Periodic solution for some parabolic degenerate equation with critical growth with respect to the gradient. (English) Zbl 1363.35193
Summary: This work is concerned with the existence of periodic solution for a parabolic degenerate equation with critical growth on the gradient and Dirichlet boundary condition. The aim will be achieved by applying some recent results. The first result that we are based on is the existence of solutions for quasilinear elliptic degenerate systems with \(L^1\) data and nonlinearity in the gradient [A. Mouida et al., Electron. J. Differ. Equ. 2013, Paper No. 142, 13 p. (2013; Zbl 1291.35094)] and the second one is the existence of weak periodic solutions of some quasilinear parabolic systems with data measures [N. Alaa and M. Iguernane, JIPAM, J. Inequal. Pure Appl. Math. 3, No. 3, Paper No. 46, 14 p. (2002; Zbl 1004.35071)].
MSC:
| 35K59 | Quasilinear parabolic equations |
| 35K57 | Reaction-diffusion equations |
| 35K55 | Nonlinear parabolic equations |
| 34C25 | Periodic solutions to ordinary differential equations |
| 74G25 | Global existence of solutions for equilibrium problems in solid mechanics (MSC2010) |
| 35D30 | Weak solutions to PDEs |
| 46T12 | Measure (Gaussian, cylindrical, etc.) and integrals (Feynman, path, Fresnel, etc.) on manifolds |
