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Regularity under general and $ p,q- $ growth conditions
On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions
Università di Cagliari, Dipartimento di Matematica e Informatica, Viale Merello 92, 09123 Cagliari, Italy |
$ \begin{equation} u_t = \Delta u^m + a\int_\Omega u^p-b u^q -c\lvert\nabla\sqrt{u}\rvert^2 \quad {\rm{in}}\quad \Omega \times I,\;\;\;\;\;\;(◇) \end{equation} $ |
$ \Omega $ |
$ \mathbb R^N $ |
$ N\geq 1 $ |
$ I = [0,t^*) $ |
$ u $ |
$ a,b,c $ |
$ m,p,q $ |
$ u $ |
$ m,p $ |
$ q $ |
$ u_0({\bf x}) $ |
$ u = u({\bf x},t) $ |
$ \Omega \times I $ |
$ \triangleright $ |
$ p>q $ |
$ t^* $ |
$ u $ |
$ L^{m(p-1)}(\Omega) $ |
$ t^* $ |
$ \triangleright $ |
$ p<q $ |
$ N $ |
$ u $ |
References:
[1] |
M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura,
Lower estimate of the attractor dimension for a chemotaxis growth system, J. London. Math. Soc. (2), 74 (2006), 453-474.
doi: 10.1112/S0024610706023015. |
[2] |
F. Andreu, J. M. Mazón, F. Simondon and J. Toledo,
Blow-up for a class of nonlinear parabolic problems, Asymptot. Anal., 29 (2002), 143-155.
|
[3] |
D. G. Aronson, The porous medium equation, Springer Berlin Heidelberg, Berlin, Heidelberg, (1986), 1–46. |
[4] |
C. Bandle and H. Brunner,
Blowup in diffusion equations: A survey, J. Comput. Appl. Math., 97 (1998), 3-22.
doi: 10.1016/S0377-0427(98)00100-9. |
[5] |
V. A. Galaktionov, A boundary value problem for the nonlinear parabolic equation $u_{t} = \Delta u^{\sigma +1}+u^{\beta}$, Differentsial'nye Uravneniya, 17 (1981), 836–842,956. |
[6] |
V. A. Galaktionov,
Blow-up for quasilinear heat equations with critical Fujita's exponents, Proc. Roy. Soc. Edinburgh: Section A Mathematics, 124 (1994), 517-525.
doi: 10.1017/S0308210500028766. |
[7] |
V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhaǐlov and A. A. Samarskiǐ,
On unbounded solutions of the Cauchy problem for the parabolic equation $u_t = \nabla (u^\sigma\nabla u)+u^\beta$, Dokl. Akad. Nauk SSSR, 252 (1980), 1362-1364.
|
[8] |
C. Grant, Theory of Ordinary Differential Equations, CreateSpace Independent Publishing Platform. |
[9] |
M. E. Gurtin and R. C. MacCamy,
On the diffusion of biological populations, Math. Biosc., 33 (1977), 35-49.
doi: 10.1016/0025-5564(77)90062-1. |
[10] |
H. Kielhöfer,
Halbgruppen und semilineare anfangs-randwertprobleme, Manuscripta Math., 12 (1974), 121-152.
doi: 10.1007/BF01168647. |
[11] |
N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, Mathematics and its Applications (Soviet Series), 7. D. Reidel Publishing Co., Dordrecht, 1987.
doi: 10.1007/978-94-010-9557-0. |
[12] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23. American Mathematical Society, 1988. |
[13] |
H. A. Levine,
The role of critical exponents in blowup theorems, SIAM Rev., 32 (1990), 262-288.
doi: 10.1137/1032046. |
[14] |
T. Li, N. Pintus and G. Viglialoro, Properties of solutions to porous medium problems with different sources and boundary conditions, Z. Angew. Math. Phys., 70: 86 (2019), 1–18.
doi: 10.1007/s00033-019-1130-2. |
[15] |
F. C. Li and C. H. Xie,
Global existence and blow-up for a nonlinear porous medium equation, Appl. Math. Lett., 16 (2003), 185-192.
