• Previous Article
    Stability and stabilization for the three-dimensional Navier-Stokes-Voigt equations with unbounded variable delay
  • EECT Home
  • This Issue
  • Next Article
    Stabilization of higher order Schrödinger equations on a finite interval: Part II
doi: 10.3934/eect.2021033
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Solvability of doubly nonlinear parabolic equation with p-laplacian

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

Received  November 2020 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.

Citation: Shun Uchida. Solvability of doubly nonlinear parabolic equation with p-laplacian. Evolution Equations & Control Theory, doi: 10.3934/eect.2021033
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.  Google Scholar

[2]

G. Akagi and U. Stefanelli, Doubly nonlinear equations as convex minimization, SIAM J. Math. Anal., 46 (2014), 1922-1945.  doi: 10.1137/13091909X.  Google Scholar

[3]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341.  doi: 10.1007/BF01176474.  Google Scholar

[4] L. AmbrosioN. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000.   Google Scholar
[5]

K. Ammar, Renormalized entropy solutions for degenerate nonlinear evolution problems, Electron. J. Differential Equations, 147 (2009), 32 pp.  Google Scholar

[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.  Google Scholar

[7]

F. AndreuJ. M. MazonJ. 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.  Google Scholar

[8]

H. Attouch, Variational Convergence for Functions and Operators, Pitman, London, 1984.  Google Scholar

[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.  Google Scholar

[10]

V. Barbu, Existence for nonlinear Volterra equations in Hilbert spaces, SIAM J. Math. Anal., 10 (1979), 552-569.  doi: 10.1137/0510052.  Google Scholar

[11]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[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.  Google Scholar

[13]

P. Bénilan, Equations d'évolution dans un Espace de Banach Quelconque et Applications, Thèse d'état, Orsay, 1972. Google Scholar

[14]

P. Bénilan and P. Wittbold, On mild and weak solutions of elliptic-parabolic problems, Adv. Differential Equations, 1 (1996), 1053-1073.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

H. Brézis, Intégrales convexes dans les espaces de Sobolev, Israel J. Math., 13 (1972), 9-23.  doi: 10.1007/BF02760227.  Google Scholar

[18]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.  Google Scholar

[19]

H. BrézisM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

J. I. Díaz, Qualitative study of nonlinear parabolic equations: An introduction, Extracta Math., 16 (2001), 303-341.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

J. DroniouR. 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.  Google Scholar

[33]

J. R. EstebanA. Rodríguez and J. L. Vázquez, A nonlinear heat equation with singular diffusivity, Comm. Partial Differential Equations, 13 (1988), 985-1039.  doi: 10.1080/03605308808820566.  Google Scholar

[34]

S. FornaroE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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)).  Google Scholar

[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)). Google Scholar

[42] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.   Google Scholar
[43]

J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1970.   Google Scholar
[49]

A. RodríguezJ.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.   Google Scholar

[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.  Google Scholar

[51]

R. E. Showalter and N. J. Walkington, A diffusion system for fluid in fractured media, Differential Integral Equations, 3 (1990), 219-236.   Google Scholar

[52]

U. Stefanelli, On a class of doubly nonlinear nonlocal evolution equations, Differential Integral Equations, 15 (2002), 897-922.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[55]

J. L. VázquezJ. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[58]

K. Yosida, Functional Analysis, 6th edition, Springer, Berlin, 1980.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

G. Akagi and U. Stefanelli, Doubly nonlinear equations as convex minimization, SIAM J. Math. Anal., 46 (2014), 1922-1945.  doi: 10.1137/13091909X.  Google Scholar

[3]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341.  doi: 10.1007/BF01176474.  Google Scholar

[4] L. AmbrosioN. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000.   Google Scholar
[5]

K. Ammar, Renormalized entropy solutions for degenerate nonlinear evolution problems, Electron. J. Differential Equations, 147 (2009), 32 pp.  Google Scholar

[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.  Google Scholar

[7]

F. AndreuJ. M. MazonJ. 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.  Google Scholar

[8]

H. Attouch, Variational Convergence for Functions and Operators, Pitman, London, 1984.  Google Scholar

[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.  Google Scholar

[10]

V. Barbu, Existence for nonlinear Volterra equations in Hilbert spaces, SIAM J. Math. Anal., 10 (1979), 552-569.  doi: 10.1137/0510052.  Google Scholar

[11]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[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.  Google Scholar

[13]

P. Bénilan, Equations d'évolution dans un Espace de Banach Quelconque et Applications, Thèse d'état, Orsay, 1972. Google Scholar

[14]

P. Bénilan and P. Wittbold, On mild and weak solutions of elliptic-parabolic problems, Adv. Differential Equations, 1 (1996), 1053-1073.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

H. Brézis, Intégrales convexes dans les espaces de Sobolev, Israel J. Math., 13 (1972), 9-23.  doi: 10.1007/BF02760227.  Google Scholar

[18]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.  Google Scholar

[19]

H. BrézisM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

J. I. Díaz, Qualitative study of nonlinear parabolic equations: An introduction, Extracta Math., 16 (2001), 303-341.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

J. DroniouR. 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.  Google Scholar

[33]

J. R. EstebanA. Rodríguez and J. L. Vázquez, A nonlinear heat equation with singular diffusivity, Comm. Partial Differential Equations, 13 (1988), 985-1039.  doi: 10.1080/03605308808820566.  Google Scholar

[34]

S. FornaroE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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)).  Google Scholar

[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)). Google Scholar

[42] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.   Google Scholar
[43]

J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1970.   Google Scholar
[49]

A. RodríguezJ.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.   Google Scholar

[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.  Google Scholar

[51]

R. E. Showalter and N. J. Walkington, A diffusion system for fluid in fractured media, Differential Integral Equations, 3 (1990), 219-236.   Google Scholar

[52]

U. Stefanelli, On a class of doubly nonlinear nonlocal evolution equations, Differential Integral Equations, 15 (2002), 897-922.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[55]

J. L. VázquezJ. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[58]

K. Yosida, Functional Analysis, 6th edition, Springer, Berlin, 1980.  Google Scholar

[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.  Google Scholar

[1]

Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2021  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 & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61

[3]

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

[4]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005

[5]

Louis Tebou. Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian. Discrete & 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 &amp; 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]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387

[9]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107

[10]

Vishal Vasan, Bernard Deconinck. Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3171-3188. doi: 10.3934/dcds.2013.33.3171

[11]

Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic & Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029

[12]

Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431

[13]

Elena Rossi. Well-posedness of general 1D initial boundary value problems for scalar balance laws. Discrete & 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 &amp; Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027

[15]

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

[16]

Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure &amp; Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058

[17]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure &amp; Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673

[18]

Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082

[19]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032

[20]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194

2020 Impact Factor: 1.081

Metrics

  • PDF downloads (38)
  • HTML views (107)
  • Cited by (0)

Other articles
by authors

[Back to Top]