Resumo: We will present existence and uniqueness results for solutions that verify an energy equality for a semilinear equation in non-cylindrical domains (cf. [1]) based on some ideas from [2, 3]. In the proof, we use a penalty method for a more regular problem and go to the limit.
[1] P. E. Kloeden, P. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations 244 (2008), 2062–2090.
[2] M. L. Bernardi, G. A. Pozzi and G. Savaré, Variational equations of Schroedinger-type in non-cylindrical domains, J. Differential Equations 171 (2001), 63–87
[3] S. Bonaccorsi and G. Guatteri, A variational approach to evolution problems with variable domains, J. Differential Equations 175 (2001), 51–70.