MATHEMATICA BOHEMICA, Vol. 140, No. 4, pp. 385-393, 2015

Existence results for a fourth order partial differential equation arising in condensed matter physics

Carlos Escudero, Filippo Gazzola, Robert Hakl, Ireneo Peral, Pedro Jose Torres

Carlos Escudero, Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, C/Francisco Tomás y Valiente 7, E-28049 Madrid, Spain, e-mail: carlos.escudero@uam.es; Filippo Gazzola, Politecnico di Milano, Sede Milano Leonardo, Edificio Nave, 5° Piano, Milan, Italy, e-mail: filippo.gazzola@polimi.it; Robert Hakl, Institute of Mathematics of the Czech Academy of Sciences, Žižkova 22, 616 62 Brno, Czech Republic, e-mail: hakl@ipm.cz; Ireneo Peral, Departamento Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, C/Francisco Tomás y Valiente 7, E-28049 Madrid, Spain, e-mail: ireneo.peral@uam.es; Pedro Jose Torres, Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, Campus de Fuentenueva, 180 71 Granada, Spain, e-mail: ptorres@ugr.es

Abstract: 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.

Keywords: higher order parabolic equation; existence of solution; blow-up in finite time; higher order elliptic equation; variational method; strongly singular boundary value problem

Classification (MSC 2010): 34B16, 35G20, 35J50, 35J60, 35K25, 35K91


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