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doi: 10.3934/dcdss.2020361

On classes of well-posedness for quasilinear diffusion equations in the whole space

Boris Andreianov 1,2,, and  Mohamed Maliki 3,

1. 

Institut Denis Poisson CNRS UMR7013, Université de Tours, Université d'Orléans, Parc Grandmont, 37200 Tours, France
2. 

Peoples’ Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
3. 

Équipe Modélisation, EDP et Analyse Numérique, FST Mohammédia, B.P. 146, Mohammédia, Morocco

* Corresponding author

Dedicated to Michel Pierre on the occasion of his 70th birthday

Received  October 2019 Published  May 2020

Well-posedness classes for degenerate elliptic problems in $ {\mathbb R}^N $ under the form $ u = \Delta {{\varphi}}(x,u)+f(x) $, with locally (in $ u $) uniformly continuous nonlinearities, are explored. While we are particularly interested in the $ L^\infty $ setting, we also investigate about solutions in $ L^1_{loc} $ and in weighted $ L^1 $ spaces. We give some sufficient conditions in order that the uniqueness and comparison properties hold for the associated solutions; these conditions are expressed in terms of the moduli of continuity of $ u\mapsto {{\varphi}}(x,u) $. Under additional restrictions on the dependency of $ {{\varphi}} $ on $ x $, we deduce the existence results for the corresponding classes of solutions and data. Moreover, continuous dependence results follow readily from the existence claim and the comparison property. In particular, we show that for a general continuous non-decreasing nonlinearity $ {{\varphi}}: {\mathbb R}\mapsto {\mathbb R} $, the space $ L^\infty $ (endowed with the $ L^1_{loc} $ topology) is a well-posedness class for the problem $ u = \Delta {{\varphi}}(u)+f(x) $.

Keywords: Quasilinear elliptic equation, porous medium equation, fast diffusion equation, stationary equation, $ L^\infty $ solutions, Kato inequality, Keller-Osserman condition, uniqueness, well-posedness, weighted contraction estimates.
Mathematics Subject Classification: Primary: 35J62, 35A02, 37L05; Secondary: 35J70, 35D30.
Citation: Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020361
References:
[1]

N. Alibaud, J. Endal and E. R. Jakobsen, Optimal and dual stability results for $L^1$ viscosity and $L^\infty$ entropy solutions, preprint, 2019, arXiv: 1812.02058. Google Scholar

[2]

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.  Google Scholar

[3]

F. Andreu, N. Igbida, J.M. Mazón and J. Toledo, Renormalized solutions for degenerate elliptic-parabolic problems with nonlinear dynamical boundary conditions and L1-data, J. Differential Equations, 244 (2008), 2764-2803.  doi: 10.1016/j.jde.2008.02.022.  Google Scholar

[4]

B. Andreianov, P. Bénilan and S. N. Kruzhkov, $L^1$ theory of scalar conservation law with continuous flux function, J. Funct. Anal., 171 (2000), 15-33.  doi: 10.1006/jfan.1999.3445.  Google Scholar

[5]

B. Andreianov and M. Brassart, Uniqueness of entropy solutions to fractional conservation laws with "fully infinite" speed of propagation, J. Differential Equations, 268 (2020), 3903-3935.  doi: 10.1016/j.jde.2019.10.008.  Google Scholar

[6]

B. Andreianov and N. Igbida, On uniqueness techniques for degenerate convection-diffusion problems, Int. J. Dyn. Syst. Differ. Equ., 4 (2012), 3-34.  doi: 10.1504/IJDSDE.2012.045992.  Google Scholar

[7]

B. Andreianov and M. Maliki, A note on uniqueness of entropy solutions to degenerate parabolic equations in $ {\mathbb R}^N$, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 109-118.  doi: 10.1007/s00030-009-0042-9.  Google Scholar

[8]

