Communications on Pure and Applied Analysis
September 2020 , Volume 19 , Issue 9
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We study a Lagrangian numerical scheme for solving a nonlinear drift diffusion equations of the form
We prove that for any
can not be achieved by any
In this paper we prove the existence of infinitely many radial solutions of
In this paper we study special properties of solutions of the initial value problem (IVP) associated to the Benjamin-Ono-Zakharov-Kuznetsov (BO-ZK) equation. We prove that if initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. In other words, the extra regularity on the data propagates in the solutions in the direction of the dispersion. The method of proof to obtain our result uses weighted energy estimates arguments combined with the smoothing properties of the solutions. Hence we need to have local well-posedness for the associated IVP via compactness method. In particular, we establish a local well-posedness in the usual
Using bifurcation theory, we prove the existence of spiky steady states and investigate the stability of bifurcating solutions of the one-dimensional continuous neighbour based chemotaxis model, in which the one-step jumping probability rate of cells is determined only by the chemoattractant concentration at the destination. These spiky steady states are crucial when we model cell aggregation, the most important phenomenon in chemotaxis.
In this paper, we consider the problem
We deal with monotonicity with respect to
In this paper, we study equations involving fully nonlinear nonlocal operators
We shall establish a maximum principle for anti-symmetric functions on any half space, and obtain key ingredients for proving the symmetry and monotonicity for positive solutions to the fully nonlinear nonlocal equations. Especially, a Liouville theorem is derived, which will be useful in carrying out the method of moving planes on unbounded domains for a variety of problems with fully nonlinear nonlocal operators.
It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by
In this work we study asymptotic properties of global solutions for an initial value problem of a second order fractional differential equation with structural damping. The evolution equation considered includes plate equation problems. We show asymptotic profiles depending on the exponents of the Laplace operators involved in the equation and optimality of the decay rates for the associated energy and the
In this work, we study the regularity criterion for the 3D incompressible MHD equations. By making use of the structure of the system, we obtain a criterion that is imposed on the magnetic vector field and only one component of the velocity vector field, both in scaling invariant spaces. Moreover, the norms imposed on the magnetic vector field are the Lebesgue and anisotropic Lebesgue norms. This improved the result of our previous blow up criterion in [
We establish the existence, uniqueness and exponential attraction properties of an invariant measure for the MHD equations with degenerate stochastic forcing acting only in the magnetic equation. The central challenge is to establish time asymptotic smoothing properties of the associated Markovian semigroup corresponding to this system. Towards this aim we take full advantage of the characteristics of the advective structure to discover a novel Hörmander-type condition which only allows for several noises in the magnetic direction.
This paper concerns with the motion of the interface for a damped hyperbolic Allen–Cahn equation, in a bounded domain of
We study the long time behaviour of the solutions for a class of nonlinear damped fractional Schrödinger type equation with anisotropic dispersion and in presence of a quadratic potential in a two dimensional unbounded domain. We prove that this behaviour is characterized by the existence of regular compact global attractor with finite fractal dimension.
In this paper, we are concerned with the large-time behavior of solutions to the Cauchy problem on the non-isentropic compressible micropolar fluid. For the initial data near the given equilibrium we prove the global well-posedness of classical solutions and obtain the optimal algebraic rate of convergence in the three-dimensional whole space. Moreover, it turns out that the density, the velocity and the temperature tend to the corresponding equilibrium state with rate
In this paper, we examine an epidemic model which is described by a system of two equations with nonlocal diffusion on the equation for the infectious agents
We study the emergent dynamics of the Cucker-Smale (C-S for brevity) ensemble under adaptive couplings. For the adaptive couplings, we basically consider two types of couplings: Hebbian vs. anti-Hebbian. When the Hebbian rule is employed, we present sufficient conditions leading to the mono-cluster flocking using the Lyapunov functional approach. On the other hand, for the anti-Hebbian rule, the possibility of mono-cluster flocking mainly depends on the integrability of the communication weight function and the regularity of the adaptive law at the origin. In addition, we perform numerical experiments and compare them with our analytic results.
We prove the existence of positive radial solutions to the problem
We study the existence of positive solutions of the following degenerate coercive quasilinear elliptic equations:
We establish uncertainty principles and Hardy inequalities for the fractional Grushin operator, which are reduced to those inequalities for the fractional generalized sublaplacian. The key ingredients to obtain them are an explicit integral representation and a ground state representation for the fractional powers of generalized sublaplacian.
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