We establish the existence and uniqueness of fundamental solutions for the fractional porous medium equation introduced in \cite{PQRV1}. They are self-similar functions of the form $u(x,t)= t^{-\alpha} f(|x|\,t^{-\beta})$ with suitable $\alpha$ and $\beta$. As a main application of this construction, we prove that the asymptotic behaviour of general solutions is represented by such special solutions. Among other interesting qualitative properties of the equation we prove an Aleksandrov reflection principle.

Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type

We study the existence of weak solutions to a Cahn-Hilliard-Darcy system coupled witha convection-reaction-diffusion equation through the fluxes, through the source terms and in Darcy’slaw. The system of equations arises from a mixture model for tumour growth accounting for transportmechanisms such as chemotaxis and active transport. We prove, via a Galerkin approximation, theexistence of global weak solutions in two and three dimensions, along with new regularity results forthe velocity field and for the pressure. Due to the coupling with the Darcy system, the time derivativeshave lower regularity compared to systems without Darcy flow, but in the two dimensional case weemploy a new regularity result for the velocity to obtain better integrability and temporal regularityfor the time derivatives. Then, we deduce the global existence of weak solutions for two variantsof the model; one where the velocity is zero and another where the chemotaxis and active transportmechanisms are absent.

Global weak solutions and asymptotic limits of a Cahn--Hilliard--Darcy system modelling tumour growth

Fundamental solution and long time behavior of the Porous Medium Equation in hyperbolic space

Forward, backward and elliptic Harnack inequalities for non-negative solutions of a class of singular, quasi-linear, parabolic equations, are established. These classes of singular equations include the p-Laplacean equation and equations of the porous medium type. Key novel points include form of a Harnack estimate backward in time, that has never been observed before, and measure theoretical proofs, as opposed to comparison principles. These Harnack estimates are established in the super-critical range (1.5) below. Such a range is optimal for a Harnack estimate to hold. Mathematics Subject Classification (2010): 35K65 (primary); 35B65, 35B45 (secondary). 1. Main results Let E be an open set in RN and for T > 0 let ET = E × (0, T ]. Let u be a weak solution u ∈ Cloc ( 0, T ; Lloc(E) ) ∩ L p loc(0, T ; W 1,p loc (E)) 1 < p < 2 (1.1) of a quasi-linear, singular parabolic equation of the type ut − div A(x, t, u, Du) = B(x, t, u, Du) weakly in ET (1.2) where the functions A : ET × RN+1 → RN and B : ET × RN+1 → R are only assumed to be measurable and subject to the structure conditions   A(x, t, u, Du) · Du ≥ Co|Du|p − C p |A(x, t, u, Du)| ≤ C1|Du|p−1 + C p−1 |B(x, t, u, Du)| ≤ C |Du|p−1 + C p a.e. in ET (1.3) This work has been partially supported by I.M.A.T.I. – C.N.R. – Pavia. DiBenedetto’s work partially supported by National Science Foundation grant DMS-0652385. Received March 9, 2009; accepted in revised form May 31, 2009.

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Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations

New perturbation methods for nonlinear parabolic problems

The paper deals with the existence of solutions of some generalized Stefan-type equation in the framework of Orlicz spaces.

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On Degenerate Parabolic Equations

This article shows how to solve the time-fractional Fisher's equation through the use of two well-known analytical methods. The techniques we propose are a modified form of the Adomian decomposition method and homotopy perturbation method with a Yang transform. To show the accuracy of the suggested techniques, illustrative examples are considered. It is confirmed that the solution we get by implementing the suggested techniques has the desired rate of convergence towards the accurate solution. The main benefit of the proposed techniques is the small number of calculations. To show the reliability of the suggested techniques, we present some graphical behaviors of the accurate and analytical results, absolute error graphs and tables that strongly agree with each other. Furthermore, it can be used for solving fractional-order physical problems in various fields of applied sciences.

Evaluation of time-fractional Fisher's equations with the help of analytical methods

The present article correlates with a fuzzy hybrid technique combined with an iterative transformation technique identified as the fuzzy new iterative transform method. With the help of Atangana-Baleanu under generalized Hukuhara differentiability, we demonstrate the consistency of this method by achieving fuzzy fractional gas dynamics equations with fuzzy initial conditions. The achieved series solution was determined and contacted the estimated value of the suggested equation. To confirm our technique, three problems have been presented, and the results were estimated in fuzzy type. The lower and upper portions of the fuzzy solution in all three examples were simulated using two distinct fractional orders between 0 and 1. Because the exponential function is present, the fractional operator is nonsingular and global. It provides all forms of fuzzy solutions occurring between 0 and 1 at any fractional-order because it globalizes the dynamical behavior of the given equation. Because the fuzzy number provides the solution in fuzzy form, with upper and lower branches, fuzziness is also incorporated in the unknown quantity. It is essential to mention that the projected methodology to fuzziness is to confirm the superiority and efficiency of constructing numerical results to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.

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Novel Analysis of Fuzzy Physical Models by Generalized Fractional Fuzzy Operators

Dynamics of lorentz force and cross-diffusion effects on ethylene glycol based hybrid nanofluid flow amidst two parallel plates with variable electrical conductivity: A multiple linear regression analysis

Heat and mass transmission in a boundary layer flow due to swimming of motile gyrotactic microorganisms with variable wall temperature over a flat plate

Mohammed Kbiri Alaoui

Papers

On Degenerate Parabolic Equations

Evaluation of time-fractional Fisher's equations with the help of analytical methods

Novel Analysis of Fuzzy Physical Models by Generalized Fractional Fuzzy Operators

Dynamics of lorentz force and cross-diffusion effects on ethylene glycol based hybrid nanofluid flow amidst two parallel plates with variable electrical conductivity: A multiple linear regression analysis

Heat and mass transmission in a boundary layer flow due to swimming of motile gyrotactic microorganisms with variable wall temperature over a flat plate