doi: 10.1016/S0893-9659(03)80030-7. |
[16] |
Y. Liu,
Blow-up phenomena for the nonlinear nonlocal porous medium equation under Robin boundary condition, Comput. Math. Appl., 66 (2013), 2092-2095.
doi: 10.1016/j.camwa.2013.08.024. |
[17] |
Y. Liu,
Lower bounds for the blow-up time in a non-local reaction diffusion problem under nonlinear boundary conditions, Math. Comput. Model., 57 (2013), 926-931.
doi: 10.1016/j.mcm.2012.10.002. |
[18] |
M. Marras, S. Vernier-Piro and G. Viglialoro,
Blow-up phenomena in chemotaxis systems with a source term, Math. Method Appl. Sci., 39 (2016), 2787-2798.
doi: 10.1002/mma.3728. |
[19] |
M. Marras and G. Viglialoro, Blow-up time of a general Keller-Segel system with source and damping terms, C. R. Acad. Bulgare Sci., 69: 6 (2016), 687–696 |
[20] |
M. Marras and G. Viglialoro,
Boundedness in a fully parabolic chemotaxis-consumption system with nonlinear diffusion and sensitivity, and logistic source, Math. Nachr., 291 (2018), 2318-2333.
doi: 10.1002/mana.201700172. |
[21] |
L. E. Payne, G. A. Philippin and S. Vernier Piro,
Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition. II, Nonlinear Anal. Theory Methods Appl., 73 (2010), 971-978.
doi: 10.1016/j.na.2010.04.023. |
[22] |
L. E. Payne, G. Philippin and P. W. Schaefer,
Bounds for blow-up time in nonlinear parabolic problems, J. Math. Anal. Appl., 338 (2008), 438-447.
doi: 10.1016/j.jmaa.2007.05.022. |
[23] |
L. E. Payne and P. W. Schaefer,
Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. Math. Anal. Appl., 328 (2007), 1196-1205.
doi: 10.1016/j.jmaa.2006.06.015. |
[24] |
L. E. Payne and P. W. Schaefer,
Blow-up in parabolic problems under Robin boundary conditions, Appl. Anal., 87 (2008), 699-707.
doi: 10.1080/00036810802189662. |
[25] |
L. E. Payne, G. A. Philippin and V. Proytcheva,
Continuous dependence on the geometry and on the initial time for a class of parabolic problems. I, Math. Methods Appl. Sci., 30 (2007), 1885-1898.
doi: 10.1002/mma.877. |
[26] |
P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73. Birkhäuser Verlag, Basel, 2007. |
[27] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Basel, 2007. |
[28] |
P. W. Schaefer,
Lower bounds for blow-up time in some porous medium problems, Dynamic Systems and Applications, Dynamic, Atlanta, GA, 5 (2008), 442-445.
|
[29] |
P. W. Schaefer,
Blow-up phenomena in some porous medium problems, Dynam. Systems Appl., 18 (2009), 103-110.
|
[30] |
J. C. Song,
Lower bounds for the blow-up time in a non-local reaction-diffusion problem, Appl. Math. Lett., 24 (2011), 793-796.
doi: 10.1016/j.aml.2010.12.042. |
[31] |
P. Souplet,
Finite time blow-up for a non-linear parabolic equation with a gradient term and applications, Math. Methods Appl. Sci., 19 (1996), 1317-1333.
doi: 10.1002/(SICI)1099-1476(19961110)19:16<1317::AID-MMA835>3.0.CO;2-M. |
[32] |
J. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.
![]() ![]() |
[33] |
G. Viglialoro,
Blow-up time of a Keller-Segel-type system with Neumann and Robin boundary conditions, Diff. Integral Equ., 29 (2016), 359-376.
|
[34] |
G. Viglialoro,
Boundedness properties of very weak solutions to a fully parabolic chemotaxis-system with logistic source, Nonlinear Anal. Real World Appl., 34 (2017), 520-535.
doi: 10.1016/j.nonrwa.2016.10.001. |
[35] |
G. Viglialoro and T. Woolley,
Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth, Discrete Continuous Dyn. Syst. Ser. B, 22 (2018), 3023-3045.
