Case Of Linear Diffusion Research Articles

Global existence of a weak solution for a reaction–diffusion system in a porous medium with membrane conditions and mass control

Abstract In this paper, we prove the global exstence of weak solutions for a porous medium dynamics of m species moving between two domains separated by a zero-thickness membrane. On this membrane, Kedem–Katchalsky conditions are considered, and the study is characterized by natural structural conditions applied to the nonlinear reactive terms. The global existence is established under the assumption that these reactive terms are bounded in $L^1$ . This problem has already been analyzed in the linear diffusion case by Ciavolella and Perthame in Ciavolella and Perthame (2021, Journal of Evolution Equations 21, 1513–1540). The present work constitutes an extension for nonlinear diffusion, particularly of the porous medium type, in the form $\partial _t v_i - \Delta v_i^{r_i} = R_i$ , for an exponent $r_i < 2$ . The case $r_i \geq 2$ remains an open problem. This paper is an adaptation of the ideas from Ciavolella and Perthame (2021, Journal of Evolution Equations 21, 1513–1540), with new strategies to overcome the appearance of nonlinearity and degeneracy in the diffusion term.

Vanishing diffusion limits for planar fronts in bistable models with saturation

We deal with planar fronts for parameter-dependent reaction-diffusion equations with bistable reaction and saturating diffusive term like u t = ε d i v ( ∇ u 1 + | ∇ u | 2 ) + f ( u ) , u = u ( x , t ) , x ∈ R n , t ∈ R , \begin{equation*}u_t=\varepsilon \, \mathrm {div}\, \left (\frac {\nabla u}{\sqrt {1+\vert \nabla u \vert ^2}}\right ) + f(u), \quad u=u(x, t), \; x \in \mathbb {R}^n, \, t \in \mathbb {R}, \end{equation*} analyzing in particular their behavior for ε → 0 \varepsilon \to 0 . First, we construct monotone and non-monotone planar traveling waves, using a change of variables allowing to analyze a two-point problem for a suitable first-order reduction of the equation above; then, we investigate the asymptotic behavior of the monotone fronts for ε → 0 \varepsilon \to 0 , showing their convergence to suitable step functions. A remarkable feature of the considered diffusive term is that the fronts connecting 0 0 and 1 1 are necessarily discontinuous (and steady, namely with 0 0 -speed) for small ε \varepsilon , so that in this case the study of the convergence concerns discontinuous steady states, differently from the linear diffusion case.

Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation

We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Ref. 29 [M. B. Short, M. R. D’Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci. 18 (2008) 1249–1267]. The focus here is on the question of how far a certain porous medium enhancement in the random diffusion of criminal agents may exert visible relaxation effects. It is shown that sufficient regularity of the non-negative source terms in the system and a sufficiently strong nonlinear enhancement ensure that a corresponding Neumann-type initial–boundary value problem, posed in a smoothly bounded planar convex domain, admits locally bounded solutions for a wide class of arbitrary initial data. Furthermore, this solution is globally bounded under mild additional conditions on the source terms. These results are supplemented by numerical evidence which illustrates smoothing effects in solutions with sharply structured initial data in the presence of such porous medium-type diffusion and support the existence of singular structures in the linear diffusion case, which is the type of diffusion proposed in Ref. 29.

Positive solutions of indefinite logistic growth models with flux-saturated diffusion

This paper analyzes the quasilinear elliptic boundary value problem driven by the mean curvature operator −div∇u∕1+|∇u|2=λa(x)f(u)inΩ,u=0on∂Ω,with the aim of understanding the effects of a flux-saturated diffusion in logistic growth models featuring spatial heterogeneities. Here, Ω is a bounded domain in RN with a regular boundary ∂Ω, λ>0 represents a diffusivity parameter, a is a continuous weight which may change sign in Ω, and f:[0,L]→R, with L>0 a given constant, is a continuous function satisfying f(0)=f(L)=0 and f(s)>0 for every s∈]0,L[. Depending on the behavior of f at zero, three qualitatively different bifurcation diagrams appear by varying λ. Typically, the solutions we find are regular as long as λ is small, while as a consequence of the saturation of the flux they may develop singularities when λ becomes larger. A rather unexpected multiplicity phenomenon is also detected, even for the simplest logistic model, f(s)=s(L−s) and a≡1, having no similarity with the case of linear diffusion based on the Fick–Fourier’s law.

Existence of ground states for aggregation-diffusion equations

We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not.

Super-critical and sub-critical bifurcations in a reaction-diffusion Schnakenberg model with linear cross-diffusion

In this paper the Turing pattern formation mechanism of a two component reaction-diffusion system modeling the Schnakenberg chemical reaction coupled to linear cross-diffusion terms is studied. The linear cross-diffusion terms favors the destabilization of the constant steady state and the mechanism of pattern formation with respect to the standard linear diffusion case, as shown in Madzvamuse et al. (J. Math. Biol. 2014). Since the subcritical Turing bifurcations of reaction-diffusion systems lead to spontaneous onset of robust, finite-amplitude localized patterns, here a detailed investigation of the Turing pattern forming region is performed to show how the diffusion coefficients for both species (the activator and the inhibitor) influence the occurrence of supercritical or subcritical bifurcations. The weakly nonlinear (WNL) multiple scales analysis is employed to derive the equations for the amplitude of the Turing patterns and to distinguish the supercritical and the subcritical pattern region, both in 1D and 2D domains. Numerical simulations are employed to confirm the WNL theoretical predictions through which a classification of the patterns (squares, rhombi, rectangle and hexagons) is obtained. In particular, due to the hysteretic nature of the subcritical bifurcation, we observe the phenomenon of pattern transition from rolls to hexagons, in agreement with the bifurcation diagram.

