Geodesic
Existence results for a fourth order partial differential equation arising in condensed matter physics
Mathematica Bohemica, Tome 140 (2015) no. 4, pp. 385-393

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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.
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.
DOI : 10.21136/MB.2015.144457
Classification : 34B16, 35G20, 35J50, 35J60, 35K25, 35K91
Keywords: higher order parabolic equation; existence of solution; blow-up in finite time; higher order elliptic equation; variational method; strongly singular boundary value problem 
Escudero, Carlos; Gazzola, Filippo; Hakl, Robert; Peral, Ireneo; Torres, Pedro José. Existence results for a fourth order partial differential equation arising in condensed matter physics. Mathematica Bohemica, Tome 140 (2015) no. 4, pp. 385-393. doi: 10.21136/MB.2015.144457
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[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

[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 | MR

[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 | MR | Zbl

[4] Escudero, C., Peral, I.: Some fourth order nonlinear elliptic problems related to epitaxial growth. J. Differ. Equations 254 (2013), 2515-2531. | DOI | MR | Zbl

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