The article is devoted to obtaining new existence and uniqueness results for weak solutions to non-homogeneous initial-boundary value problems for parabolic equations of the form
\[ \begin{aligned} \frac{\partial u}{\partial t} - \nabla_x\cdot A(x,t,\nabla_x\,u) &= f \quad\text{in }{\mathcal D}'(Q_+),\\ u &= g \quad\text{on } (\Omega \times\{0\})\cup(\partial\Omega\times\mathbb R_+), \end{aligned} \]
where \(\Omega\) is an open and bounded set in \(\mathbb R^n\) and \(Q_+ = \Omega\times \mathbb R_+\). The model of these equations is the usual \(p\)-Laplacian with \(1<p<\infty\).
The author is interested in finding the “largest possible” classes of boundary and source data such that the above-mentioned problem has a good meaning and is uniquely solvable. To get the optimal solution-space, the appropriate distribution theory allowing for a fractional differential in the time direction of general \(L^p\)-functions, having half order time derivatives in \(L^2(Q_+)\) and first order space derivatives in \(L^p(Q_+)\), is developed. This analytical framework makes it possible to give a precise meaning to fractional integration by parts for the time derivatives that is one of the key tools in the author’s method. The author shows that the solution space \(X^{1,1/2}(Q_+)\) really is a true analog of the space \(W^{1,p}(\Omega)\) for the elliptic \(p\)-Laplacian in the sense that: given \(g \in X^{1,1/2}(Q_+)\), there exists a unique solution \(u \in X^{1,1/2}(Q_+)\) to the \(p\)-Laplacian equation such that \(u - g\) belongs to the closure of \({\mathcal D}(\Omega_+)\) in the \(X^{1,1/2}(Q_+)\)-norm topology. Furthermore, the source data \(f\) can be taken as sums of first order space derivatives of \(L^{p/(p-1)}(Q_+)\)-functions and half-a-time derivatives of \(L^2(Q_+)\)-functions. Finally, the optimal results in the linear case (\(s = 1\)) and a complete description of the space of solutions are given and the precise structural conditions for \(A\) are presented.