Local existence and regularity of solutions for some partial functional integrodifferential equations with infinite delay in Banach spaces
The authors study the existence and regularity of solutions for the following partial functional integrodifferential equations with infinite delay \[ \begin{cases} u'(t)=Au(t)+\displaystyle\int_0^t\gamma(t-s)u(s)\,ds+f(u_t)\qquad \text{for}\quad t\geq0\\ u_0=\varphi\in\mathcal{B} \end{cases} \] in Banach spaces: \(A\;:\;D(A)\rightarrow X\) is a closed linear operator on a Banach space \(X\), \(\gamma(t)\) for \(t\geq0\) is a closed linear operator with domain \(D(\gamma)\supset D(A)\), being time-independent. \(\mathcal{B}\) is a linear space of functions \(\psi\;:\;]-\infty,0]\rightarrow X\) satisfying some axioms. The history \(u_t\in\mathcal{B}\), \(t\geq0\), is defined by \(u_t(\theta)=u(t+\theta)\), \(\theta\in]-\infty,0]\). \(f\) is a continuous function. The existence of a resolvent operator for the homogeneous equation (1), with \(f=0\), is admitted in the sense of \textit{R. Grimmer} [Trans. Am. Math. Soc. 273, 333--349 (1982; Zbl 0493.45015)]. The delayed part is assumed to be locally Lipschitz. Local existence of the mild solutions of equation (1), their global continuation and their possible blow up in finite time is proved. Later is proved the existence of the strict solutions and also the existence of the mild solution in nonautonomous case. The results are applied to partial functional integrodifferential equations with diffusion.
partial functional integrodifferential equations
scientific article; zbMATH DE number 5539219