January  2012, 17(1): 101-126. doi: 10.3934/dcdsb.2012.17.101

Periodic solutions of a non-divergent diffusion equation with nonlinear sources

1. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

2. 

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received  July 2010 Revised  March 2011 Published  October 2011

This paper is concerned with the existence of nontrivial and nonnegative periodic solutions of a doubly degenerate and singular parabolic equation in non-divergent form with nonlinear sources. We will determine exact classification for the exponent values of the source, and so, for the nonexistence of nontrivial periodic solutions, as well as the existence of those solutions with compact support, and the existence of positive periodic solutions.
Citation: Chunhua Jin, Jingxue Yin. Periodic solutions of a non-divergent diffusion equation with nonlinear sources. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 101-126. doi: 10.3934/dcdsb.2012.17.101
References:
[1]

L. J. S. Allen, Persistence and extinction in single-species reaction-diffusion models, Bull. Math. Biol., 45 (1983), 209-227.

[2]

R. Dal Passo and S. Luckhaus, A degenerate diffusion problem not in divergence form, J. Differential Equations, 69 (1987), 1-14.

[3]

B. C. Low, Resistive diffusion of force-free magnetic fields in a passive medium, Astrophys. J., 81 (1973), 209-226. doi: 10.1086/152042.

[4]

S. Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geom., 33 (1991), 601-633.

[5]

C. H. Jin and J. X. Yin, Non-extinct shrinking self-similar solutions for a class of non-divergence equations, in press.

[6]

C. T. Taam, On nonlinear diffusion equations, J. Differential Equations, 3 (1967), 482-499.

[7]

S. J. Farlow, An existence theorem for periodic solutions of a parabolic boundary value problem of the second kind, SIAM J. Appl. Math., 16 (1968), 1223-1226. doi: 10.1137/0116102.

[8]

Y. Giga and N. Mizoguchi, On time periodic solutions of the Dirichlet problem for degenerate parabolic equations of nondivergence type, J. Math. Anal. Appl., 201 (1996), 396-416. doi: 10.1006/jmaa.1996.0263.

[9]

N. Hirano and N. Mizoguchi, Positive unstable periodic solutions for superlinear parabolic equations, Proceedings of the American Mathematical Society, 123 (1995), 1487-1495. doi: 10.1090/S0002-9939-1995-1234627-2.

[10]

F. Browder, "Periodic Solutions of Nonlinear Equations of Evolution in Infinite Dimensional Spaces," 1969 Lectures in Differential Equations, Vol. 1, Van Nostrand, New York, (1969), 71-96.

[11]

B. A. Ton, Periodic solutions of nonlinear evolution equations in Banach spaces, Canad. J. Math., 23 (1971), 189-196. doi: 10.4153/CJM-1971-018-x.

[12]

Y. Wang, J. Yin and Z. Wu, Periodic solutions of evolution p-laplacian equations with nonlinear sources, J. Math. Anal. Appl., 219 (1998), 76-96. doi: 10.1006/jmaa.1997.5783.

[13]

A. Beltramo and P. Hess, On the principal eigenvalue of a periodic-parabolic operator, Comm. Partial Differential Equations, 9 (1984), 919-941.

[14]

M. J. Esteban, On periodic solutions of superlinear parabolic problems, Tran. Amer. Math. Society, 293 (1986), 171-189. doi: 10.1090/S0002-9947-1986-0814919-8.

[15]

M. J. Esteban, A remark on the existence of positive periodic solutions of superlinear parabolic problems, Proceedings of the American Mathematical Society, 102 (1988), 131-136. doi: 10.1090/S0002-9939-1988-0915730-7.

[16]

P. Quittner, Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems, NoDEA Nonlinear Differ. Equ. Appl., 11 (2004), 237-258.

[17]

T. I. Seidman, Periodic solutions of a non-linear parabolic equation, J. Differential Equations, 19 (1975), 242-257.

[18]

M. Nakao, On boundedness, periodicity, and almost periodicity of solutions of some nonlinear parabolic equations, J. Differential Equations, 19 (1975), 371-385.

[19]

N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432. doi: 10.1512/iumj.1995.44.1994.

[20]

J. X. Yin and C. H. Jin, Periodic solutions of the evolutionary $p$-Laplacian with nonlinear sources, J. Math. Anal. Appl., 368 (2010), 604-622. doi: 10.1016/j.jmaa.2010.03.006.

[21]

C. H. Jin and J. X. Yin, The asymptotic behavior of a doubly degenerate parabolic equation not in divergence form, in press.

[22]

C. H. Jin and J. X. Yin, Critical exponent of a doubly degenerate parabolic equation in non-divergence form with reaction sources, Chinese Ann. Math. Ser. A, 30 (2009), 525-538; translation in Chinese J. Contemp. Math., 30 (2009), 311-328.

[23]

J. García-Melián and J. Sabina de Lis, Maximum and comparison principles for operators involving the $p$-Laplacian, J. Math. Anal. Appl., 218 (1998), 49-65. doi: 10.1006/jmaa.1997.5732.

[24]

C. Azizieh and Ph. Clément, A priori estimates and continuation methods for positive solutions of $p$-Laplace equations, J. Differential Equations, 179 (2002), 213-245.

[25]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645.

[26]

È. Mitidieri and S. I. Pokhozhaev, Absence of global positive solutions of quasilinear elliptic inequalities, (Russian), Dokl. Akad. Nauk., 359 (1998), 456-460.

[27]

È. Mitidieri and S. I. Pokhozhaev, Absence of positive solutions for quasilinear elliptic problems in $\mathbb R^N$, Proc. Steklov Inst. Math., 227 (1999), 186-216.

[28]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.

