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We consider existence and uniqueness issues for the initial value problem ofparabolic equations $\partial_{t} u = {\rm div} A \nabla u$ on the upper halfspace, with initial data in $L^p$ spaces. The coefficient matrix $A$ is assumedto be uniformly elliptic, but merely bounded measurable in space and time. Forreal coefficients and a single equation, this is an old topic for which acomprehensive theory is available, culminating in the work of Aronson. Muchless is understood for complex coefficients or systems of equations except forthe work of Lions, mainly because of the failure of maximum principles. In thispaper, we come back to this topic with new methods that do not rely on maximumprinciples. This allows us to treat systems in this generality when $p\geq 2$,or under certain assumptions such as bounded variation in the time variable (amuch weaker assumption that the usual H\"older continuity assumption) when $p<2$. We reobtain results for real coefficients, and also complement them. Forinstance, we obtain uniqueness for arbitrary $L^p$ data, $1\leq p \leq \infty$,in the class $L^\infty(0,T; L^p({\mathbb{R}}^n))$. Our approach to theexistence problem relies on a careful construction of propagators for anappropriate energy space, encompassing previous constructions. Our approach tothe uniqueness problem, the most novel aspect here, relies on a parabolicversion of the Kenig-Pipher maximal function, used in the context of ellipticequations on non-smooth domains. We also prove comparison estimates involvingconical square functions of Lusin type and prove some Fatou type results aboutnon-tangential convergence of solutions. Recent results on maximal regularityoperators in tent spaces that do not require pointwise heat kernel bounds arekey tools in this study.

On existence and uniqueness for non-autonomous parabolic Cauchy problems with rough coefficients