Global and blow-up solutions for the nonlocal $p$-Laplacian evolution equation with weighted nonlinear nonlocal boundary condition

In this paper, we investigate global existence and blow-up properties of nonnegative solutions to a nonlocal $p$-Laplacian evolution equation with weighted nonlinear nonlocal boundary condition. By using the method of upper and lower solutions, we consider some effects of weight function and nonlinear exponent on the global and blow-up solutions. In addition, we show the blow-up rate estimate, blow-up profile and blow-up set for linear diffusion case.

Turing Instability and Pattern Formation for the Lengyel–Epstein System with Nonlinear Diffusion

In this work we study the effect of density dependent nonlinear diffusion on pattern formation in the Lengyel--Epstein system. Via the linear stability analysis we determine both the Turing and the Hopf instability boundaries and we show how nonlinear diffusion intensifies the tendency to pattern formation; %favors the mechanism of pattern formation with respect to the classical linear diffusion case; in particular, unlike the case of classical linear diffusion, the Turing instability can occur even when diffusion of the inhibitor is significantly slower than activator's one. In the Turing pattern region we perform the WNL multiple scales analysis to derive the equations for the amplitude of the stationary pattern, both in the supercritical and in the subcritical case. Moreover, we compute the complex Ginzburg-Landau equation in the vicinity of the Hopf bifurcation point as it gives a slow spatio-temporal modulation of the phase and amplitude of the homogeneous oscillatory solution.

Global existence and blow-up of solutions for p-Laplacian evolution equation with nonlinear memory term and nonlocal boundary condition

In this paper, we deal with an initial boundary value problem for a p-Laplacian evolution equation with nonlinear memory term and inner absorption term subject to a weighted linear nonlocal boundary condition. We find the effects of a weighted function as regards determining blow-up of nonnegative solutions or not and establish the precise blow-up estimate for the linear diffusion case under some suitable conditions.

The Keller–Segel Model for Chemotaxis with Prevention of Overcrowding: Linear vs. Nonlinear Diffusion

The aim of this paper is to discuss the effects of linear and nonlinear diffusion in the large time asymptotic behavior of the Keller–Segel model of chemotaxis with volume filling effect. In the linear diffusion case we provide several sufficient conditions for the diffusion part to dominate and yield decay to zero solutions. We also provide an explicit decay rate towards self–similarity. Moreover, we prove that no stationary solutions with positive mass exist. In the nonlinear diffusion case we prove that the asymptotic behavior is fully determined by whether the diffusivity constant in the model is larger or smaller than the threshold value $\varepsilon =1$. Below this value we have existence of nondecaying solutions and their convergence (along subsequences) to stationary solutions. For $\varepsilon >1$ all compactly supported solutions are proved to decay asymptotically to zero, unlike in the classical models with linear diffusion, where the asymptotic behavior depends on the initial mass.

Travelling wave solutions of the degenerate KolmogorovPetrovskiPiskunov equation

We prove the existence of a family of Travelling Wave (TW) solutions for a large class of scalar reaction-diffusion equations with degenerate, nonlinear diffusion coefficients and monostable nonlinear reaction terms. We also investigate stability. Specifically, we show that, as in the linear diffusion case [6], the slowest TW in the family yields the asymptotic rate of the propagation of disturbances from the unstable rest state in these systems. In addition, we give conditions on the reaction term and diffusion coefficient ensuring the existence of interfaces.

Characterizing the existence of large solutions for a class of sublinear problems with nonlinear diffusion

In this paper we characterize the existence of large solutions for a general class of sublinear elliptic problems of logistic type related to the porous media equation. Our main result shows that large solutions do exist if, and only if, the nonlinear diffusion is not too large. As a byproduct of the general theory developed by the authors in [3], those large solutions must be unstable with respect to the positive solutions of the parabolic counterpart of the elliptic model. This seems to be the first result of this nature available in the literature. Most precisely, as the diffusion becomes non-linear the metasolutions become unstable, so arising a classical steady-state gaining the stability lost by the metasolution. In particular, a dynamical bifurcation occurs from the linear diffusion case.

LARGE TIME ASYMPTOTICS OF NONLINEAR DRIFT-DIFFUSION SYSTEMS WITH POISSON COUPLING

We study the asymptotic behavior as t→ +∞ of a system of densities of charged particles satisfying nonlinear drift-diffusion equations coupled by a damped Poisson equation for the drift-potential. In plasma physics applications the damping is caused by a spatio-temporal rescaling of an “unconfined” problem, which introduces a harmonic external potential of confinement. We present formal calculations (valid for smooth solutions) which extend the results known in the linear diffusion case to nonlinear diffusion of e.g. Fermi–Dirac or fast diffusion/porous media type.

Global Existence for Nonlinear Diffusion Equations

Solutions of nonlinear (possibly degenerate) reaction-diffusion models are known to exist for all time if the asymptotic growth of the reaction term is not greater than that of the diffusion term. Via concavity methods, it is also known that negating such a condition results in solutions which blow up in finite time. On the other hand, two recent studies regarding the case of linear diffusion have reported on the ability for convective terms in a model to create global existence of solutions where no such result is true in their absence. In this paper, we develop a theory of global existence for a general class of reaction-diffusion-convection models which only requires a similar balance of diffusion and reaction. Our result extends previous work to models including convection, but it does not reveal the exact role of convection in yielding global solutions. Further analysis of a slightly simplified model establishes the global existence of all solutions if the reaction term grows asymptotically at a rate less than that of either diffusion or convection.