Well‐posedness for parabolic equations of arbitrary order

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.