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higher order parabolic equation; existence of solution; blow-up in finite time; higher order elliptic equation; variational method; strongly singular boundary value problem
We study a higher order parabolic partial differential equation that arises in the context of condensed matter physics. It is a fourth order semilinear equation which nonlinearity is the determinant of the Hessian matrix of the solution. We consider this model in a bounded domain of the real plane and study its stationary solutions both when the geometry of this domain is arbitrary and when it is the unit ball and the solution is radially symmetric. We also consider the initial-boundary value problem for the full parabolic equation. We summarize our results on existence of solutions in these cases and propose an open problem related to the existence of self-similar solutions.
[1] Escudero, C., Gazzola, F., Peral, I.: Global existence versus blow-up results for a fourth order parabolic PDE involving the Hessian. J. Math. Pures Appl. (9) 103 (2015), 924-957. MR 3318175
[2] Escudero, C., Hakl, R., Peral, I., Torres, P. J.: Existence and nonexistence results for a singular boundary value problem arising in the theory of epitaxial growth. Math. Methods Appl. Sci. 37 (2014), 793-807. DOI 10.1002/mma.2836 | MR 3188526
[3] Escudero, C., Hakl, R., Peral, I., Torres, P. J.: On radial stationary solutions to a model of non-equilibrium growth. Eur. J. Appl. Math. 24 (2013), 437-453. DOI 10.1017/S0956792512000484 | MR 3053654 | Zbl 1266.91051
[4] Escudero, C., Peral, I.: Some fourth order nonlinear elliptic problems related to epitaxial growth. J. Differ. Equations 254 (2013), 2515-2531. DOI 10.1016/j.jde.2012.12.012 | MR 3016212 | Zbl 1284.35435
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