On the maximal L_p-L_q regularity of solutions to a general linear parabolic system

We show the existence of solution in the maximal L_p−L_q regularity framework to a class of symmetric parabolic problems on a uniformly C² domain in Rⁿ. Our approach consist in showing R - boundedness of families of solution operators to corresponding resolvent problems first in the whole space, then in half-space, perturbed half-space and finally, using localization arguments, on the domain. Assuming additionally boundedness of the domain we also show exponential decay of the solution. In particular, our approach does not require assuming a priori the uniform Lopatinskii - Shapiro condition.