[29]

J. L. Vásquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.

[30]

M. Ôtani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Func. Anal., 76 (1988), 140-159. doi: 10.1016/0022-1236(88)90053-5.

[31]

Peter Lindqvist, On the equation div $(|\nabla u|^{p-2}\nabla u)+ \lambda| u|^{p-2}u=0$, Proc. Amer. Math. Soc., 109 (1990), 157-164. doi: 10.1090/S0002-9939-1990-1007505-7.

show all references

References:
[1]

L. J. S. Allen, Persistence and extinction in single-species reaction-diffusion models, Bull. Math. Biol., 45 (1983), 209-227.

[2]

R. Dal Passo and S. Luckhaus, A degenerate diffusion problem not in divergence form, J. Differential Equations, 69 (1987), 1-14.

[3]

B. C. Low, Resistive diffusion of force-free magnetic fields in a passive medium, Astrophys. J., 81 (1973), 209-226. doi: 10.1086/152042.

[4]

S. Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geom., 33 (1991), 601-633.

[5]

C. H. Jin and J. X. Yin, Non-extinct shrinking self-similar solutions for a class of non-divergence equations, in press.

[6]

C. T. Taam, On nonlinear diffusion equations, J. Differential Equations, 3 (1967), 482-499.

[7]

S. J. Farlow, An existence theorem for periodic solutions of a parabolic boundary value problem of the second kind, SIAM J. Appl. Math., 16 (1968), 1223-1226. doi: 10.1137/0116102.

[8]

Y. Giga and N. Mizoguchi, On time periodic solutions of the Dirichlet problem for degenerate parabolic equations of nondivergence type, J. Math. Anal. Appl., 201 (1996), 396-416. doi: 10.1006/jmaa.1996.0263.

[9]

N. Hirano and N. Mizoguchi, Positive unstable periodic solutions for superlinear parabolic equations, Proceedings of the American Mathematical Society, 123 (1995), 1487-1495. doi: 10.1090/S0002-9939-1995-1234627-2.

[10]

F. Browder, "Periodic Solutions of Nonlinear Equations of Evolution in Infinite Dimensional Spaces," 1969 Lectures in Differential Equations, Vol. 1, Van Nostrand, New York, (1969), 71-96.

[11]

B. A. Ton, Periodic solutions of nonlinear evolution equations in Banach spaces, Canad. J. Math., 23 (1971), 189-196. doi: 10.4153/CJM-1971-018-x.

[12]

Y. Wang, J. Yin and Z. Wu, Periodic solutions of evolution p-laplacian equations with nonlinear sources, J. Math. Anal. Appl., 219 (1998), 76-96. doi: 10.1006/jmaa.1997.5783.

[13]

A. Beltramo and P. Hess, On the principal eigenvalue of a periodic-parabolic operator, Comm. Partial Differential Equations, 9 (1984), 919-941.

[14]

M. J. Esteban, On periodic solutions of superlinear parabolic problems, Tran. Amer. Math. Society, 293 (1986), 171-189. doi: 10.1090/S0002-9947-1986-0814919-8.

[15]

M. J. Esteban, A remark on the existence of positive periodic solutions of superlinear parabolic problems, Proceedings of the American Mathematical Society, 102 (1988), 131-136. doi: 10.1090/S0002-9939-1988-0915730-7.

[16]

P. Quittner, Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems, NoDEA Nonlinear Differ. Equ. Appl., 11 (2004), 237-258.

[17]

T. I. Seidman, Periodic solutions of a non-linear parabolic equation, J. Differential Equations, 19 (1975), 242-257.

[18]

M. Nakao, On boundedness, periodicity, and almost periodicity of solutions of some nonlinear parabolic equations, J. Differential Equations, 19 (1975), 371-385.

[19]

N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432. doi: 10.1512/iumj.1995.44.1994.

[20]

J. X. Yin and C. H. Jin, Periodic solutions of the evolutionary $p$-Laplacian with nonlinear sources, J. Math. Anal. Appl., 368 (2010), 604-622. doi: 10.1016/j.jmaa.2010.03.006.

[21]

C. H. Jin and J. X. Yin, The asymptotic behavior of a doubly degenerate parabolic equation not in divergence form, in press.

[22]

C. H. Jin and J. X. Yin, Critical exponent of a doubly degenerate parabolic equation in non-divergence form with reaction sources, Chinese Ann. Math. Ser. A, 30 (2009), 525-538; translation in Chinese J. Contemp. Math., 30 (2009), 311-328.

[23]

J. García-Melián and J. Sabina de Lis, Maximum and comparison principles for operators involving the $p$-Laplacian, J. Math. Anal. Appl., 218 (1998), 49-65. doi: 10.1006/jmaa.1997.5732.

[24]

C. Azizieh and Ph. Clément, A priori estimates and continuation methods for positive solutions of $p$-Laplace equations, J. Differential Equations, 179 (2002), 213-245.

[25]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645.

[26]

È. Mitidieri and S. I. Pokhozhaev, Absence of global positive solutions of quasilinear elliptic inequalities, (Russian), Dokl. Akad. Nauk., 359 (1998), 456-460.

[27]

È. Mitidieri and S. I. Pokhozhaev, Absence of positive solutions for quasilinear elliptic problems in $\mathbb R^N$, Proc. Steklov Inst. Math., 227 (1999), 186-216.

[28]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.

[29]

J. L. Vásquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.

[30]

M. Ôtani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Func. Anal., 76 (1988), 140-159. doi: 10.1016/0022-1236(88)90053-5.

[31]

Peter Lindqvist, On the equation div $(|\nabla u|^{p-2}\nabla u)+ \lambda| u|^{p-2}u=0$, Proc. Amer. Math. Soc., 109 (1990), 157-164. doi: 10.1090/S0002-9939-1990-1007505-7.

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