Well-posedness of a nonlinear second-order anisotropic reaction-diffusion problem with nonlinear and inhomogeneous dynamic boundary conditions

The paper is concerned with a qualitative analysis for a nonlinear second-order boundary value problem, endowed with nonlinear and inhomogeneous dynamic boundary conditions, extending the types of bounday conditions already studied. Under certain assumptions on the input data: $f_{_1}(t,x)$, $w(t,x)$ and $u_0(x)$, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space $W^{1,2}_p(Q)$. This extends previous works concerned with nonlinear dynamic boundary conditions, allowing to the present mathematical model to better approximate the real physical phenomena (the anisotropy effects, phase change in $\Omega$ and at the boundary $\partial\Omega$, etc.).

A Gradient Discretisation Method for Anisotropic Reaction–Diffusion Models with Applications to the Dynamics of Brain Tumors

A gradient discretisation method (GDM) is an abstract setting that designs the unified convergence analysis of several numerical methods for partial differential equations and their corresponding models. In this paper, we study the GDM for anisotropic reaction–diffusion problems, based on a general reaction term, with Neumann boundary condition. With natural regularity assumptions on the exact solution, the framework enables us to provide proof of the existence of weak solutions for the problem, and to obtain a uniform-in-time convergence for the discrete solution and a strong convergence for its discrete gradient. It also allows us to apply non-conforming numerical schemes to the model on a generic grid (the non-conforming ℙ ⁢ 1 {\mathbb{P}1} finite element scheme and the hybrid mixed mimetic (HMM) methods). Numerical experiments using the HMM method are performed to assess the accuracy of the proposed scheme and to study the growth of glioma tumors in heterogeneous brain environment. The dynamics of their highly diffusive nature is also measured using the fraction anisotropic measure. The validity of the HMM is examined further using four different mesh types. The results indicate that the dynamics of the brain tumor is still captured by the HMM scheme, even in the event of a highly heterogeneous anisotropic case performed on the mesh with extreme distortions.

Rigorous Mathematical Investigation of a Nonlocal and Nonlinear Second-Order Anisotropic Reaction-Diffusion Model: Applications on Image Segmentation

In this paper we are addressing two main topics, as follows. First, a rigorous qualitative study is elaborated for a second-order parabolic problem, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as non-homogeneous Cauchy-Neumann boundary conditions. Under certain assumptions on the input data: f(t,x), w(t,x) and v0(x), we prove the well-posedness (the existence, a priori estimates, regularity, uniqueness) of a solution in the Sobolev space Wp1,2(Q), facilitating for the present model to be a more complete description of certain classes of physical phenomena. The second topic refers to the construction of two numerical schemes in order to approximate the solution of a particular mathematical model (local and nonlocal case). To illustrate the effectiveness of the new mathematical model, we present some numerical experiments by applying the model to image segmentation tasks.

Spatio-temporal Investigations of the Incomplete Spin Transition in a Single Crystal of [Fe(2-pytrz)2{Pt(CN)4}]·3H2O: Experiment and Theory

Optical microscopy technique is used to investigate the thermal and the spatio-temporal properties of the spin-crossover single crystal [Fe(2-pytrz) 2 {Pt(CN) 4 }]·3H 2 O, which exhibits a first-order spin transition from a full high-spin (HS) state at high temperature to an intermediate, high-spin low-spin (HS-LS) state, below 153 K, where only one of the two crystallographic Fe(II) centers switches from the HS to HS-LS state. In comparison with crystals undergoing a complete spin transition, the present transformation involves smaller volume changes at the transition, which helps to preserving the crystal’s integrity. By analyzing the spatio-temporal properties of this spin transition, we evidenced a direct correlation between the orientation and shape of HS/HS-LS domain wall with the crystal’s shape. Thanks to the small volume change accompanying this spin transition, the analysis of the experimental data by an anisotropic reaction-diffusion model becomes very relevant and leads to an excellent agreement with the experimental observations.