In this work, we study the regularity of the Cauchy problem for the free transport equation, and by using the inverse Wigner transformation, we reduce this problem to the Cauchy problem of a class of linear homogeneous hyperbolic Schro & uml;dinger equation. We prove firstly the analytical smoothing effect of Cauchy problem for Schro & uml;dinger type equation if the initial datum is exponential decay. Finally we prove the directional propagation of the exponential decay and also analytic regularity for free transport equation.
We address the problem given by the following partial differential equation: some semi-Linear parabolic equations with uniformly elliptic non-local operators in Half-Space. Initially, we establish a generalized weighted average inequality and a maximum principle in unbounded domains, which are crucial for the sliding method. Then, we employ sliding to demonstrate the monotonicity of bounded positive solutions. In this paper, we will remove the monotonicity assumption of the kernel function $a(x)$ by using the sliding method. The techniques employed in the process of this method have applications to other problems related to uniformly elliptic operators.
In this paper, we consider an eigenvalue problem for a class of nonlinear operators containing p(& centerdot;)-Laplacian and mean curvature operator with mixed boundary conditions. More precisely, we are concerned with the problem with the Dirichlet condition on a part of the boundary and the Steklov boundary condition on an another part of the boundary. We show that the eigenvalue problem has infinitely many eigenpairs by using the celebrated Ljusternik-Schnirelmann principle of the calculus of variation. Moreover, in a variable exponent Sobolev space, there are two cases where the infimum of all eigenvalues is equal to zero and is positive.
In this paper, we consider the structure of the $L^2$-spectrum of the sub-Laplacian on 2-step stratified Lie groups by using the theory of unitary irreducible representations and Hermit functions. We extend the results for Heisenberg group and H-type Lie group to more general 2-step stratified Lie groups without the Moore-Wolf condition.
This note is devoted to investigating regularity criteria for 3D liquid crystal flows in Besov space.
This paper studies the nonlinear fractional Helmholtz equation (-triangle)(s) u-k(2)u=Q(x)|u|(p-2)u, in R-N, N >= 3, (0.1) where N/N+1 < s < N/2, 2(N+1)/N-1 infinity) Q(x) < sup(x is an element of R)(N) Q(x). (0.2) For sufficiently large k>0, the existence of real-valued solutions to (0.1) is established. Furthermore, as k -> infinity, it is shown that the sequence of solutions associated with the ground states of a dual equation concentrates, after rescaling, at global maximum points of the function Q(x).
In this work, we study the existence of one-sign solutions for the following problem: $ \begin{cases} -\Delta u = \lambda a(x) f(u), & \text{in } \mathbb{R}^N, \\ u(x) \to 0, & \text{as } |x| \to +\infty. \end{cases} $ where $N≥3$, $λ$ is a real parameter and $a∈C_{loc}^α$ ($\mathbb{R}^N$,$\mathbb{R}$) for some $α∈(0,1)$ is a weighted function, $f$:$\mathbb{R}$→$\mathbb{R}$ is a Hölder continuous function with exponent $α$ such that $f(s)s$>0 for any $s\ne0$. We determine the intervals of $λ$ for the existence, exact multiplicity of one-sign solutions for this problem. We use bifurcation techniques and the approximation of connected components to prove our main results.
Recently,qualitative analysis of peaked solutions of Lane-Emden problem in dimension two has been widely considered.In particular,the Morse index of con-centrated solutions with a single peak or multi peaks has been computed in[15,16]separately.In this paper,we continue to consider the qualitative properties of the eigenvalues and eigenfunctions of the linearized Lane-Emden problem associated to peak solutions.Here we establish the fine behaviors of the first m eigenvalues and eigenfunctions of the linearized Lane-Emden problem in dimension two,and corre-spondingly the number of concentrated points of the first m eigenfunctions are given.
In this work, we study the existence of one-sign solutions for the following problem: { -triangle u=Aa(x)f(u), in R-N, u(x)-> 0, as |x|->+infinity, where N >= 3, lambda is a real parameter and a is an element of C-loc(alpha)(R-N,R) for some c is an element of(0,1) is a weighted function, f : R -> R is a Holder continuous function with exponent c such that f (s)s>0 for any s not equal 0. We determine the intervals of A for the existence, exact multiplicity of one-sign solutions for this problem. We use bifurcation techniques and the approximation of connected components to prove our main results.
