No Arabic abstract
We investigate a class of systems of partial differential equations with nonlinear cross-diffusion and nonlocal interactions, which are of interest in several contexts in social sciences, finance, biology, and real world applications. Assuming a uniform coerciveness assumption on the diffusion part, which allows to consider a large class of systems with degenerate cross-diffusion (i.e. of porous medium type) and relaxes sets of assumptions previously considered in the literature, we prove global-in-time existence of weak solutions by means of a semi-implicit version of the Jordan-Kinderlehrer-Otto scheme. Our approach allows to consider nonlocal interaction terms not necessarily yielding a formal gradient flow structure.
We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but here the population dispersal is described by nonlocal diffusion instead of local diffusion. We prove that such a nonlocal diffusion problem with free boundary has a unique global solution, and for models with Lotka-Volterra type competition or predator-prey growth terms, we show that a spreading-vanishing dichotomy holds, and obtain criteria for spreading and vanishing; moreover, for the weak competition case and for the weak predation case, we can determine the long-time asymptotic limit of the solution when spreading happens. Compared with the single species free boundary model with nonlocal diffusion considered recently in cite{CDLL}, and the two species cases with local diffusion extensively studied in the literature, the situation considered in this paper involves several extra difficulties, which are overcome by the use of some new techniques.
We consider an epidemic model with nonlocal diffusion and free boundaries, which describes the evolution of an infectious agents with nonlocal diffusion and the infected humans without diffusion, where humans get infected by the agents, and infected humans in return contribute to the growth of the agents. The model can be viewed as a nonlocal version of the free boundary model studied by Ahn, Beak and Lin cite{ABL2016}, with its origin tracing back to Capasso et al. cite{CP1979, CM1981}. We prove that the problem has a unique solution defined for all $t>0$, and its long-time dynamical behaviour is governed by a spreading-vanishing dichotomy. Sharp criteria for spreading and vanishing are also obtained, which reveal significant differences from the local diffusion model in cite{ABL2016}. Depending on the choice of the kernel function in the nonlocal diffusion operator, it is expected that the nonlocal model here may have accelerated spreading, which would contrast sharply to the model of cite{ABL2016}, where the spreading has finite speed whenever spreading happens cite{ZLN2019}.
We study a Cahn-Hilliard-Hele-Shaw (or Cahn-Hilliard-Darcy) system for an incompressible mixture of two fluids. The relative concentration difference $varphi$ is governed by a convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential. The volume averaged fluid velocity $mathbf{u}$ obeys a Darcys law depending on the so-called Korteweg force $mu abla varphi$, where $mu$ is the nonlocal chemical potential. In addition, the kinematic viscosity $eta$ may depend on $varphi$. We establish first the existence of a global weak solution which satisfies the energy identity. Then we prove the existence of a strong solution. Further regularity results on the pressure and on $mathbf{u}$ are also obtained. Weak-strong uniqueness is demonstrated in the two dimensional case. In the three-dimensional case, uniqueness of weak solutions holds if $eta$ is constant. Otherwise, weak-strong uniqueness is shown by assuming that the pressure of the strong solution is $alpha$-H{o}lder continuous in space for $alphain (1/5,1)$.
Conditions for the existence and uniqueness of weak solutions for a class of nonlinear nonlocal degenerate parabolic equations are established. The asymptotic behaviour of the solutions as time tends to infinity are also studied. In particular, the finite time extinction and polynomial decay properties are proved.
In this paper, the finite time extinction of solutions to the fast diffusion system $u_t=mathrm{div}(| abla u|^{p-2} abla u)+v^m$, $v_t=mathrm{div}(| abla v|^{q-2} abla v)+u^n$ is investigated, where $1<p,q<2$, $m,n>0$ and $Omegasubset mathbb{R}^N (Ngeq1)$ is a bounded smooth domain. After establishing the local existence of weak solutions, the authors show that if $mn>(p-1)(q-1)$, then any solution vanishes in finite time provided that the initial data are ``comparable; if $mn=(p-1)(q-1)$ and $Omega$ is suitably small, then the existence of extinction solutions for small initial data is proved by using the De Giorgi iteration process and comparison method. On the other hand, for $1<p=q<2$ and $mn<(p-1)^2$, the existence of at least one non-extinction solution for any positive smooth initial data is proved.