B. Andreianov, K. Sbihi and P. Wittbold, On uniqueness and existence of entropy solutions for a nonlinear parabolic problem with absorption, J. Evol. Equ., 8 (2008), 449-490.  doi: 10.1007/s00028-008-0365-8.  Google Scholar

[9]

B. Andreianov and P. Wittbold, Convergence of approximate solutions to an elliptic-parabolic equation without the structure condition, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 695-717.  doi: 10.1007/s00030-011-0148-8.  Google Scholar

[10]

D. G. Aronson, The porous medium equation, in Nonlinear Diffusion Problems, Vol. 1224, Springer, Berlin, 1986, 1–46. doi: 10.1007/BFb0072687.  Google Scholar

[11]

P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble), 34 (1984), 185-206.   Google Scholar

[12]

P. Bénilan, Équations d'évolution dans un espace de Banach quelconque et applications, Thèse d'état, 1972. Google Scholar

[13]

P. Bénilan, H. Brezis and M. G. Crandall, Asemilinear equation in $L^{1}(R^{N})$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (1975), 523-555.   Google Scholar

[14]

P. Bénilan and M. G. Crandall, The continuous dependence on $\phi$ of solutions of $u_t-\Delta\phi(u) = 0$, Indiana Univ. Math. J., 30 (1981), 161-177.  doi: 10.1512/iumj.1981.30.30014.  Google Scholar

[15]

P. Bénilan, M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Preprint book. Google Scholar

[16]

P. Bénilan, M. G. Crandall and M. Pierre, Solutions of the porous medium equation in $ {\mathbb R}^N$ under optimal conditions on initial values, Indiana Univ. Math. J., 33 (1984), 51-87.  doi: 10.1512/iumj.1984.33.33003.  Google Scholar

[17]

P. Bénilan and S. N. Kruzhkov, Conservation laws with continuous flux functions, NoDEA Nonlinear Differential Equations Appl., 3 (1996), 395-419.  doi: 10.1007/BF01193828.  Google Scholar

[18]

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.  Google Scholar

[19]

N. M. Bokalo, Uniqueness of the solution of the Fourier problem for quasilinear equations of unsteady filtration type, Uspekhi Mat. Nauk, 39 (1984), 139-140.   Google Scholar

[20]

H. Brézis, Semilinear equations in $R^ N$ without condition at infinity, Appl. Math. Optim., 12 (1984), 271-282.  doi: 10.1007/BF01449045.  Google Scholar

[21]

H. Brézis and M. G. Crandall, Uniqueness of solutions of the initial-value problem for $u_{t}-\Delta \varphi (u) = 0$, J. Math. Pures Appl., 58 (1979), 153-163.   Google Scholar

[22]

J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147 (1999), 269-361.  doi: 10.1007/s002050050152.  Google Scholar

[23]

B. E. J. Dahlberg and C. Kenig, Nonnegative solutions of the porous medium equation, Comm. Partial Differential Equations, 9 (1984), 409-437.  doi: 10.1080/03605308408820336.  Google Scholar

[24]

P. Daskalopoulos and C. Kenig, Degenerate diffusions. Initial value problems and local regularity theory, European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/033.  Google Scholar

[25]

F. del Teso, J. Endal and E. R. Jakobsen, Uniqueness and properties of distributional solutions of non-local equations of porous medium type, Adv. Math., 305 (2017), 78-143.  doi: 10.1016/j.aim.2016.09.021.  Google Scholar

[26]

F. del Teso, J. Endal and E. R. Jakobsen, On distributional solutions of local and non-local problems of porous medium type, C. R. Math. Acad. Sci. Paris, 355 (2017), 1154-1160.  doi: 10.1016/j.crma.2017.10.010.  Google Scholar

[27]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[28]

J. Endal and E. R. Jakobsen, $L^1$ Contraction for bounded (nonintegrable) solutions of degenerate parabolic equations, SIAM J. Math. Anal., 46 (2014), 3957-3982.  doi: 10.1137/140966599.  Google Scholar

[29]

