23 (1970), 123-144. doi: 10.1002/cpa.3160230107.
[20]	J. Berryman and C. J. Holland, Asymptotic behavior of the nonlinear diffusion equation $n_t = $ $(n^{-1}n _ x)_x$, J. Math. Phys., 23 (1982), 983-987. doi: 10.1063/1.525466.
[21]	D. Blanchard and G. A. Francfort, Study of a doubly nonlinear heat equation with no growth assumptions on the parabolic term, SIAM J. Math. Anal., 19 (1988), 1032-1056. doi: 10.1137/0519070.
[22]	D. Blanchard and A. Porretta, Stefan problems with nonlinear diffusion and convection, J. Differential Equations, 210 (2005), 383-428. doi: 10.1016/j.jde.2004.06.012.
[23]	J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Rational Mech. Anal., 147 (1999), 269-361. doi: 10.1007/s002050050152.
[24]	J. Carrillo and P. Wittbold, Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems, J. Differential Equations, 156 (1999), 93-121. doi: 10.1006/jdeq.1998.3597.
[25]	M.G. Crandall and T. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298. doi: 10.2307/2373376.
[26]	P. Daskalopoulos and M. A. del Pino, On the Cauchy problem for $u_t = \Delta \log u$ in higher dimensions, Math. Ann., 313 (1999), 189-206. doi: 10.1007/s002080050257.
[27]	J. I. Díaz, Qualitative study of nonlinear parabolic equations: An introduction, Extracta Math., 16 (2001), 303-341.
[28]	G. Díaz and I. Díaz, Finite extinction time for a class of nonlinear parabolic equations, Comm. Partial Differential Equations, 4 (1979), 1213-1231. doi: 10.1080/03605307908820126.
[29]	J. I. Díaz and J. F. Padial, Uniqueness and existence of solutions in the $BV _t(Q)$ space to a doubly nonlinear parabolic problem, Publ. Mat., 40 (1996), 527-560. doi: 10.5565/PUBLMAT_40296_18.
[30]	E. DiBenedetto and A. Friedman, The ill-posed Hele-Shaw model and the Stefan problem for supercooled water, Trans. Amer. Math. Soc., 282 (1984), 183-204. doi: 10.1090/S0002-9947-1984-0728709-6.
[31]	E. DiBenedetto and R. E. Showalter, Implicit degenerate evolution equations and applications, SIAM J. Math. Anal., 12 (1981), 731-751. doi: 10.1137/0512062.
[32]	J. Droniou, R. Eymard and K. S. Talbot, Convergence in $C([0, T];L^ 2(\Omega))$ of weak solutions to perturbed doubly degenerate parabolic equations, J. Differential Equations, 260 (2016), 7821-7860. doi: 10.1016/j.jde.2016.02.004.
[33]	J. R. Esteban, A. Rodríguez and J. L. Vázquez, A nonlinear heat equation with singular diffusivity, Comm. Partial Differential Equations, 13 (1988), 985-1039. doi: 10.1080/03605308808820566.
[34]	S. Fornaro, E. Henriques and V. Vespri, Harnack type inequalities for the parabolic logarithmic $p$-Laplacian equation, Matematiche (Catania), 75 (2020), 277-311. doi: 10.4418/2020.75.1.13.
[35]	O. Grange and F. Mignot, Sur la résolution d'une équation et d'une inéquation paraboliques non linéaires, J. Functional Analysis, 11 (1972), 77-92. doi: 10.1016/0022-1236(72)90080-8.
[36]	V. M. Hokkanen, An implicit nonlinear time dependent equation has a solution, J. Math. Anal. Appl., 161 (1991), 117-141. doi: 10.1016/0022-247X(91)90364-6.
[37]	N. Igbida and J. M. Urbano, Uniqueness for nonlinear degenerate problems, NoDEA Nonlinear Differential Equations Appl., 10 (2003), 287-307. doi: 10.1007/s00030-003-1030-0.
[38]	N. Kenmochi and I. Pawƚow, A class of nonlinear elliptic-parabolic equations with time-dependent constraints, Nonlinear Anal., 10 (1986), 1181-1202. doi: 10.1016/0362-546X(86)90058-1.
[39]	K. Kobayasi, The equivalence of weak solutions and entropy solutions of nonlinear degenerate second-order equations, J. Differential Equations, 189 (2003), 383-395. doi: 10.1016/S0022-0396(02)00069-4.
[40]	S. N. Kružkov, Generalized solutions of the Cauchy problem in the large for non-linear equations of first order, Dokl. Akad. Nauk SSSR, 187 (1969), 29–32; (English transl. in Soviet Math. Dokl., 10 (1969)).
