Preprint
Inserted: 25 sep 2026
Last Updated: 25 sep 2026
Pages: 23
Year: 2026
Abstract:
We establish Morrey-type conditions ensuring that a finite signed
Radon measure belongs to the dual of the energy space
of solutions to nonlinear parabolic equations of $p$-Laplacian type. More precisely, for the
parabolic cylinders $Q_{r,r^p}(z):=B_r(x)\times(t-r^p,t+r^p)$,
we prove that
\[
\mu
\bigl(Q_{r,r^p}(z)\cap(\Omega\times (0,T))\bigr)
\leq Mr^{n+p-\vartheta},
\qquad
\vartheta<p,
\]
implies duality, and this threshold is sharp in general. More generally, for cylinders with
time length proportional to $r^q$, $q>1$, the sufficient threshold is
$\vartheta<\min\{p,q\}$. Finally, we discuss the resulting variational, energy, and
renormalized solution theories for nonlinear parabolic equations with
measure data, relating our conclusions to the existing literature.
Keywords: measure data, duality, Morrey measures, parabolic capacity, variational solutions
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