Spectrum of Fractional and Fractional Prabhakar Sturm–Liouville Problems with Homogeneous Dirichlet Boundary Conditions

In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm–Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both cases, we study the spectral properties of Sturm–Liouville operators on function space restricted by homogeneous Dirichlet boundary conditions. Fractional and fractional Prabhakar Sturm–Liouville problems are converted into the equivalent integral ones. Afterwards, the integral Sturm–Liouville operators are rewritten as Hilbert–Schmidt operators determined by kernels, which are continuous under the corresponding assumptions. In particular, the range of fractional order is here restricted to interval (1/2,1]. Applying the spectral Hilbert–Schmidt theorem, we prove that the spectrum of integral Sturm–Liouville operators is discrete and the system of eigenfunctions forms a basis in the corresponding Hilbert space. Then, equivalence results for integral and differential versions of respective eigenvalue problems lead to the main theorems on the discrete spectrum of differential fractional and fractional Prabhakar Sturm–Liouville operators.

Blow-up Properties of a Coupled System of Reaction-Diffusion Equations

This paper is concerned with a Coupled Reaction-diffusion system defined in a ball with homogeneous Dirichlet boundary conditions. Firstly, we studied the blow-up set showing that, under some conditions, the blow-up in this problem occurs only at a single point. Secondly, under some restricted assumptions on the reaction terms, we established the upper (lower) blow-up rate estimates. Finally, we considered the Ignition system in general dimensional space as an application to our results.

Optimal error estimate for a space-time discretization for incompressible generalized Newtonian fluids: the Dirichlet problem

In this paper we prove optimal error estimates for solutions with natural regularity of the equations describing the unsteady motion of incompressible shear-thinning fluids. We consider a full space-time semi-implicit scheme for the discretization. The main novelty, with respect to previous results, is that we obtain the estimates directly without introducing intermediate semi-discrete problems, which enables the treatment of homogeneous Dirichlet boundary conditions.

A 3D Non-Stationary Micropolar Fluids Equations with Navier Slip Boundary Conditions

Micropolar fluids are fluids with microstructure and belong to a class of fluids with asymmetric stress tensor that called Polar fluids, and include, as a special case, the well-established Navier–Stokes model. In this work we study a 3D micropolar fluids model with Navier boundary conditions without friction for the velocity field and homogeneous Dirichlet boundary conditions for the angular velocity. Using the Galerkin method, we prove the existence of weak solutions and establish a Prodi–Serrin regularity type result which allow us to obtain global-in-time strong solutions at finite time.

Shape Perturbation of Grushin Eigenvalues

We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of $${\mathbb {R}}^N$$ R N . We prove that the symmetric functions of the eigenvalues depend real analytically upon domain perturbations and we prove an Hadamard-type formula for their shape differential. In the case of perturbations depending on a single scalar parameter, we prove a Rellich–Nagy-type theorem which describes the bifurcation phenomenon of multiple eigenvalues. As corollaries, we characterize the critical shapes under isovolumetric and isoperimetric perturbations in terms of overdetermined problems and we deduce a new proof of the Rellich–Pohozaev identity for the Grushin eigenvalues.

Homogenization of the Poisson equation in a non-periodically perforated domain

We study the Poisson equation in a perforated domain with homogeneous Dirichlet boundary conditions. The size of the perforations is denoted by ε > 0, and is proportional to the distance between neighbouring perforations. In the periodic case, the homogenized problem (obtained in the limit ε → 0) is well understood (see (Rocky Mountain J. Math. 10 (1980) 125–140)). We extend these results to a non-periodic case which is defined as a localized deformation of the periodic setting. We propose geometric assumptions that make precise this setting, and we prove results which extend those of the periodic case: existence of a corrector, convergence to the homogenized problem, and two-scale expansion.

Reaction-advection-diffusion competition models under lethal boundary conditions

<p style='text-indent:20px;'>In this study, we consider a Lotka–Volterra reaction–diffusion–advection model for two competing species under homogeneous Dirichlet boundary conditions, describing a hostile environment at the boundary. In particular, we deal with the case in which one species diffuses at a constant rate, whereas the other species has a constant rate diffusion rate with a directed movement toward a better habitat in a heterogeneous environment with a lethal boundary. By analyzing linearized eigenvalue problems from the system, we conclude that the species dispersion in the advection direction is not always beneficial, and survival may be determined by the convexity of the environment. Further, we obtain the coexistence of steady-states to the system under the instability conditions of two semi-trivial solutions and the uniqueness of the coexistence steady states, implying the global asymptotic stability of the positive steady-state.</p>

Local null controllability of the penalized Boussinesq system with a reduced number of controls

<p style='text-indent:20px;'>In this paper we consider the Boussinesq system with homogeneous Dirichlet boundary conditions, defined in a regular domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb R^N $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ N = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ N = 3 $\end{document}</tex-math></inline-formula>. The incompressibility condition of the fluid is replaced by its approximation by penalization with a small parameter <inline-formula><tex-math id="M4">\begin{document}$ \varepsilon &gt; 0 $\end{document}</tex-math></inline-formula>. We prove that our system is locally null controllable using a control with a restricted number of components, localized in an open set <inline-formula><tex-math id="M5">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> contained in <inline-formula><tex-math id="M6">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. We also show that the control cost is bounded uniformly with respect to <inline-formula><tex-math id="M7">\begin{document}$ \varepsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>. The proof is based on a linearization argument. The null controllability of the linearized system is obtained by proving a new Carleman estimate for the adjoint system. This inequality is derived by exploiting the coercivity of some second order differential operator involving crossed derivatives.</p>

Pullback attractors for 2D MHD equations on time-varying domains

<p style='text-indent:20px;'>In the present paper, we consider the asymptotic dynamics of 2D MHD equations defined on the time-varying domains with homogeneous Dirichlet boundary conditions. First we introduce some coordinate transformations to construct the invariance of the divergence operators in any <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional spaces and establish some equivalent estimates of the vectors between the time-varying domains and the cylindrical domains. Then, we apply these estimates to overcome the difficulties caused by the variations of the spatial domains, including the processing of the pressure <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula> and the definition of weak solutions. Detailed arguments of converting the equations on the time-varying domains into the corresponding equations on the cylindrical domains are presented. Finally, we show the well-posedness of weak solutions and the existence of a compact pullback attractor for the 2D MHD equations.</p>