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Unidirectional evolution equations of diffusion type
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This paper is concerned with the uniqueness, existence, comparison principle and long-time behavior of solutions to the initial-boundary value problem for a unidirectional diffusion equation. The unidirectional evolution often appears in Damage Mechanics due to the strong irreversibility of crack propagation or damage evolution. The existence of solutions is proved in an L^2-framework by introducing a peculiar discretization of the unidirectional diffusion equation by means of variational inequities of obstacle type and by developing a regularity theory for such variational inequalities. The novel discretization argument will be also applied to prove the comparison principle as well as to investigate the long-time behavior of solutions.
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The Lewy-Stampacchia Inequality for the Fractional Laplacian and Its Application to Anomalous Unidirectional Diffusion Equations
For the spectral fractional Laplacian on a bounded domain, obstacle solutions satisfy the Lewy-Stampacchia inequality f ≤ Au ≤ max{f, Aψ}, which yields well-posedness of a fractional unidirectional diffusion equation.
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