Differential and Integral Equations

Existence of entropy solutions to a doubly nonlinear integro-differential equation

Martin Scholtes and Petra Wittbold

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Abstract

We consider a class of doubly nonlinear history-dependent problems associated with the equation $$ \partial_{t}k\ast(b(v)- b(v_{0})) = \text{div}\, a(x,Dv) + f . $$ Our assumptions on the kernel $k$ include the case $k(t) = t^{-\alpha}/\Gamma(1-\alpha)$, in which case the left-hand side becomes the fractional derivative of order $\alpha\in (0,1)$ in the sense of Riemann-Liouville. Existence of entropy solutions is established for general $L^{1}-$data and Dirichlet boundary conditions. Uniqueness of entropy solutions has been shown in a previous work.

Article information

Source
Differential Integral Equations Volume 31, Number 5/6 (2018), 465-496.

Dates
First available in Project Euclid: 23 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.die/1516676439

Subjects
Primary: 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 47J35: Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 47H20, 58D25] 45D05: Volterra integral equations [See also 34A12] 35D99: None of the above, but in this section

Citation

Scholtes, Martin; Wittbold, Petra. Existence of entropy solutions to a doubly nonlinear integro-differential equation. Differential Integral Equations 31 (2018), no. 5/6, 465--496. https://projecteuclid.org/euclid.die/1516676439


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