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\-->infinity (see conditions (A(1))-(A(4)) 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.
Well-posedness for parabolic equations of arbitrary order
MARCHI, CLAUDIO
2003
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\-->infinity (see conditions (A(1))-(A(4)) 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.
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