T. Gallouët and J.-M. Morel, Resolution of a semilinear equation in $L^1$, Proc. Roy. Soc. Edinburgh Sect. A, 96 (1984), 275-288.  doi: 10.1017/S0308210500025403.  Google Scholar

[30]

T. Gallouët and J.-M. Morel, The equation $-\Delta u +|u|^{\alpha-1}u = f$, for $0\leq \alpha\leq 1$, Nonlinear Anal., 11 (1987), 893-912.  doi: 10.1016/0362-546X(87)90059-9.  Google Scholar

[31]

M. A. Herrero and M. Pierre, The Cauchy problem for $u_t = \Delta u^m$ when $0 <m<1$, Trans. Amer. Math. Soc., 291 (1985), 145-158.  doi: 10.1090/S0002-9947-1985-0797051-0.  Google Scholar

[32]

A. S. Kalashnikov, Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Uspekhi Mat. Nauk, 42 (1987), 135-254.   Google Scholar

[33]

T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148.  doi: 10.1007/BF02760233.  Google Scholar

[34]

J. B. Keller, On solutions of $\Delta u = f(u)$., Comm. Pure Appl. Math., 10 (1957), 503-510.  doi: 10.1002/cpa.3160100402.  Google Scholar

[35]

S. N. Kruzhkov and E. Y. Panov, First-order quasilinear conservation laws with infinite initial data dependence area., Dokl. Akad. Nauk URSS, 314 (1990), 79-84.   Google Scholar

[36]

S. N. Kruzhkov and E. Y. Panov, Osgood's type conditions for uniqueness of entropy solutions to Cauchy problem for quasilinear conservation laws of the first order, Ann. Univ. Ferrara Sez. VII (N.S.), 40 (1994), 31-54.   Google Scholar

[37]

M. Maliki and H. Touré, Uniqueness of entropy solutions for nonlinear degenerate parabolic problem, J. Evol. Equ., 3 (2003), 603-622.  doi: 10.1007/s00028-003-0105-z.  Google Scholar

[38]

R. Osserman, On the inequality $\Delta u \geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.   Google Scholar

[39]

F. Otto, $L^1$ contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Differential Equations, 131 (1996), 20-38.  doi: 10.1006/jdeq.1996.0155.  Google Scholar

[40]

A. Ouédraogo, Explicit conditions for the uniqueness of solutions for parabolic degenerate problems, Int. J. Dyn. Syst. Differ. Equ., 6 (2016), 75-86.  doi: 10.1504/IJDSDE.2016.074582.  Google Scholar

[41]

M. Pierre, Uniqueness of the solutions of $u_t-\Delta \varphi(u) = 0$ with initial datum a measure, Nonlinear Anal., 6 (1982), 175-187.  doi: 10.1016/0362-546X(82)90086-4.  Google Scholar

[42] J. L. Vázquez, The Porous Medium Equation. Mathematical theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.   Google Scholar
[43]

Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations, World Scientific Publishing Co., River Edge, New Jersey, 2001. doi: 10.1142/9789812799791.  Google Scholar

show all references

References:
[1]

N. Alibaud, J. Endal and E. R. Jakobsen, Optimal and dual stability results for $L^1$ viscosity and $L^\infty$ entropy solutions, preprint, 2019, arXiv: 1812.02058. Google Scholar

[2]

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.  Google Scholar

[3]

F. Andreu, N. Igbida, J.M. Mazón and J. Toledo, Renormalized solutions for degenerate elliptic-parabolic problems with nonlinear dynamical boundary conditions and L1-data, J. Differential Equations, 244 (2008), 2764-2803.  doi: 10.1016/j.jde.2008.02.022.  Google Scholar

[4]

B. Andreianov, P. Bénilan and S. N. Kruzhkov, $L^1$ theory of scalar conservation law with continuous flux function, J. Funct. Anal., 171 (2000), 15-33.  doi: 10.1006/jfan.1999.3445.  Google Scholar

[5]

