AN L-p-LIPSCHITZ THEORY FOR PARABOLIC EQUATIONS WITH TIME MEASURABLE PSEUDO-DIFFERENTIAL OPERATORS

Abstract

In this article we prove the existence and uniqueness of a (weak) solution u in L-p ((0; T); Lambda(gamma) +m) to the Cauchy problem partial derivative u/partial derivative t (t; x) = psi(t; i del) u (t; x) + f (t; x); (t; x) is an element of (0; T) R-d u (0; x) = 0; (1) where d is an element of N, p is an element of (1;infinity], Lambda(gamma+m) 2 (0;1), Lambda(gamma+m) is the Lipschitz space on R-d whose order is (gamma+m), f is an element of L-p ((0; T);Lambda(gamma) ), and psi(t; i del) is a time measurable pseudo-di ff erential operator whose symbol is (t; ), i. e. (t; ir) u (t; x) = F [(t; ) F [u (t; )] ()] (x); with the assumptions < [(t; )] and jDff (t; ) j : Furthermore, we show Z T 0 ku (t; ) k Lambda(gamma+m) dt <= N Z T 0 kf (t; ) k p m dt; (2) where N is a positive constant depending only on d,Lambda(gamma+m), and T, The unique solvability of equation (1) in Lp -H older space is also considered. More precisely, for any f 2 Lp ((0; T); Cn+ff), there exists a unique solution u 2 Lp ((0; T); C+n+ff (Rd)) to equation (1) and for this solution u, Z T 0 ku (t; ) k p C+n+ff dt N Z T 0 kf (t; ) k p Cn+ff dt; (3) where n is an element of Z(+), alpha is an element of (0; 1), and gamma + alpha is not an element of Z(+)