A 2D climate energy balance model coupled with a 3D deep ocean model.
A blow up condition for a nonautonomous semilinear system.
A blow-up result for nonlinear diffusion equations
A boundary control problem with a nonlinear reaction term.
A class of kinetic models for chemotaxis with threshold to prevent overcrowding.
A class of one-dimensional degenerate parabolic equations
A combustion model with unbounded thermal conductivity and reactant diffusivity in non-smooth domains.
A double -shaped bifurcation curve for a reaction-diffusion model with nonlinear boundary conditions.
A Finite Element Model Based on Discontinuous Galerkin Methods on Moving Grids for Vertebrate Limb Pattern Formation
Skeletal patterning in the vertebrate limb, i.e., the spatiotemporal regulation of cartilage differentiation (chondrogenesis) during embryogenesis and regeneration, is one of the best studied examples of a multicellular developmental process. Recently [Alber et al., The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb, Bulletin of Mathematical Biology, 2008, v70, pp. 460-483], a simplified two-equation reaction-diffusion system was developed to describe the interaction...
A free boundary problem for a predator-prey model with nonlinear prey-taxis
This paper deals with a reaction-diffusion system modeling a free boundary problem of the predator-prey type with prey-taxis over a one-dimensional habitat. The free boundary represents the spreading front of the predator species. The global existence and uniqueness of classical solutions to this system are established by the contraction mapping principle. With an eye on the biological interpretations, numerical simulations are provided which give a real insight into the behavior of the free boundary...
A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions
A generalized solution to a Cahn-Hilliard/Allen-Cahn system.
A global attractor in a general diffusive food chain model.
A Hopf bifurcation of interfaces in a Dirichlet problem.
A Hybrid Model Describing Different Morphologies of Tumor Invasion Fronts
The invasive capability is fundamental in determining the malignancy of a solid tumor. Revealing biomedical strategies that are able to partially decrease cancer invasiveness is therefore an important approach in the treatment of the disease and has given rise to multiple in vitro and in silico models. We here develop a hybrid computational framework, whose aim is to characterize the effects of the different cellular and subcellular mechanisms involved...
A linear scheme to approximate nonlinear cross-diffusion systems*
This paper proposes a linear discrete-time scheme for general nonlinear cross-diffusion systems. The scheme can be regarded as an extension of a linear scheme based on the nonlinear Chernoff formula for the degenerate parabolic equations, which proposed by Berger et al. [RAIRO Anal. Numer.13 (1979) 297–312]. We analyze stability and convergence of the linear scheme. To this end, we apply the theory of reaction-diffusion system approximation. After discretizing the scheme in space, we obtain a versatile,...
A linear scheme to approximate nonlinear cross-diffusion systems*
This paper proposes a linear discrete-time scheme for general nonlinear cross-diffusion systems. The scheme can be regarded as an extension of a linear scheme based on the nonlinear Chernoff formula for the degenerate parabolic equations, which proposed by Berger et al. [RAIRO Anal. Numer.13 (1979) 297–312]. We analyze stability and convergence of the linear scheme. To this end, we apply the theory of reaction-diffusion system approximation. After discretizing the scheme in space, we obtain a versatile,...
A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations
An extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived...
A mechanochemical model of angiogenesis and vasculogenesis
Vasculogenesis and angiogenesis are two different mechanisms for blood vessel formation. Angiogenesis occurs when new vessels sprout from pre-existing vasculature in response to external chemical stimuli. Vasculogenesis occurs via the reorganization of randomly distributed cells into a blood vessel network. Experimental models of vasculogenesis have suggested that the cells exert traction forces onto the extracellular matrix and that these forces may play an important role in the network forming...
A mechanochemical model of angiogenesis and vasculogenesis
Vasculogenesis and angiogenesis are two different mechanisms for blood vessel formation. Angiogenesis occurs when new vessels sprout from pre-existing vasculature in response to external chemical stimuli. Vasculogenesis occurs via the reorganization of randomly distributed cells into a blood vessel network. Experimental models of vasculogenesis have suggested that the cells exert traction forces onto the extracellular matrix and that these forces may play an important role in the network forming...