B. Andreianov and M. Brassart, Uniqueness of entropy solutions to fractional conservation laws with "fully infinite" speed of propagation, J. Differential Equations, 268 (2020), 3903-3935.  doi: 10.1016/j.jde.2019.10.008.  Google Scholar

[6]

B. Andreianov and N. Igbida, On uniqueness techniques for degenerate convection-diffusion problems, Int. J. Dyn. Syst. Differ. Equ., 4 (2012), 3-34.  doi: 10.1504/IJDSDE.2012.045992.  Google Scholar

[7]

B. Andreianov and M. Maliki, A note on uniqueness of entropy solutions to degenerate parabolic equations in $ {\mathbb R}^N$, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 109-118.  doi: 10.1007/s00030-009-0042-9.  Google Scholar

[8]

B. Andreianov, K. Sbihi and P. Wittbold, On uniqueness and existence of entropy solutions for a nonlinear parabolic problem with absorption, J. Evol. Equ., 8 (2008), 449-490.  doi: 10.1007/s00028-008-0365-8.  Google Scholar

[9]

B. Andreianov and P. Wittbold, Convergence of approximate solutions to an elliptic-parabolic equation without the structure condition, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 695-717.  doi: 10.1007/s00030-011-0148-8.  Google Scholar

[10]

D. G. Aronson, The porous medium equation, in Nonlinear Diffusion Problems, Vol. 1224, Springer, Berlin, 1986, 1–46. doi: 10.1007/BFb0072687.  Google Scholar

[11]

P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble), 34 (1984), 185-206.   Google Scholar

[12]

P. Bénilan, Équations d'évolution dans un espace de Banach quelconque et applications, Thèse d'état, 1972. Google Scholar

[13]

P. Bénilan, H. Brezis and M. G. Crandall, Asemilinear equation in $L^{1}(R^{N})$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (1975), 523-555.   Google Scholar

[14]

P. Bénilan and M. G. Crandall, The continuous dependence on $\phi$ of solutions of $u_t-\Delta\phi(u) = 0$, Indiana Univ. Math. J., 30 (1981), 161-177.  doi: 10.1512/iumj.1981.30.30014.  Google Scholar

[15]

P. Bénilan, M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Preprint book. Google Scholar

[16]

P. Bénilan, M. G. Crandall and M. Pierre, Solutions of the porous medium equation in $ {\mathbb R}^N$ under optimal conditions on initial values, Indiana Univ. Math. J., 33 (1984), 51-87.  doi: 10.1512/iumj.1984.33.33003.  Google Scholar

[17]

P. Bénilan and S. N. Kruzhkov, Conservation laws with continuous flux functions, NoDEA Nonlinear Differential Equations Appl., 3 (1996), 395-419.  doi: 10.1007/BF01193828.  Google Scholar

[18]

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.  Google Scholar

[19]

N. M. Bokalo, Uniqueness of the solution of the Fourier problem for quasilinear equations of unsteady filtration type, Uspekhi Mat. Nauk, 39 (1984), 139-140.   Google Scholar

[20]

H. Brézis, Semilinear equations in $R^ N$ without condition at infinity, Appl. Math. Optim., 12 (1984), 271-282.  doi: 10.1007/BF01449045.  Google Scholar

[21]

H. Brézis and M. G. Crandall, Uniqueness of solutions of the initial-value problem for $u_{t}-\Delta \varphi (u) = 0$, J. Math. Pures Appl., 58 (1979), 153-163.   Google Scholar

[22]

J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147 (1999), 269-361.  doi: 10.1007/s002050050152.  Google Scholar

[23]

B. E. J. Dahlberg and C. Kenig, Nonnegative solutions of the porous medium equation, Comm. Partial Differential Equations, 9 (1984), 409-437.  doi: 10.1080/03605308408820336.  Google Scholar

[24]

P. Daskalopoulos and C. Kenig, Degenerate diffusions. Initial value problems and local regularity theory, European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/033.  Google Scholar

