# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [2] R. Dal Passo and S. Luckhaus, A degenerate diffusion problem not in divergence form, J. Differential Equations, 69 (1987), 1-14.  Google Scholar [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.  Google Scholar [4] S. Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geom., 33 (1991), 601-633.  Google Scholar [5] C. H. Jin and J. X. Yin, Non-extinct shrinking self-similar solutions for a class of non-divergence equations,, in press., ().   Google Scholar [6] C. T. Taam, On nonlinear diffusion equations, J. Differential Equations, 3 (1967), 482-499.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] A. Beltramo and P. Hess, On the principal eigenvalue of a periodic-parabolic operator, Comm. Partial Differential Equations, 9 (1984), 919-941.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] T. I. Seidman, Periodic solutions of a non-linear parabolic equation, J. Differential Equations, 19 (1975), 242-257.  Google Scholar [18] M. Nakao, On boundedness, periodicity, and almost periodicity of solutions of some nonlinear parabolic equations, J. Differential Equations, 19 (1975), 371-385.  Google Scholar [19] N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432. doi: 10.1512/iumj.1995.44.1994.  Google Scholar [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.  Google Scholar [21] C. H. Jin and J. X. Yin, The asymptotic behavior of a doubly degenerate parabolic equation not in divergence form,, in press., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] È. Mitidieri and S. I. Pokhozhaev, Absence of global positive solutions of quasilinear elliptic inequalities, (Russian), Dokl. Akad. Nauk., 359 (1998), 456-460.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] J. L. Vásquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [2] R. Dal Passo and S. Luckhaus, A degenerate diffusion problem not in divergence form, J. Differential Equations, 69 (1987), 1-14.  Google Scholar [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.  Google Scholar [4] S. Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geom., 33 (1991), 601-633.  Google Scholar [5] C. H. Jin and J. X. Yin, Non-extinct shrinking self-similar solutions for a class of non-divergence equations,, in press., ().   Google Scholar [6] C. T. Taam, On nonlinear diffusion equations, J. Differential Equations, 3 (1967), 482-499.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] A. Beltramo and P. Hess, On the principal eigenvalue of a periodic-parabolic operator, Comm. Partial Differential Equations, 9 (1984), 919-941.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] T. I. Seidman, Periodic solutions of a non-linear parabolic equation, J. Differential Equations, 19 (1975), 242-257.  Google Scholar [18] M. Nakao, On boundedness, periodicity, and almost periodicity of solutions of some nonlinear parabolic equations, J. Differential Equations, 19 (1975), 371-385.  Google Scholar [19] N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432. doi: 10.1512/iumj.1995.44.1994.  Google Scholar [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.  Google Scholar [21] C. H. Jin and J. X. Yin, The asymptotic behavior of a doubly degenerate parabolic equation not in divergence form,, in press., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] È. Mitidieri and S. I. Pokhozhaev, Absence of global positive solutions of quasilinear elliptic inequalities, (Russian), Dokl. Akad. Nauk., 359 (1998), 456-460.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] J. L. Vásquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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