[41]	S. N. Kružkov, First order quasilinear equations in several independent variables, Mat. Sb., 81 (1970), 228–255; (English transl. in Math. USSR Sb., 10 (1970)).
[42]	O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.
[43]	J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.
[44]	A. Matas and J. Merker, On doubly nonlinear evolution equations with non-potential or dynamic relation between the state variables, J. Evol. Equ., 17 (2017), 869-881. doi: 10.1007/s00028-016-0342-6.
[45]	H. Miyoshi and M. Tsutsumi, Convergence of hydrodynamical limits for generalized Carleman models, Funkcial. Ekvac., 59 (2016), 351-382. doi: 10.1619/fesi.59.351.
[46]	Y. W. Qi, Existence and non-existence of a fast diffusion equation in $ \mathbb{R}^n $, J. Differential Equations, 136 (1997), 378-393. doi: 10.1006/jdeq.1996.3246.
[47]	P. A. Raviart, Sur la résolution de certaines equations paraboliques non linéaires, J. Functional Analysis, 5 (1970), 299-328. doi: 10.1016/0022-1236(70)90031-5.
[48]	R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1970.
[49]	A. Rodríguez, J.L. Vázquez and J. R. Esteban, The maximal solution of the logarithmic fast diffusion equation in two space dimensions, Adv. Differential Equations, 2 (1997), 867-894.
[50]	R. E. Showalter, Mathematical formulation of the Stefan problem, Internat. J. Engrg. Sci., 20 (1982), 909-912. doi: 10.1016/0020-7225(82)90109-4.
[51]	R. E. Showalter and N. J. Walkington, A diffusion system for fluid in fractured media, Differential Integral Equations, 3 (1990), 219-236.
[52]	U. Stefanelli, On a class of doubly nonlinear nonlocal evolution equations, Differential Integral Equations, 15 (2002), 897-922.
[53]	M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. Appl., 132 (1988), 187-212. doi: 10.1016/0022-247X(88)90053-4.
[54]	J. L. Vázquez, Finite-time blow-down in the evolution of point masses by planar logarithmic diffusion, Discrete Contin. Dyn. Syst., 19 (2007), 1-35. doi: 10.3934/dcds.2007.19.1.
[55]	J. L. Vázquez, J. R. Esteban and A. Rodríguez, The fast diffusion equation with logarithmic nonlinearity and the evolution of conformal metrics in the plane, Adv. Differential Equations, 1 (1996), 21-50.
[56]	L. F. Wu, A new result for the porous medium equation derived from the Ricci flow, Bull. Amer. Math. Soc. (N.S.), 28 (1993), 90-94. doi: 10.1090/S0273-0979-1993-00336-7.
[57]	N. Yamazaki, Almost periodic stability for doubly nonlinear evolution equations generated by subdifferentials, Nonlinear Anal., 47 (2001), 1725-1736. doi: 10.1016/S0362-546X(01)00305-4.
[58]	K. Yosida, Functional Analysis, 6th edition, Springer, Berlin, 1980.
[59]	W. P. Ziemer, Weakly Differentiable Functions, Sobolev Spaces and Functions of Bounded Variation, Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

show all references

References:

[1]	S. Aizicovici and V. M. Hokkanen, Doubly nonlinear equations with unbounded operators, Nonlinear Anal., 58 (2004), 591-607. doi: 10.1016/j.na.2003.10.029.
[2]	G. Akagi and U. Stefanelli, Doubly nonlinear equations as convex minimization, SIAM J. Math. Anal., 46 (2014), 1922-1945. doi: 10.1137/13091909X.
[3]	H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. doi: 10.1007/BF01176474.
[4]	L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000.
[5]	K. Ammar, Renormalized entropy solutions for degenerate nonlinear evolution problems, Electron. J. Differential Equations, 147 (2009), 32 pp.
[6]	K. Ammar and P. Wittbold, Existence of renormalized solutions of degenerate elliptic-parabolic problems, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 477-496. doi: 10.1017/S0308210500002493.
[7]	F. Andreu, J. M. Mazon, J. Toledo and N. Igbida, A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions, Interfaces Free Bound., 8 (2006), 447-479. doi: 10.4171/IFB/151.
[8]	H. Attouch, Variational Convergence for Functions and Operators, Pitman, London, 1984.
[9]	A. Bamberger, Étude d'une équation doublement non linéaire, J. Functional Analysis, 24 (1977), 148-155. doi: 10.1016/0022-1236(77)90051-9.
