- Previous Article
- EECT Home
- This Issue
-
Next Article
Local stabilization of viscous Burgers equation with memory
Solvability of doubly nonlinear parabolic equation with p-laplacian
Faculty of Science and Technology, Oita University, 700 Dannoharu, Oita, 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.
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. |
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. |
[1] |
Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3749-3778. doi: 10.3934/dcdsb.2021205 |
[2] |
Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control and Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61 |
[3] |
Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations and Control Theory, 2022, 11 (2) : 399-414. doi: 10.3934/eect.2021005 |
[4] |
Goro Akagi, Kei Matsuura. Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian. Conference Publications, 2011, 2011 (Special) : 22-31. doi: 10.3934/proc.2011.2011.22 |
[5] |
Louis Tebou. Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315 |
[6] |
Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure and Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319 |
[7] |
Jun Zhou. Initial boundary value problem for a inhomogeneous pseudo-parabolic equation. Electronic Research Archive, 2020, 28 (1) : 67-90. doi: 10.3934/era.2020005 |
[8] |
Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107 |
[9] |
Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387 |
[10] |
Vishal Vasan, Bernard Deconinck. Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3171-3188. doi: 10.3934/dcds.2013.33.3171 |
[11] |
Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431 |
[12] |
Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic and Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029 |
[13] |
Elena Rossi. Well-posedness of general 1D initial boundary value problems for scalar balance laws. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3577-3608. doi: 10.3934/dcds.2019147 |
[14] |
Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027 |
[15] |
Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058 |
[16] |
Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763 |
[17] |
Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194 |
[18] |
Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure and Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673 |
[19] |
Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082 |
[20] |
Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic and Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032 |
2021 Impact Factor: 1.169
Tools
Metrics
Other articles
by authors
[Back to Top]