February  2021, 14(2): 505-531. doi: 10.3934/dcdss.2020361

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

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  February 2021 Early access  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) $.

Citation: Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. 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.

[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.

[3]

F. AndreuN. IgbidaJ.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.

[4]

B. AndreianovP. 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.

[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.

[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.

[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.

[8]

B. AndreianovK. 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.

[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.

[10]

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

[11]

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

[12]

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

[13]

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

[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.

[15]

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

[16]

P. BénilanM. 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.

[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.

[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.

[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. 

[20]

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

[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. 

[22]

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

[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.

[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.

[25]

F. del TesoJ. 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.

[26]

F. del TesoJ. 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.

[27]

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

[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.

[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.

[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.

[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.

[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. 

[33]

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

[34]

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

[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. 

[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. 

[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.

[38]

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

[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.

[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.

[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.

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

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.

[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.

[3]

F. AndreuN. IgbidaJ.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.

[4]

B. AndreianovP. 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.

[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.

[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.

[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.

[8]

B. AndreianovK. 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.

[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.

[10]

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

[11]

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

[12]

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

[13]

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

[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.

[15]

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

[16]

P. BénilanM. 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.

[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.

[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.

[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. 

[20]

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

[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. 

[22]

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

[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.

[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.

[25]

F. del TesoJ. 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.

[26]

F. del TesoJ. 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.

[27]

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

[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.

[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.

[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.

[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.

[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. 

[33]

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

[34]

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

[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. 

[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. 

[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.

[38]

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

[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.

[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.

[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.

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

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