[10]	V. Barbu, Existence for nonlinear Volterra equations in Hilbert spaces, SIAM J. Math. Anal., 10 (1979), 552-569. doi: 10.1137/0510052.
[11]	V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.
[12]	V. Barbu and A. Favini, Existence for an implicit nonlinear differential equation, Nonlinear Anal., 32 (1998), 33-40. doi: 10.1016/S0362-546X(97)00450-1.
[13]	P. Bénilan, Equations d'évolution dans un Espace de Banach Quelconque et Applications, Thèse d'état, Orsay, 1972.
[14]	P. Bénilan and P. Wittbold, On mild and weak solutions of elliptic-parabolic problems, Adv. Differential Equations, 1 (1996), 1053-1073.
[15]	F. Bernis, Existence results for doubly nonlinear higher order parabolic equations on unbounded domains, Math. Ann., 279 (1988), 373-394. doi: 10.1007/BF01456275.
[16]	H. Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in Contributions to Nonlinear Functional Analysis (ed. E. H. Zarantonello), Academic Press, New York, (1971), 101–156. doi: 10.1016/B978-0-12-775850-3.50009-1.
[17]	H. Brézis, Intégrales convexes dans les espaces de Sobolev, Israel J. Math., 13 (1972), 9-23. doi: 10.1007/BF02760227.
[18]	H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.
[19]	H. Brézis, M. G. Crandall and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl. Math., 23 (1970), 123-144. doi: 10.1002/cpa.3160230107.
[20]	J. Berryman and C. J. Holland, Asymptotic behavior of the nonlinear diffusion equation $n_t = $ $(n^{-1}n _ x)_x$, J. Math. Phys., 23 (1982), 983-987. doi: 10.1063/1.525466.
[21]	D. Blanchard and G. A. Francfort, Study of a doubly nonlinear heat equation with no growth assumptions on the parabolic term, SIAM J. Math. Anal., 19 (1988), 1032-1056. doi: 10.1137/0519070.
[22]	D. Blanchard and A. Porretta, Stefan problems with nonlinear diffusion and convection, J. Differential Equations, 210 (2005), 383-428. doi: 10.1016/j.jde.2004.06.012.
[23]	J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Rational Mech. Anal., 147 (1999), 269-361. doi: 10.1007/s002050050152.
[24]	J. Carrillo and P. Wittbold, Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems, J. Differential Equations, 156 (1999), 93-121. doi: 10.1006/jdeq.1998.3597.
[25]	M.G. Crandall and T. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298. doi: 10.2307/2373376.
[26]	P. Daskalopoulos and M. A. del Pino, On the Cauchy problem for $u_t = \Delta \log u$ in higher dimensions, Math. Ann., 313 (1999), 189-206. doi: 10.1007/s002080050257.
[27]	J. I. Díaz, Qualitative study of nonlinear parabolic equations: An introduction, Extracta Math., 16 (2001), 303-341.
[28]	G. Díaz and I. Díaz, Finite extinction time for a class of nonlinear parabolic equations, Comm. Partial Differential Equations, 4 (1979), 1213-1231. doi: 10.1080/03605307908820126.
[29]	J. I. Díaz and J. F. Padial, Uniqueness and existence of solutions in the $BV _t(Q)$ space to a doubly nonlinear parabolic problem, Publ. Mat., 40 (1996), 527-560. doi: 10.5565/PUBLMAT_40296_18.
[30]	E. DiBenedetto and A. Friedman, The ill-posed Hele-Shaw model and the Stefan problem for supercooled water, Trans. Amer. Math. Soc., 282 (1984), 183-204. doi: 10.1090/S0002-9947-1984-0728709-6.
[31]	E. DiBenedetto and R. E. Showalter, Implicit degenerate evolution equations and applications, SIAM J. Math. Anal., 12 (1981), 731-751. doi: 10.1137/0512062.
[32]	J. Droniou, R. Eymard and K. S. Talbot, Convergence in $C([0, T];L^ 2(\Omega))$ of weak solutions to perturbed doubly degenerate parabolic equations, J. Differential Equations, 260 (2016), 7821-7860. doi: 10.1016/j.jde.2016.02.004.
[33]	J. R. Esteban, A. Rodríguez and J. L. Vázquez, A nonlinear heat equation with singular diffusivity, Comm. Partial Differential Equations, 13 (1988), 985-1039. doi: 10.1080/03605308808820566.
[34]	S. Fornaro, E. Henriques and V. Vespri, Harnack type inequalities for the parabolic logarithmic $p$-Laplacian equation, Matematiche (Catania), 75 (2020), 277-311. doi: 10.4418/2020.75.1.13.
