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Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity. (English) Zbl 1358.35063

The authors investigate the existence and stability of \(T\)-periodic solutions of \[ u_t-k\partial_t \Delta u=m(t)u\log(\left|u\right|),\quad x\in\Omega,\;t\in\mathbb{R}; \qquad u(x,t)=0, \quad x\in\partial\Omega,\;t\in\mathbb{R} \] under the following assumptions: \(T>0\), \(k\in\{0,1\}\) (\(k=1\) in pseudo-parabolic case), \(\Omega\subset \mathbb{R}^n\) bounded domain with smooth boundary, \(m\) is a smooth, positive and T -periodic function. They find that existence results are similar to those for right hand sides \(m(t)u^p\), \(p>1\), whereas there are differences between the instability results for the linear case, the ‘logarithmic case’, and the ‘power case’. The existence proofs rely on regularity results, a priori estimates, and topological degree arguments; the stability results use the comparison method.

MSC:

35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
35K58 Semilinear parabolic equations
35B10 Periodic solutions to PDEs
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[1] Beltramo, A.; Hess, P., On the principal eigenvalue of a periodic-parabolic operator, Comm. Partial Differential Equations, 9, 919-941 (1984) · Zbl 0563.35033
[2] Cao, Y.; Yin, J. X., Small perturbation of a semilinear pseudo-parabolic equation, Discrete Contin. Dyn. Syst., 36, 2, 631-642 (2016) · Zbl 1322.35090
[3] Cao, Y.; Yin, J. X.; Jin, C. H., A periodic problem of a semilinear pseudoparabolic equation, Abstr. Appl. Anal., 2011, Article 363579 pp. (2011), 27 pp · Zbl 1252.35171
[4] Cao, Y.; Yin, J. X.; Wang, C. P., Cauchy problems of semilinear pseudo-parabolic equations, J. Differential Equations, 246, 4568-4590 (2009) · Zbl 1179.35178
[5] Chen, H.; Luo, P.; Liu, G., Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422, 84-98 (2015) · Zbl 1302.35071
[6] Chen, H.; Tian, S. Y., Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 285, 4424-4442 (2015) · Zbl 1370.35190
[7] Esteban, M. J., On periodic solutions of superlinear parabolic problems, Trans. Amer. Math. Soc., 293, 171-189 (1986) · Zbl 0619.35058
[8] Esteban, M. J., A remark on the existence of positive periodic solutions of superlinear parabolic problems, Proc. Amer. Math. Soc., 102, 131-136 (1988) · Zbl 0653.35039
[9] Hirano, N.; Mizoguchi, N., A priori bounds for positive solutions of non-linear elliptic equations, Comm. Partial Differential Equations, 6, 883-901 (1981)
[10] Gidas, B.; Spruck, J., Positive unstable periodic solutions for superlinear parabolic equations, Proc. Amer. Math. Soc., 123, 5, 1487-1495 (1995) · Zbl 0828.35063
[11] Kaikina, E. I.; Naumkin, P. I.; Shishmarev, I. A., The Cauchy problem for a Sobolev type equation with power like nonlinearity, Izv. Math., 69, 1, 59-111 (2005) · Zbl 1079.35024
[12] Khomrutai, S., Global well-posedness and grow-up rate of solutions for a sublinear pseudoparabolic equation, J. Differential Equations, 260, 4, 3598-3657 (2015) · Zbl 1336.35204
[13] Ladyženskaya, O. A.; Solonnikov, V. A.; Uraltseva, N. N., Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr., vol. 23 (1968), AMS: AMS Providence, RI
[14] Li, Y. H.; Cao, Y.; Yin, J. X.; Wang, Y. F., Time periodic solutions for a viscous diffusion equation with nonlinear periodic sources, Electron. J. Qual. Theory Differ. Equ., 2011, Article 10 pp. (2011), 19 pp
[15] Li, Z. P.; Du, W. J., Cauchy problems of pseudo-parabolic equations with inhomogeneous terms, Z. Angew. Math. Phys., 66, 3181-3203 (2015) · Zbl 1330.35225
[16] Lieb, E.; Loss, M., Analysis, Graduate Studies in Mathematics, vol. 14 (2001)
[17] Matahashi, T.; Tsutsumi, M., On a periodic problem for pseudo-parabolic equations of Sobolev-Galpern type, Math. Jpn., 22, 535-553 (1978) · Zbl 0384.35006
[18] Ôtani, M., Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Funct. Anal., 76, 140-159 (1988) · Zbl 0662.35047
[19] Quittner, P., Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems, NoDEA Nonlinear Differential Equations Appl., 11, 237-258 (2004) · Zbl 1058.35120
[20] Xu, R. Z.; Su, J., Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264, 2732-2763 (2013) · Zbl 1279.35065
[21] Yang, C. X.; Cao, Y.; Zheng, S. N., Life span and second critical exponent for semilinear pseudo-parabolic equation, J. Differential Equations, 253, 3286-3303 (2012) · Zbl 1278.35140
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