July  2020, 13(7): 2033-2045. doi: 10.3934/dcdss.2020156

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

*Corresponding author: giuseppe.viglialoro@unica.it

The authors dedicate this paper to Professor Patrizia Pucci on the occasion of her sixty-fifth birthday

Received  August 2018 Revised  December 2018 Published  November 2019

In this paper we analyze the porous medium equation
$ \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} $
where
$ \Omega $
is a bounded and smooth domain of
$ \mathbb R^N $
, with
$ N\geq 1 $
, and
$ I = [0,t^*) $
is the maximal interval of existence for
$ u $
. The constants
$ a,b,c $
are positive,
$ m,p,q $
proper real numbers larger than 1 and the equation is complemented with nonlinear boundary conditions involving the outward normal derivative of
$ u $
. Under some hypotheses on the data, including intrinsic relations between
$ m,p $
and
$ q $
, and assuming that for some positive and sufficiently regular function
$ u_0({\bf x}) $
the Initial Boundary Value Problem (IBVP) associated to (◇) possesses a positive classical solution
$ u = u({\bf x},t) $
on
$ \Omega \times I $
:
$ \triangleright $
when
$ p>q $
and in 2- and 3-dimensional domains, we determine a lower bound of
$ t^* $
for those
$ u $
becoming unbounded in
$ L^{m(p-1)}(\Omega) $
at such
$ t^* $
;
$ \triangleright $
when
$ p<q $
and in
$ N $
-dimensional settings, we establish a global existence criterion for
$ u $
.
Citation: Monica Marras, Nicola Pintus, Giuseppe Viglialoro. On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 2033-2045. doi: 10.3934/dcdss.2020156
References:
[1]

M. AidaT. TsujikawaM. EfendievA. 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.  Google Scholar

[2]

F. AndreuJ. M. MazónF. Simondon and J. Toledo, Blow-up for a class of nonlinear parabolic problems, Asymptot. Anal., 29 (2002), 143-155.   Google Scholar

[3]

D. G. Aronson, The porous medium equation, Springer Berlin Heidelberg, Berlin, Heidelberg, (1986), 1–46. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

V. A. GalaktionovS. P. KurdyumovA. 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.   Google Scholar

[8]

C. Grant, Theory of Ordinary Differential Equations, CreateSpace Independent Publishing Platform. Google Scholar

[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.  Google Scholar

[10]

H. Kielhöfer, Halbgruppen und semilineare anfangs-randwertprobleme, Manuscripta Math., 12 (1974), 121-152.  doi: 10.1007/BF01168647.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[13]

H. A. Levine, The role of critical exponents in blowup theorems, SIAM Rev., 32 (1990), 262-288.  doi: 10.1137/1032046.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. MarrasS. 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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[21]

L. E. PayneG. 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.  Google Scholar

[22]

L. E. PayneG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

L. E. PayneG. 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.  Google Scholar

[26]

P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73. Birkhäuser Verlag, Basel, 2007.  Google Scholar

[27]

P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Basel, 2007.  Google Scholar

[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.   Google Scholar

[29]

P. W. Schaefer, Blow-up phenomena in some porous medium problems, Dynam. Systems Appl., 18 (2009), 103-110.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32] J. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.   Google Scholar
[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

M. AidaT. TsujikawaM. EfendievA. 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.  Google Scholar

[2]

F. AndreuJ. M. MazónF. Simondon and J. Toledo, Blow-up for a class of nonlinear parabolic problems, Asymptot. Anal., 29 (2002), 143-155.   Google Scholar

[3]

D. G. Aronson, The porous medium equation, Springer Berlin Heidelberg, Berlin, Heidelberg, (1986), 1–46. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

V. A. GalaktionovS. P. KurdyumovA. 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.   Google Scholar

[8]

C. Grant, Theory of Ordinary Differential Equations, CreateSpace Independent Publishing Platform. Google Scholar

[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.  Google Scholar

[10]

H. Kielhöfer, Halbgruppen und semilineare anfangs-randwertprobleme, Manuscripta Math., 12 (1974), 121-152.  doi: 10.1007/BF01168647.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[13]

H. A. Levine, The role of critical exponents in blowup theorems, SIAM Rev., 32 (1990), 262-288.  doi: 10.1137/1032046.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. MarrasS. 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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[21]

L. E. PayneG. 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.  Google Scholar

[22]

L. E. PayneG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

L. E. PayneG. 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.  Google Scholar

[26]

P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73. Birkhäuser Verlag, Basel, 2007.  Google Scholar

[27]

P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Basel, 2007.  Google Scholar

[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.   Google Scholar

[29]

P. W. Schaefer, Blow-up phenomena in some porous medium problems, Dynam. Systems Appl., 18 (2009), 103-110.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32] J. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.   Google Scholar
[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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