doi: 10.3934/dcdsb.2021280
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Exponential decay for quasilinear parabolic equations in any dimension

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

2. 

Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 41566, Republic of Korea

* Corresponding author

Received  September 2021 Early access November 2021

We estimate decay rates of solutions to the initial-boundary value problem for a class of quasilinear parabolic equations in any dimension. Such decay rates depend only on the constitutive relations, spatial domain, and range of the initial function.

Citation: Jian-Wen Sun, Seonghak Kim. Exponential decay for quasilinear parabolic equations in any dimension. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021280
References:
[1]

X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations, 78 (1989), 160-190.  doi: 10.1016/0022-0396(89)90081-8.

[2]

M. D. Donsker and S. R. S. Varadhan, On the principal eigenvalue of second-order elliptic differential operators, Comm. Pure Appl. Math., 29 (1976), 595-621.  doi: 10.1002/cpa.3160290606.

[3]

L. C. Evans, Partial Differential Equations. Second Edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[4]

E. Feireisl and F. Simondon, Convergence for degenerate parabolic equations, J. Differential Equations, 152 (1999), 439-466.  doi: 10.1006/jdeq.1998.3545.

[5]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.

[6]

V. A. Galaktionov and J. L. Vázquez, Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations, Arch. Rational Mech. Anal., 129 (1995), 225-244.  doi: 10.1007/BF00383674.

[7]

M. Gokieli and F. Simondon, Convergence to equilibrium for a parabolic problem with mixed boundary conditions in one space dimension, J. Evol. Equ., 3 (2003), 523-548.  doi: 10.1007/s00028-003-0085-z.

[8]

J. K. Hale and G. Raugel, Convergence in gradient-like systems with applications to PDE, Z. Angew. Math. Phys., 43 (1992), 63-124.  doi: 10.1007/BF00944741.

[9]

A. HarauxM. Jendoubi and O. Kavian, Rate of decay to equilibrium in some semilinear parabolic equations, J. Evol. Equ., 3 (2003), 463-484.  doi: 10.1007/s00028-003-1112-8.

[10]

A. Haraux and P. Poláčik, Convergence to a positive equilibrium for some nonlinear evolution equations in a ball, Acta Math. Univ. Comenian. (N.S.), 61 (1992), 129-141. 

[11]

S. Kim, Rate of convergence for one-dimensional quasilinear parabolic problem and its applications, J. Differential Equations, 264 (2018), 82-97.  doi: 10.1016/j.jde.2017.08.069.

[12]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type. (Russian), Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968.

[13]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, N.J., 1996. doi: 10.1142/3302.

[14]

H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ., 18 (1978), 221-227.  doi: 10.1215/kjm/1250522572.

[15]

L. E. Payne and G. A. Philippin, Decay bounds for solutions of second order parabolic problems and their derivatives, Math. Models Methods Appl. Sci., 5 (1995), 95-110.  doi: 10.1142/S0218202595000061.

[16]

L. E. Payne and G. A. Philippin, Decay bounds for solutions of second order parabolic problems and their derivatives. Ⅱ, Math. Inequal. Appl., 7 (2004), 543-549.  doi: 10.7153/mia-07-55.

[17]

L. E. Payne and G. A. Philippin, Decay bounds for solutions of second order parabolic problems and their derivatives. Ⅲ, Z. Anal. Anwendungen, 23 (2004), 809-818.  doi: 10.4171/ZAA/1224.

[18]

L. E. PayneG. A. Philippin and S. Vernier Piro, Decay bounds for solutions of second order parabolic problems and their derivatives. Ⅳ, Appl. Anal., 85 (2006), 293-302.  doi: 10.1080/00036810500276530.

[19]

M. P. VishnevskiĭT. I. Zelenyak and M. M. Lavrent'ev Jr., The behavior of solutions of nonlinear parabolic equations for a large time value, Siberian Math. J., 36 (1995), 435-453.  doi: 10.1007/BF02109832.

[20]

T. I. Zelenjak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable. (Russian), Differencial'nye Uravnenija, 4 (1968), 34-45. 

show all references

References:
[1]

X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations, 78 (1989), 160-190.  doi: 10.1016/0022-0396(89)90081-8.

[2]

M. D. Donsker and S. R. S. Varadhan, On the principal eigenvalue of second-order elliptic differential operators, Comm. Pure Appl. Math., 29 (1976), 595-621.  doi: 10.1002/cpa.3160290606.

[3]

L. C. Evans, Partial Differential Equations. Second Edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[4]

E. Feireisl and F. Simondon, Convergence for degenerate parabolic equations, J. Differential Equations, 152 (1999), 439-466.  doi: 10.1006/jdeq.1998.3545.

[5]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.

[6]

V. A. Galaktionov and J. L. Vázquez, Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations, Arch. Rational Mech. Anal., 129 (1995), 225-244.  doi: 10.1007/BF00383674.

[7]

M. Gokieli and F. Simondon, Convergence to equilibrium for a parabolic problem with mixed boundary conditions in one space dimension, J. Evol. Equ., 3 (2003), 523-548.  doi: 10.1007/s00028-003-0085-z.

[8]

J. K. Hale and G. Raugel, Convergence in gradient-like systems with applications to PDE, Z. Angew. Math. Phys., 43 (1992), 63-124.  doi: 10.1007/BF00944741.

[9]

A. HarauxM. Jendoubi and O. Kavian, Rate of decay to equilibrium in some semilinear parabolic equations, J. Evol. Equ., 3 (2003), 463-484.  doi: 10.1007/s00028-003-1112-8.

[10]

A. Haraux and P. Poláčik, Convergence to a positive equilibrium for some nonlinear evolution equations in a ball, Acta Math. Univ. Comenian. (N.S.), 61 (1992), 129-141. 

[11]

S. Kim, Rate of convergence for one-dimensional quasilinear parabolic problem and its applications, J. Differential Equations, 264 (2018), 82-97.  doi: 10.1016/j.jde.2017.08.069.

[12]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type. (Russian), Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968.

[13]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, N.J., 1996. doi: 10.1142/3302.

[14]

H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ., 18 (1978), 221-227.  doi: 10.1215/kjm/1250522572.

[15]

L. E. Payne and G. A. Philippin, Decay bounds for solutions of second order parabolic problems and their derivatives, Math. Models Methods Appl. Sci., 5 (1995), 95-110.  doi: 10.1142/S0218202595000061.

[16]

L. E. Payne and G. A. Philippin, Decay bounds for solutions of second order parabolic problems and their derivatives. Ⅱ, Math. Inequal. Appl., 7 (2004), 543-549.  doi: 10.7153/mia-07-55.

[17]

L. E. Payne and G. A. Philippin, Decay bounds for solutions of second order parabolic problems and their derivatives. Ⅲ, Z. Anal. Anwendungen, 23 (2004), 809-818.  doi: 10.4171/ZAA/1224.

[18]

L. E. PayneG. A. Philippin and S. Vernier Piro, Decay bounds for solutions of second order parabolic problems and their derivatives. Ⅳ, Appl. Anal., 85 (2006), 293-302.  doi: 10.1080/00036810500276530.

[19]

M. P. VishnevskiĭT. I. Zelenyak and M. M. Lavrent'ev Jr., The behavior of solutions of nonlinear parabolic equations for a large time value, Siberian Math. J., 36 (1995), 435-453.  doi: 10.1007/BF02109832.

[20]

T. I. Zelenjak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable. (Russian), Differencial'nye Uravnenija, 4 (1968), 34-45. 

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