Quasilinear equations with a sectorial set of operators at Gerasimov-Caputo derivatives. (English. Russian original) Zbl 1522.35553
Proc. Steklov Inst. Math. 321, Suppl. 1, S78-S89 (2023); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 29, No. 2, 248-259 (2023).
Summary: The issues of unique solvability of the Cauchy problem are studied for a quasilinear equation solved with respect to the highest fractional Gerasimov-Caputo derivative in a Banach space with closed operators from the class \(A_{\alpha,G}^n\) in the linear part and with a nonlinear operator continuous in the graph norm. A theorem on the local existence and uniqueness of a solution to the Cauchy problem is proved in the case of a locally Lipschitz nonlinear operator. Under the nonlocal Lipschitz condition for the nonlinear operator, the existence of a unique solution on a predetermined interval is shown. Abstract results are illustrated by examples of initial-boundary value problems for partial differential equations with Gerasimov-Caputo time derivatives.
MSC:
| 35R11 | Fractional partial differential equations |
| 35G31 | Initial-boundary value problems for nonlinear higher-order PDEs |
| 34G20 | Nonlinear differential equations in abstract spaces |
Keywords:
Gerasimov-Caputo fractional derivative; Cauchy problem; sectorial set of operators; resolving family of operators; quasilinear equation; local solution; nonlocal solution; initial-boundary value problemReferences:
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