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In this paper we investigate existence of solutions for the system: begin{equation*} left{ begin{array}{l} D^{alpha}_tu=textrm{div}(u abla p), D^{alpha}_tp=-(-Delta)^{s}p+u^{2}, end{array} right. end{equation*} in $mathbb{T}^3$ for $0< s leq 1$, and $0< alpha le 1$. The term $D^alpha_t u$ denotes the Caputo derivative, which models memory effects in time. The fractional Laplacian $(-Delta)^{s}$ represents the L{e}vy diffusion. We prove global existence of nonnegative weak solutions that satisfy a variational inequality. The proof uses several approximations steps, including an implicit Euler time discretization. We show that the proposed discrete Caputo derivative satisfies several important properties, including positivity preserving, convexity and rigorous convergence towards the continuous Caputo derivative. Most importantly, we give a strong compactness criteria for piecewise constant functions, in the spirit of Aubin-Lions theorem, based on bounds of the discrete Caputo derivative.
This paper deals with the investigation of the computational solutions of an unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized fra
We obtain $L_p$ estimates for fractional parabolic equations with space-time non-local operators $$ partial_t^alpha u - Lu= f quad mathrm{in} quad (0,T) times mathbb{R}^d,$$ where $partial_t^alpha u$ is the Caputo fractional derivative of order $alph
We prove uniqueness for weak solutions to abstract parabolic equations with the fractional Marchaud or Caputo time derivative. We consider weak solutions in time for divergence form equations when the fractional derivative is transferred to the test function.
We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain $Omega$ of $R^N$ and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the entire complement of $Omega$). After sett
In this paper we investigate the solution of generalized distributed order diffusion equations with composite time fractional derivative by using the Fourier-Laplace transform method. We represent solutions in terms of infinite series in Fox $H$-func