Some properties of solutions of the Cauchy problem for evolution equations
The authors study solvability of the Cauchy problem \[ \begin{aligned} u_t-Lu &=f,\qquad t\in [0,T],\quad T<\infty,\\ u|_{t=0} &=u_0,\end{aligned} \] where \(L : E\to E\) is the generator of an analytic semigroup in a Banach space \(E\). In contrast to many other papers on the subject, the authors assume that the Cauchy data \(u_0\) and the right-hand side \(f\) as \(t\to 0\) are singular in some sense: \(u_0\) belongs to some negative space (space of distributions) and \(f\) belongs to the space \(L_p\) with weight \(t^{\alpha}\). Sharp weighted estimates for the solutions and existence theorems in the case of arbitrary weight function and singular initial data are obtained. The results are used to study initial boundary value problems for parabolic equations with operator-valued coefficients.
parabolic equations with operator-valued coefficients
scientific article; zbMATH DE number 6065264