Asymptotic Analysis - Volume 35, issue 1

Authors: Borisov, Denis I.

Article Type: Research Article

Abstract: Model two‐dimensional singular perturbed eigenvalue problem for Laplacian with frequently alternating type of boundary condition is considered. Complete two‐parametrical asymptotics for the eigenelements are constructed.

Citation: Asymptotic Analysis, vol. 35, no. 1, pp. 1-26, 2003

Authors: Cicalese, Marco | Trombetti, Cristina

Article Type: Research Article

Citation: Asymptotic Analysis, vol. 35, no. 1, pp. 27-40, 2003

Authors: Marchi, Claudio

Article Type: Research Article

Abstract: This paper deals with the well‐posedness of the Cauchy problem for higher order parabolic equations. Our aim is to show existence and uniqueness of the solution belonging to a suitable weighted Sobolev space, provided that the weight function satisfies some appropriate differential inequality (the “dual” one). Under some restrictions on the growth of the coefficients as |x|→∞ (see conditions (A1 )–(A4 ) below), we obtain a simplified dual inequality; we deduce a well‐posedness result which extends results known in literature. In Appendix, dropping any growth condition on the coefficients, we extend our result, but the dual inequality is complicated.

Citation: Asymptotic Analysis, vol. 35, no. 1, pp. 41-64, 2003

Authors: Akhmetov, Denis R. | Lavrentiev, Jr., Mikhail M. | Spigler, Renato

Article Type: Research Article

Abstract: Linear parabolic partial differential equations with a small parameter multiplying some of the higher space derivatives are considered, in the limiting case when such parameter vanishes. A number of different boundary‐value problems for singularly perturbed equations are examined. Such problems are unified by the rather unexpected property that no boundary‐layers are required, despite of the presence of the small vanishing parameter. The high points are the following. First, solvability theorems for some classes of ultraparabolic problems have been established. Second, the boundary conditions to be imposed to obtain well‐posed problems do not depend on the sign of the coefficient multiplying …the time‐like derivative. Third, all coefficients are allowed to depend on all space and time variables. These results, in part, have been established by imposing suitably generalized compatibility conditions on coefficients and data. Show more

Citation: Asymptotic Analysis, vol. 35, no. 1, pp. 65-89, 2003