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A doubly nonlinear evolution problem involving the fractional p-Laplacian
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abstract
In this article, we focus on a doubly nonlinear nonlocal parabolic initial boundary value problem driven by the fractional $p$-Laplacian equipped with homogeneous Dirichlet boundary conditions on a domain in $\mathbb{R}^{d}$ and composed with a continuous, strictly increasing function. We establish well-posedness in $L^1$ in the sense of mild solutions, a comparison principle, and for restricted initial data we obtain that mild solutions of the inhomogeneous evolution problem are strong. We obtain $L^{q}$-$L^{\infty}$ regularity estimates for mild solutions, implying decay estimates and extending the property of strong solutions for more initial data. Moreover, we prove local and global H\"older continuity results as well as a comparison principle that yields extinction in finite time of mild solutions to the homogeneous evolution equation.
Forward citations
Cited by 3 Pith papers
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Existence and global behaviour of solutions of a parabolic problem involving the fractional $p$-Laplacian in porous medium
For the porous fractional p-Laplacian problem with power source, the authors prove existence and uniqueness of weak-mild solutions and characterize global stabilization, finite-time extinction, and blow-up.
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Integral Harnack estimates and the rate of extinction of singular fractional diffusion
Singular fractional p-Laplacian diffusion satisfies local integral Harnack estimates that imply finite-time extinction with (T*-t)^{1/(2-p)} decay.
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Doubly nonlinear parabolic equation involving a mixed local-nonlocal operator and a convection term
For a doubly nonlinear parabolic equation with a mixed local-nonlocal operator and convection, the paper establishes weak-mild well-posedness and conditions for stabilization, finite-time extinction, and blow-up.
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