[25]

F. del Teso, J. Endal and E. R. Jakobsen, Uniqueness and properties of distributional solutions of non-local equations of porous medium type, Adv. Math., 305 (2017), 78-143.  doi: 10.1016/j.aim.2016.09.021.  Google Scholar

[26]

F. del Teso, J. Endal and E. R. Jakobsen, On distributional solutions of local and non-local problems of porous medium type, C. R. Math. Acad. Sci. Paris, 355 (2017), 1154-1160.  doi: 10.1016/j.crma.2017.10.010.  Google Scholar

[27]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[28]

J. Endal and E. R. Jakobsen, $L^1$ Contraction for bounded (nonintegrable) solutions of degenerate parabolic equations, SIAM J. Math. Anal., 46 (2014), 3957-3982.  doi: 10.1137/140966599.  Google Scholar

[29]

T. Gallouët and J.-M. Morel, Resolution of a semilinear equation in $L^1$, Proc. Roy. Soc. Edinburgh Sect. A, 96 (1984), 275-288.  doi: 10.1017/S0308210500025403.  Google Scholar

[30]

T. Gallouët and J.-M. Morel, The equation $-\Delta u +|u|^{\alpha-1}u = f$, for $0\leq \alpha\leq 1$, Nonlinear Anal., 11 (1987), 893-912.  doi: 10.1016/0362-546X(87)90059-9.  Google Scholar

[31]

M. A. Herrero and M. Pierre, The Cauchy problem for $u_t = \Delta u^m$ when $0 <m<1$, Trans. Amer. Math. Soc., 291 (1985), 145-158.  doi: 10.1090/S0002-9947-1985-0797051-0.  Google Scholar

[32]

A. S. Kalashnikov, Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Uspekhi Mat. Nauk, 42 (1987), 135-254.   Google Scholar

[33]

T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148.  doi: 10.1007/BF02760233.  Google Scholar

[34]

J. B. Keller, On solutions of $\Delta u = f(u)$., Comm. Pure Appl. Math., 10 (1957), 503-510.  doi: 10.1002/cpa.3160100402.  Google Scholar

[35]

S. N. Kruzhkov and E. Y. Panov, First-order quasilinear conservation laws with infinite initial data dependence area., Dokl. Akad. Nauk URSS, 314 (1990), 79-84.   Google Scholar

[36]

S. N. Kruzhkov and E. Y. Panov, Osgood's type conditions for uniqueness of entropy solutions to Cauchy problem for quasilinear conservation laws of the first order, Ann. Univ. Ferrara Sez. VII (N.S.), 40 (1994), 31-54.   Google Scholar

[37]

M. Maliki and H. Touré, Uniqueness of entropy solutions for nonlinear degenerate parabolic problem, J. Evol. Equ., 3 (2003), 603-622.  doi: 10.1007/s00028-003-0105-z.  Google Scholar

[38]

R. Osserman, On the inequality $\Delta u \geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.   Google Scholar

[39]

F. Otto, $L^1$ contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Differential Equations, 131 (1996), 20-38.  doi: 10.1006/jdeq.1996.0155.  Google Scholar

[40]

A. Ouédraogo, Explicit conditions for the uniqueness of solutions for parabolic degenerate problems, Int. J. Dyn. Syst. Differ. Equ., 6 (2016), 75-86.  doi: 10.1504/IJDSDE.2016.074582.  Google Scholar

[41]

M. Pierre, Uniqueness of the solutions of $u_t-\Delta \varphi(u) = 0$ with initial datum a measure, Nonlinear Anal., 6 (1982), 175-187.  doi: 10.1016/0362-546X(82)90086-4.  Google Scholar

[42] J. L. Vázquez, The Porous Medium Equation. Mathematical theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.   Google Scholar
[43]

Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations, World Scientific Publishing Co., River Edge, New Jersey, 2001. doi: 10.1142/9789812799791.  Google Scholar

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