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Periodic \(p\)-Laplacian with nonlocal terms. (English) Zbl 1110.35039

The authors study the existence of non-trivial solutions for a periodic \(p\)-Laplacian with nonlocal term of the form \[ u_t - \text{div }(| \nabla u| ^{p-2}\nabla u) = (m - \Phi [u])u,\quad \text{in } Q_T \] with the boundary value condition \[ u(x,t) = 0,\quad \text{for } (x,t) \in \partial\Omega \times (0,T) \] and periodic in time \[ u(x,0) = u(x,T),\quad \text{for } x \in \Omega, \] where \(p>2\), \(\Omega\) is a bound domain in \(\mathbb R^n\) with smooth boundary, \(Q_T = \Omega\times (0,T)\). The nonlocal term \(\Phi [u]: L^2 \to R\) is a bounded continuous functional. Under suitable assumptions on the nonlocal term \(\Phi\), the authors prove the existence of a generalized solution to the problem based on the theory of Leray-Schauder degree.

MSC:

35K65 Degenerate parabolic equations
35B10 Periodic solutions to PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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[1] Allegretto, W.; Nistri, P., Existence and optimal control for periodic parabolic equations with nonlocal term, IMA J. Math. Control Inform., 16, 1, 43-58 (1999) · Zbl 0926.49002
[2] Wang, Y.; Yin, J.; Wu, Z., Periodic solutions of evolution \(p\)-Laplacian equations with nonlinear sources, J. Math. Anal. Appl., 219, 1, 76-96 (1998) · Zbl 0932.35127
[3] Wu, Z. Q.; Zhao, Q. N.; Yin, J. X.; Li, H. L., Nonlinear Diffusion Equations (2001), World Scientific: World Scientific Singapore
[4] Dibenedetto, E., Degenerate Parabolic Equations (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0794.35090
[5] Wang, Y.; Wu, Z.; Yin, J., Periodic solutions of evolution \(p\)-Laplacian equations with weakly nonlinear sources, Int. J. Math. Game Theory Algebra, 10, 1, 67-77 (2000) · Zbl 1031.35085
[6] Pao, C. V., Nonlinear Parabolic and Elliptic Equations (1992), Plenum Press: Plenum Press New York · Zbl 0780.35044
[7] Calsina, A.; Perello, C., Equations for biological evolution, Proc. Roy. Soc. Edinburgh Sect. A, 125, 939-958 (1995) · Zbl 0834.92018
[8] Calsina, A.; Perello, C.; Saldana, J., Non-local reaction-diffusion equations modelling predator-prey coevolution, Publ. Mat., 32, 315-325 (1994) · Zbl 0834.92019
[9] Crema, J.; Boldrini, J. L., More on forced periodic solutions of quasi-parabolic equations, Cadernos de Matemática, 01, 71-88 (2000)
[10] Boldrini, J. L.; Crema, J., On forced periodic solutions of superlinear quasi-parabolic problems, Electron. J. Differential Equations, 1998, 14, 1-18 (1998) · Zbl 0899.35051
[11] Crema, J.; Boldrini, J. L., Periodic solutions of quasilinear equations with discontinuous perturbations, Cadernos de Matemática, 01, 53-69 (2000)
[12] Húska, J., Periodic solutions in superlinear parabolic problems, Acta Math. Univ. Comenianae, LXXI, 1, 19-26 (2002) · Zbl 1049.35026
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