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We consider density solutions for gradient flow equations of the form ut = ∇ · (γ(u)∇ N(u)), where N is the Newtonian repulsive potential in the whole space ℝd with the nonlinear convex mobility γ(u) = uα, and α > 1. We show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility case γ(u) = u. For linear mobility it was shown that there is a special solution in the form of a disk vortex of constant intensity in space u = c1t−1 supported in a ball that spreads in time like c2t1/d, thus showing a discontinuous leading front or shock. Our present results are in sharp contrast with the case of concave mobilities of the form γ(u) = uα, with 0 < α < 1 studied in [10]. There, we developed a well-posedness theory of viscosity solutions that are positive everywhere and moreover display a fat tail at infinity. Here, we also develop a well-posedness theory of viscosity solutions that in the radial case leads to a very detailed analysis allowing us to show a waiting time phenomena. This is a typical behaviour for nonlinear degenerate diffusion equations such as the porous medium equation. We will also construct explicit self-similar solutions exhibiting similar vortex-like behaviour characterizing the long time asymptotics of general radial solutions under certain assumptions. Convergent numerical schemes based on the viscosity solution theory are proposed analysing their rate of convergence. We complement our analytical results with numerical simulations illustrating the proven results and showcasing some open problems.

The main goal of this paper is to characterize the change of structural behavior (i.e. the appearance of the so-called 'strange terms') arising in the homogenization process when applied to distributed microscopic chemical reactions taking place on fixed-bed nanoreactors, at the microscopic level, on the boundary of the particles of critical size. The presence of non-homogeneous distributed functions of the reaction kinetics may be originated by many different reasons. The case of quick oscillation is often due to own structure of the fixed bed reactor since the flux of the fluid acts on each particle in a non-homogeneous way. In some other cases, the non-homogeneous distributed functions of the reaction kinetics is artificially provoked in order to control a certain desired global effect. Our main result gives a complete classification of the strange terms according the assumed periodicity on the distributed functions of the reaction kinetic. [ABSTRACT FROM AUTHOR]

Abstract We study the Dirichlet problem for the stationary Schrödinger fractional Laplacian equation (− Δ) s u + V u = f posed in bounded domain Ω ⊂ R n with zero outside conditions. We consider general nonnegative potentials V ∈ L l o c 1 (Ω) and prove well-posedness of very weak solutions when the data are chosen in an optimal class of weighted integrable functions f. Important properties of the solutions, such as its boundary behaviour, are derived. The case of super singular potentials that blow up near the boundary is given special consideration since it leads to so-called flat solutions. We comment on related literature. [ABSTRACT FROM AUTHOR]

We perform a unified analysis for the boundary behaviour of solutions to nonlocal fractional equations posed in bounded domains. Based on previous findings for some models of the fractional Laplacian operator, we show how it strongly differs from the boundary behaviour of solutions to elliptic problems modelled upon the Laplace-Poisson equation with zero boundary data. In the classical case it is known that, at least in a suitable weak sense, solutions of the homogeneous Dirichlet problem with a forcing term tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that, for equations driven by a wide class of nonlocal fractional operators, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal operators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Green function., (© 2022 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society.)

Background: Peritumoral inflammation-a response mainly involving polimorphonuclear neutrophils-has traditionally been thought protumoral in its effects. In recent years, however, a number of studies have indicated that it may play an important antitumoral role. This discrepancy has been difficult to explain., Methods and Findings: This work describes a tool for simulating tumor growth that obeys the universal model of tumor growth dynamics, and shows through its use that low intensity peritumoral inflammation exerts a protumoral effect, while high intensity inflammation exerts a potent antitumoral effect. Indeed, the simulation results obtained indicate that a sufficiently strong antitumoral effect can reverse tumor growth, as has been suggested several times in the clinical literature., Conclusions: The present result indicate that an 'immunological threshold' must exist, marking the boundary between states in which peritumoral inflammation is either harmful or beneficial. These findings lend support to the idea that stimulating intense peritumoral inflammation could be used as a treatment against solid tumors., Competing Interests: The authors have declared that no competing interests exist.