[35]	O. Grange and F. Mignot, Sur la résolution d'une équation et d'une inéquation paraboliques non linéaires, J. Functional Analysis, 11 (1972), 77-92. doi: 10.1016/0022-1236(72)90080-8.
[36]	V. M. Hokkanen, An implicit nonlinear time dependent equation has a solution, J. Math. Anal. Appl., 161 (1991), 117-141. doi: 10.1016/0022-247X(91)90364-6.
[37]	N. Igbida and J. M. Urbano, Uniqueness for nonlinear degenerate problems, NoDEA Nonlinear Differential Equations Appl., 10 (2003), 287-307. doi: 10.1007/s00030-003-1030-0.
[38]	N. Kenmochi and I. Pawƚow, A class of nonlinear elliptic-parabolic equations with time-dependent constraints, Nonlinear Anal., 10 (1986), 1181-1202. doi: 10.1016/0362-546X(86)90058-1.
[39]	K. Kobayasi, The equivalence of weak solutions and entropy solutions of nonlinear degenerate second-order equations, J. Differential Equations, 189 (2003), 383-395. doi: 10.1016/S0022-0396(02)00069-4.
[40]	S. N. Kružkov, Generalized solutions of the Cauchy problem in the large for non-linear equations of first order, Dokl. Akad. Nauk SSSR, 187 (1969), 29–32; (English transl. in Soviet Math. Dokl., 10 (1969)).
[41]	S. N. Kružkov, First order quasilinear equations in several independent variables, Mat. Sb., 81 (1970), 228–255; (English transl. in Math. USSR Sb., 10 (1970)).
[42]	O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.
[43]	J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.
[44]	A. Matas and J. Merker, On doubly nonlinear evolution equations with non-potential or dynamic relation between the state variables, J. Evol. Equ., 17 (2017), 869-881. doi: 10.1007/s00028-016-0342-6.
[45]	H. Miyoshi and M. Tsutsumi, Convergence of hydrodynamical limits for generalized Carleman models, Funkcial. Ekvac., 59 (2016), 351-382. doi: 10.1619/fesi.59.351.
[46]	Y. W. Qi, Existence and non-existence of a fast diffusion equation in $ \mathbb{R}^n $, J. Differential Equations, 136 (1997), 378-393. doi: 10.1006/jdeq.1996.3246.
[47]	P. A. Raviart, Sur la résolution de certaines equations paraboliques non linéaires, J. Functional Analysis, 5 (1970), 299-328. doi: 10.1016/0022-1236(70)90031-5.
[48]	R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1970.
[49]	A. Rodríguez, J.L. Vázquez and J. R. Esteban, The maximal solution of the logarithmic fast diffusion equation in two space dimensions, Adv. Differential Equations, 2 (1997), 867-894.
[50]	R. E. Showalter, Mathematical formulation of the Stefan problem, Internat. J. Engrg. Sci., 20 (1982), 909-912. doi: 10.1016/0020-7225(82)90109-4.
[51]	R. E. Showalter and N. J. Walkington, A diffusion system for fluid in fractured media, Differential Integral Equations, 3 (1990), 219-236.
[52]	U. Stefanelli, On a class of doubly nonlinear nonlocal evolution equations, Differential Integral Equations, 15 (2002), 897-922.
[53]	M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. Appl., 132 (1988), 187-212. doi: 10.1016/0022-247X(88)90053-4.
[54]	J. L. Vázquez, Finite-time blow-down in the evolution of point masses by planar logarithmic diffusion, Discrete Contin. Dyn. Syst., 19 (2007), 1-35. doi: 10.3934/dcds.2007.19.1.
[55]	J. L. Vázquez, J. R. Esteban and A. Rodríguez, The fast diffusion equation with logarithmic nonlinearity and the evolution of conformal metrics in the plane, Adv. Differential Equations, 1 (1996), 21-50.
[56]	L. F. Wu, A new result for the porous medium equation derived from the Ricci flow, Bull. Amer. Math. Soc. (N.S.), 28 (1993), 90-94. doi: 10.1090/S0273-0979-1993-00336-7.
[57]	N. Yamazaki, Almost periodic stability for doubly nonlinear evolution equations generated by subdifferentials, Nonlinear Anal., 47 (2001), 1725-1736. doi: 10.1016/S0362-546X(01)00305-4.
[58]	K. Yosida, Functional Analysis, 6th edition, Springer, Berlin, 1980.
[59]	W. P. Ziemer, Weakly Differentiable Functions, Sobolev Spaces and Functions of Bounded Variation, Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

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