doi: 10.3934/dcdsb.2017199. |
[36] |
G. Viglialoro and T. E. Woolley,
Boundedness in a parabolic-elliptic chemotaxis system with nonlinear diffusion and sensitivity and logistic source, Math. Methods Appl. Sci., 41 (2018), 1809-1824.
doi: 10.1002/mma.4707. |
[37] |
M. X. Wang and Y. M. Wang,
Properties of positive solutions for non-local reaction-diffusion problems, Math. Method. Appl. Sc., 19 (1996), 1141-1156.
doi: 10.1002/(SICI)1099-1476(19960925)19:14<1141::AID-MMA811>3.0.CO;2-9. |
show all references
References:
[1] |
M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura,
Lower estimate of the attractor dimension for a chemotaxis growth system, J. London. Math. Soc. (2), 74 (2006), 453-474.
doi: 10.1112/S0024610706023015. |
[2] |
F. Andreu, J. M. Mazón, F. Simondon and J. Toledo,
Blow-up for a class of nonlinear parabolic problems, Asymptot. Anal., 29 (2002), 143-155.
|
[3] |
D. G. Aronson, The porous medium equation, Springer Berlin Heidelberg, Berlin, Heidelberg, (1986), 1–46. |
[4] |
C. Bandle and H. Brunner,
Blowup in diffusion equations: A survey, J. Comput. Appl. Math., 97 (1998), 3-22.
doi: 10.1016/S0377-0427(98)00100-9. |
[5] |
V. A. Galaktionov, A boundary value problem for the nonlinear parabolic equation $u_{t} = \Delta u^{\sigma +1}+u^{\beta}$, Differentsial'nye Uravneniya, 17 (1981), 836–842,956. |
[6] |
V. A. Galaktionov,
Blow-up for quasilinear heat equations with critical Fujita's exponents, Proc. Roy. Soc. Edinburgh: Section A Mathematics, 124 (1994), 517-525.
doi: 10.1017/S0308210500028766. |
[7] |
V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhaǐlov and A. A. Samarskiǐ,
On unbounded solutions of the Cauchy problem for the parabolic equation $u_t = \nabla (u^\sigma\nabla u)+u^\beta$, Dokl. Akad. Nauk SSSR, 252 (1980), 1362-1364.
|
[8] |
C. Grant, Theory of Ordinary Differential Equations, CreateSpace Independent Publishing Platform. |
[9] |
M. E. Gurtin and R. C. MacCamy,
On the diffusion of biological populations, Math. Biosc., 33 (1977), 35-49.
doi: 10.1016/0025-5564(77)90062-1. |
[10] |
H. Kielhöfer,
Halbgruppen und semilineare anfangs-randwertprobleme, Manuscripta Math., 12 (1974), 121-152.
doi: 10.1007/BF01168647. |
[11] |
N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, Mathematics and its Applications (Soviet Series), 7. D. Reidel Publishing Co., Dordrecht, 1987.
doi: 10.1007/978-94-010-9557-0. |
[12] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23. American Mathematical Society, 1988. |
[13] |
H. A. Levine,
The role of critical exponents in blowup theorems, SIAM Rev., 32 (1990), 262-288.
doi: 10.1137/1032046. |
[14] |
T. Li, N. Pintus and G. Viglialoro, Properties of solutions to porous medium problems with different sources and boundary conditions, Z. Angew. Math. Phys., 70: 86 (2019), 1–18.
doi: 10.1007/s00033-019-1130-2. |
[15] |
F. C. Li and C. H. Xie,
Global existence and blow-up for a nonlinear porous medium equation, Appl. Math. Lett., 16 (2003), 185-192.
doi: 10.1016/S0893-9659(03)80030-7. |
[16] |
Y. Liu,
Blow-up phenomena for the nonlinear nonlocal porous medium equation under Robin boundary condition, Comput. Math. Appl., 66 (2013), 2092-2095.
doi: 10.1016/j.camwa.2013.08.024. |
[17] |
Y. Liu,
Lower bounds for the blow-up time in a non-local reaction diffusion problem under nonlinear boundary conditions, Math. Comput. Model., 57 (2013), 926-931.
doi: 10.1016/j.mcm.2012.10.002. |
[18] |
M. Marras, S. Vernier-Piro and G. Viglialoro,
Blow-up phenomena in chemotaxis systems with a source term, Math. Method Appl. Sci., 39 (2016), 2787-2798.
