Well‐posedness for parabolic equations of arbitrary order

Article type: Research Article

Authors: Marchi, Claudio

Affiliations: Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Piazzale A. Moro 5, 00185 Rome, Italy E‐mail: marchi@math.unipd.it, claudiomarchi@tiscali.it

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.

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