This paper is concerned with the asymptotic behavior of solutions of non-autonomous reaction-diffusion equation with delays. The well-posedness theory of equation for the initial data belonging to C-L (R)(Omega)(1
We consider periodic solutions of the following nonlinear system associated with the fractional Laplacian (-partial derivative xx)su(x)+del F(u(x)) = 0 in R, where F : R2 -> R is a smooth double-well potential. For the case that F is even in its two variables we obtain the existence of more and more periodic solutions with large period, by using Clark's theorem. For the case that F is only even in its the second variable and the origin is a saddle critical point of F, we give two periodic solutions by using Morse theory.
In this article, we investigate the well-posedness of the initial-boundary value problem (I-B-V problem) for the fifth-order KdV equation posed on a finite domain with nonlinear boundary conditions. Firstly, we establish various a priori estimates, including Kato smoothing effects, sharp trace regularity, and nonlinear estimates. Subsequently, we demonstrate that the initial-boundary value problem of the fifth-order KdV equation with quadratic boundary feedbacks is locally well-posed for the appropriately chosen initial value and boundary values.
In this paper,we study the existence of at least two,three and infinitely many solutions for nonlinear problems on the Sierpiński gasket,modelling some phys-ical phenomena such as reaction-diffusion problems,elastic properties of fractal media and flow through fractal non-smooth domains.We will obtain the existence of two weak solutions when nonlinear term f(x,t)is non-negative,when it is non-positive in neighbored of zero and otherwise is positive we will show the existence of three weak solutions,and when it is odd we will get the existence of infinitely many solutions.The results are proved by using some critical point theorems.
In this paper, we consider the following critical Choquard equation -triangle u=mu f (x) |u|(p-2)u+ ( integral(Omega) (g(y)|u(y)|6-nu) (|x-y|nu) dy( ))( )g( )(x)|u|(4-nu)u, x is an element of ohm, where mu >0 is a parameter, nu is an element of(0,3), p is an element of(4,6) and f,g are continuous functions. For mu small enough, by using Lusternik-Schnirelmann category theory, we establish a relationship between the number of solutions and the category of the global maximum set of g.
In this paper,we obtain a parameter type logarithmic Sobolev inequality with weight and a modified parameter type logarithmic Sobolev inequality with weight on Euclidean space based on the parameter type logarithmic Caffarelli-Kohn-Nirenberg inequality on Euclidean space,respectively.By virtue of the convexity of a special function equivalent to a logarithmic Hölder inequality,and combining the Sobolev in-equality and Gagliardo-Nirenberg inequality on Hörmander's vector fields,we also derive a logarithmic Sobolev inequality and a parameter type logarithmic Gagliardo-Nirenberg inequality on Hörmander's vector fields,respectively.In addition,a pa-rameter type logarithmic Sobolev inequality on Hörmander's vector fields is given by suitable stretching transformation.
Recently, qualitative analysis of peaked solutions of Lane-Emden problem in dimension two has been widely considered. In particular, the Morse index of concentrated solutions with a single peak or multi peaks has been computed in [15, 16] separately. In this paper, we continue to consider the qualitative properties of the eigenvalues and eigenfunctions of the linearized Lane-Emden problem associated to peak solutions. Here we establish the fine behaviors of the first m eigenvalues and eigenfunctions of the linearized Lane-Emden problem in dimension two, and corre spondingly the number of concentrated points of the first m eigenfunctions are given.
This paper considers initial boundary value to a pseudo-parabolic equation with singular potential u(t)/|x|(s)-Delta u(t)-Delta u=|u|(p-2)u with 2
In this paper, we are concerned about a food chain model with a protection zone for the prey species. Dynamical behavior, nonexistence and existence of positive steady states are obtained and there exist several critical values determined by the pa-rameters and the protection zone for the growth rate of the prey species. The results reveal that the protection zone is effective for the survival of the prey species and bene ficial for the coexistence of multiple species. Moreover, different properties of positive steady states from those of the two-species models are shown. The introduction of the prey or the top predator can be either favorable or unfavorable for the coexistence of multiple species.
This paper is devoted to estimates on weighted L-q-norms of the nonstation-ary 3D Navier-Stokes flow in an exterior domain. By multiplying the Navier-Stokes equation with a well selected vector field, an integral equation is derived, from which, we establish the weighted estimate |parallel to parallel to|x|(alpha)u(t)parallel to(q)<= C(1+t(alpha/2+epsilon))t(-3/2(1-1/q)),t>0, where 013(0), |x|gL (2) with ||U-0||(3) sufficiently small. With the aid of the representation of the flow, we also prove that if in addition u(0)is an element of D-a(1-1/b,b) for some 6/5 <= a<3/2 and 10 holds, where alpha>0 and 1
