March  2021, 41(3): 1297-1318. doi: 10.3934/dcds.2020318

Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium

1. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro - RJ, 21941-909, Brazil

2. 

Departamento de Matemática, Universidade de Brasília, Brasília - DF, 70910-900, Brazil

* Corresponding author: Juliana Fernandes

Received  December 2019 Revised  May 2020 Published  March 2021 Early access  August 2020

Fund Project: The first author was partially supported by FAPERJ. The second author was partially supported by FAPDF, CAPES, and CNPq grant 308378/2017 -2

The present paper is on the existence and behaviour of solutions for a class of semilinear parabolic equations, defined on a bounded smooth domain and assuming a nonlinearity asymptotically linear at infinity. The behavior of the solutions when the initial data varies in the phase space is analyzed. Global solutions are obtained, which may be bounded or blow-up in infinite time (grow-up). The main tools are the comparison principle and variational methods. In particular, the Nehari manifold is used to separate the phase space into regions of initial data where uniform boundedness or grow-up behavior of the semiflow may occur. Additionally, some attention is paid to initial data at high energy level.

Citation: Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.

[2]

J. M. ArrietaA. N. Carvalho and A. Rodríguez-Bernal, Attractors for parabolic problems with nonlinear boundary conditions. Uniform bounds, Comm. Partial Differential Equations, 25 (2000), 1-37.  doi: 10.1080/03605300008821506.

[3]

A. V. Babin and M. I. Vishik, Attractor in Evolutionary Equations, Studies in Mathemathics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[4]

P. BartoloV. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong" resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012.  doi: 10.1016/0362-546X(83)90115-3.

[5]

N. Ben-Gal, Grow-Up Solutions and Heteroclinics to Infinity for Scalar Parabolic PDEs, Ph.D thesis, Brown University, 2010.

[6]

A. Biswas and S. Konar, Introduction to Non-Kerr Law Optical Solitons, Applied Mathematics and Nonlinear Science Series, Chapman & Hall/CRC, Boca Raton, FL, 2007 doi: 10.1201/9781420011401.

[7]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[8]

G. Cerami, Un criterio di esistenza per i punti critici su varietà illimitate, Rend. Accad. Sc. Lett. Inst. Lombardo, 112 (1978), 332-336. 

[9]

M. ChenX.-Y. Chen and J. K. Hale, Structural stability for time periodic one-dimensional parabolic equations, J. Differential Equations, 96 (1992), 355-418.  doi: 10.1016/0022-0396(92)90159-K.

[10]

V. V. Chepyzhov and A. Y. Goritskiĭ, Unbounded attractors of evolution equations, in Properties of Global Attractors of Partial Differential Equations, , Adv. Soviet Math., 10, Amer. Math. Soc., Providence, RI, 1992, 85–128.

[11]

M. Clapp and L. A. Maia, A positive bound state for an asymptotically linear or superlinear Schrödinger equation, J. Differential Equations, 260 (2016), 3173-3192.  doi: 10.1016/j.jde.2015.09.059.

[12]

D. G. Costa and C. A. Magalhães, Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal., 23 (1994), 1401-1412.  doi: 10.1016/0362-546X(94)90135-X.

[13]

E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations, 74 (1988), 120-156.  doi: 10.1016/0022-0396(88)90021-6.

[14]

F. DicksteinN. MizoguchiP. Souplet and F. Weissler, Transversality of stable and Nehari manifolds for a semilinear heat equation, Calc. Var. Partial Differential Equations, 42 (2011), 547-562.  doi: 10.1007/s00526-011-0397-8.

[15]

F. Gazolla and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differential Integral Equations, 18 (2005), 961-990. 

[16]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Fundamental Principles of Mathematical Sciences, 224, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[17]

M. Grossi, A uniqueness result for a semilinear elliptic equation in symmetric domains, Adv. Differential Equations, 5 (2000), 193-121. 

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089647.

[19]

H. Hofer, The topological degree at a critical point of mountain-pass type, in Nonlinear Functional Analysis and its Applications, Proc. Sympos. Pure Math., 45, Amer. Math. Soc., Providence, RI, 1986,501–509.

[20]

H. Hoshino and Y. Yamada, Solvability and soothing effect for semilinear parabolic equations, Funkcial. Ekvac., 34 (1991), 475-492. 

[21] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418.
[22]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.

[23]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[24]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[25]

J. Pimentel and C. Rocha, A permutation related to non-compact global attractors for slowly non-dissipative systems, J. Dynam. Differential Equations, 28 (2016), 1-28.  doi: 10.1007/s10884-014-9414-x.

[26]

P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem, Acta Math. Univ. Comenian. (N.S.), 68 (1999), 195-203. 

[27]

P. Quittner, Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, Houston J. Math., 29 (2003), 757-799. 

[28]

C. W. Steele, Numerical Computation of Electric and Magnetic Fields, Chapman & Hall, New York; International Thomson Publishing, London, 1997. doi: 10.1007/978-1-4615-6035-7.