doi: 10.1002/mma.3728. |
[19] |
M. Marras and G. Viglialoro, Blow-up time of a general Keller-Segel system with source and damping terms, C. R. Acad. Bulgare Sci., 69: 6 (2016), 687–696 |
[20] |
M. Marras and G. Viglialoro,
Boundedness in a fully parabolic chemotaxis-consumption system with nonlinear diffusion and sensitivity, and logistic source, Math. Nachr., 291 (2018), 2318-2333.
doi: 10.1002/mana.201700172. |
[21] |
L. E. Payne, G. A. Philippin and S. Vernier Piro,
Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition. II, Nonlinear Anal. Theory Methods Appl., 73 (2010), 971-978.
doi: 10.1016/j.na.2010.04.023. |
[22] |
L. E. Payne, G. Philippin and P. W. Schaefer,
Bounds for blow-up time in nonlinear parabolic problems, J. Math. Anal. Appl., 338 (2008), 438-447.
doi: 10.1016/j.jmaa.2007.05.022. |
[23] |
L. E. Payne and P. W. Schaefer,
Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. Math. Anal. Appl., 328 (2007), 1196-1205.
doi: 10.1016/j.jmaa.2006.06.015. |
[24] |
L. E. Payne and P. W. Schaefer,
Blow-up in parabolic problems under Robin boundary conditions, Appl. Anal., 87 (2008), 699-707.
doi: 10.1080/00036810802189662. |
[25] |
L. E. Payne, G. A. Philippin and V. Proytcheva,
Continuous dependence on the geometry and on the initial time for a class of parabolic problems. I, Math. Methods Appl. Sci., 30 (2007), 1885-1898.
doi: 10.1002/mma.877. |
[26] |
P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73. Birkhäuser Verlag, Basel, 2007. |
[27] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Basel, 2007. |
[28] |
P. W. Schaefer,
Lower bounds for blow-up time in some porous medium problems, Dynamic Systems and Applications, Dynamic, Atlanta, GA, 5 (2008), 442-445.
|
[29] |
P. W. Schaefer,
Blow-up phenomena in some porous medium problems, Dynam. Systems Appl., 18 (2009), 103-110.
|
[30] |
J. C. Song,
Lower bounds for the blow-up time in a non-local reaction-diffusion problem, Appl. Math. Lett., 24 (2011), 793-796.
doi: 10.1016/j.aml.2010.12.042. |
[31] |
P. Souplet,
Finite time blow-up for a non-linear parabolic equation with a gradient term and applications, Math. Methods Appl. Sci., 19 (1996), 1317-1333.
doi: 10.1002/(SICI)1099-1476(19961110)19:16<1317::AID-MMA835>3.0.CO;2-M. |
[32] |
J. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.
![]() ![]() |
[33] |
G. Viglialoro,
Blow-up time of a Keller-Segel-type system with Neumann and Robin boundary conditions, Diff. Integral Equ., 29 (2016), 359-376.
|
[34] |
G. Viglialoro,
Boundedness properties of very weak solutions to a fully parabolic chemotaxis-system with logistic source, Nonlinear Anal. Real World Appl., 34 (2017), 520-535.
doi: 10.1016/j.nonrwa.2016.10.001. |
[35] |
G. Viglialoro and T. Woolley,
Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth, Discrete Continuous Dyn. Syst. Ser. B, 22 (2018), 3023-3045.
doi: 10.3934/dcdsb.2017199. |
[36] |
G. Viglialoro and T. E. Woolley,
Boundedness in a parabolic-elliptic chemotaxis system with nonlinear diffusion and sensitivity and logistic source, Math. Methods Appl. Sci., 41 (2018), 1809-1824.
doi: 10.1002/mma.4707. |
[37] |
M. X. Wang and Y. M. Wang,
Properties of positive solutions for non-local reaction-diffusion problems, Math. Method. Appl. Sc., 19 (1996), 1141-1156.
doi: 10.1002/(SICI)1099-1476(19960925)19:14<1141::AID-MMA811>3.0.CO;2-9. |
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