[29]

G. I. StegemanD. N. Christodoulides and M. Segev, Optical spatial solitons: Historical Perspectives, IEEE J. Selected Topics Quantum Electronics, 6 (2000), 1419-1427.  doi: 10.1109/2944.902197.

[30]

M. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Japon., 17 (1972), 173-193. 

[31]

F. B. Weissler, Semilinear evolution equations in Banach spaces, J. Functional Analysis, 32 (1979), 277-296.  doi: 10.1016/0022-1236(79)90040-5.

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.

[2]

J. M. ArrietaA. N. Carvalho and A. Rodríguez-Bernal, Attractors for parabolic problems with nonlinear boundary conditions. Uniform bounds, Comm. Partial Differential Equations, 25 (2000), 1-37.  doi: 10.1080/03605300008821506.

[3]

A. V. Babin and M. I. Vishik, Attractor in Evolutionary Equations, Studies in Mathemathics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[4]

P. BartoloV. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong" resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012.  doi: 10.1016/0362-546X(83)90115-3.

[5]

N. Ben-Gal, Grow-Up Solutions and Heteroclinics to Infinity for Scalar Parabolic PDEs, Ph.D thesis, Brown University, 2010.

[6]

A. Biswas and S. Konar, Introduction to Non-Kerr Law Optical Solitons, Applied Mathematics and Nonlinear Science Series, Chapman & Hall/CRC, Boca Raton, FL, 2007 doi: 10.1201/9781420011401.

[7]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[8]

G. Cerami, Un criterio di esistenza per i punti critici su varietà illimitate, Rend. Accad. Sc. Lett. Inst. Lombardo, 112 (1978), 332-336. 

[9]

M. ChenX.-Y. Chen and J. K. Hale, Structural stability for time periodic one-dimensional parabolic equations, J. Differential Equations, 96 (1992), 355-418.  doi: 10.1016/0022-0396(92)90159-K.

[10]

V. V. Chepyzhov and A. Y. Goritskiĭ, Unbounded attractors of evolution equations, in Properties of Global Attractors of Partial Differential Equations, , Adv. Soviet Math., 10, Amer. Math. Soc., Providence, RI, 1992, 85–128.

[11]

M. Clapp and L. A. Maia, A positive bound state for an asymptotically linear or superlinear Schrödinger equation, J. Differential Equations, 260 (2016), 3173-3192.  doi: 10.1016/j.jde.2015.09.059.

[12]

D. G. Costa and C. A. Magalhães, Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal., 23 (1994), 1401-1412.  doi: 10.1016/0362-546X(94)90135-X.

[13]

E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations, 74 (1988), 120-156.  doi: 10.1016/0022-0396(88)90021-6.

[14]

F. DicksteinN. MizoguchiP. Souplet and F. Weissler, Transversality of stable and Nehari manifolds for a semilinear heat equation, Calc. Var. Partial Differential Equations, 42 (2011), 547-562.  doi: 10.1007/s00526-011-0397-8.

[15]

F. Gazolla and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differential Integral Equations, 18 (2005), 961-990. 

[16]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Fundamental Principles of Mathematical Sciences, 224, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[17]

M. Grossi, A uniqueness result for a semilinear elliptic equation in symmetric domains, Adv. Differential Equations, 5 (2000), 193-121. 

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089647.

[19]

H. Hofer, The topological degree at a critical point of mountain-pass type, in Nonlinear Functional Analysis and its Applications, Proc. Sympos. Pure Math., 45, Amer. Math. Soc., Providence, RI, 1986,501–509.

[20]

H. Hoshino and Y. Yamada, Solvability and soothing effect for semilinear parabolic equations, Funkcial. Ekvac., 34 (1991), 475-492. 

[21] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418.
[22]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.

[23]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[24]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[25]

J. Pimentel and C. Rocha, A permutation related to non-compact global attractors for slowly non-dissipative systems, J. Dynam. Differential Equations, 28 (2016), 1-28.  doi: 10.1007/s10884-014-9414-x.

[26]

P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem, Acta Math. Univ. Comenian. (N.S.), 68 (1999), 195-203. 

[27]

P. Quittner, Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, Houston J. Math., 29 (2003), 757-799. 

[28]

C. W. Steele, Numerical Computation of Electric and Magnetic Fields, Chapman & Hall, New York; International Thomson Publishing, London, 1997. doi: 10.1007/978-1-4615-6035-7.

[29]

G. I. StegemanD. N. Christodoulides and M. Segev, Optical spatial solitons: Historical Perspectives, IEEE J. Selected Topics Quantum Electronics, 6 (2000), 1419-1427.  doi: 10.1109/2944.902197.

[30]

M. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Japon., 17 (1972), 173-193. 

[31]

F. B. Weissler, Semilinear evolution equations in Banach spaces, J. Functional Analysis, 32 (1979), 277-296.  doi: 10.1016/0022-1236(79)90040-5.

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