Communications on Pure & Applied Analysis
November 2019 , Volume 18 , Issue 6
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We prove a global Calderón-Zygmund type estimate in the framework of Lorentz spaces for a variable power of the gradients of weak solution pair (u,P ) to the Dirichlet problem of stationary Stokes system. It is mainly assumed that the leading coefficients are merely measurable in one spatial variable and have sufficiently small bounded mean oscillation (BMO) seminorm in the other variables, the boundary of underlying domain is Reifenberg flat, and the variable exponents p(x) satisfy the so-called log-Hölder continuity.
This paper studies a porous elasticity system with past history
By introducing a new variable, we establish an explicit and a general decay of energy for the case of equal-speed wave propagation as well as for the nonequalspeed case. To establish our results, we mainly adopt the method developed by Guesmia, Messaoudi and Soufyane [Electron. J. Differ. Equa. 2012(2012), 1-45] and some properties of convex functions developed by Alabau-Boussouira and Cannarsa [C. R. Acad. Sci. Paris Ser. I, 347(2009), 867-872], Lasiecka and Tataru [Differ. Inte. Equa., 6(1993), 507-533]. In addition we remove the assumption that b is positive constant in [J. Math. Anal. Appl., 469(2019), 457-471] and hence improve the result.
In this paper we deal with uniqueness of unbounded solutions to the following problem
This paper is concerned with the Camassa-Holm-KP equation, which is a model for shallow water waves. By using the analysis of the phase space, we obtain some qualitative properties of equilibrium points and existence results of solitary wave solutions for the Camassa-Holm-KP equation without delay. Furthermore we show the existence of solitary wave solutions for the equation with a special local delay convolution kernel by combining the geometric singular perturbation theory and invariant manifold theory. In addition, we discuss the existence of solitary wave solutions for the Camassa-Holm-KP equation with strength
This paper is concerned with asymptotic spreading for a time-periodic predator-prey system where both species synchronously invade a new habitat. Under two different conditions, we show the bounds of spreading speeds of the predator and the prey, which is proved by the theory of asymptotic spreading of scalar equations, comparison principle and generalized eigenvalue. We show either the predator or the prey has a spreading speed that is determined by the linearized equation at the trivial steady state while the spreading speed of the other also depends on the interspecific nonlinearity. From the viewpoint of population dynamics, our results imply that the predator may play a negative effect on the spreading of the prey while the prey may play a positive role on the spreading of the predator.
In this paper, we show a new regularity result on the transport density
We focus on the spreading properties of solutions of monostable equations with fast diffusion. The nonlinear reaction term involves a weak Allee effect, which tends to slow down the propagation. We complete the picture of [
In this paper, by using Fourier splitting method and the properties of decay character
The mixed boundary value problem for a compressible Stokes system of partial differential equations in a bounded domain is reduced to two different systems of segregated direct Boundary-Domain Integral Equations (BDIEs) expressed in terms of surface and volume parametrix-based potential type operators. Equivalence of the BDIE systems to the mixed BVP and invertibility of the matrix operators associated with the BDIE systems are proved in appropriate Sobolev spaces.
In this work we study the existence of positive solutions for the following class of coupled elliptic systems involving nonlinear Schrödinger equations
In this paper, we investigate bi-parameter modulation spaces on the product of two Euclidean spaces
This work is concerned with the invariant measure of a stochastic fractional Burgers equation with degenerate noise on one dimensional bounded domain. Due to the disturbance and influence of the fractional Laplacian operator on a bounded interval interacting with the degenerate noise, the study of the system becomes more complicated. In order to get over the difficulties caused by the fractional Laplacian operator, the usual Hilbert space does not fit the system, we introduce an appropriate weighted space to study it. Meanwhile, we apply the asymptotically strong Feller property instead of the usually strong Feller property to overcome the trouble caused by the degenerate noise, the corresponding Malliavin operator is not invertible. We finally derive the uniqueness of the invariant measure which further implies the ergodicity of the stochastic system.
In this paper we study subquadratic elliptic systems in divergence form with VMO leading coefficients in
This paper deals with the nonlinear phase field system
in a general domain
In this paper, we study the following fractional Kirchhoff equation with critical nonlinearity
In this paper, we prove the existence of bounded positive solutions for a class of semilinear degenerate elliptic equations involving supercritical cone Sobolev exponents. We also obtain the existence of multiple solutions by the Ljusternik-Schnirelman theory.
We are concerned with a class of nonlocal elliptic equations as follows:
In this note, we study the mean curvature flow and the prescribed mean curvature type equation with general capillary-type boundary condition, which is
In this paper, we study the global bifurcation curves and the exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem
We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schrödinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for the solution of the linear nonhomogenenous problem by using the Fokas method (also known as the unified transform method). We use this representation formula to prove space and time estimates on the solutions of the linear model in fractional Sobolev spaces by using Fourier analysis. Secondly, we consider the nonlinear model with a power type nonlinearity and prove the local wellposedness by means of a classical contraction argument. We obtain Strichartz estimates to treat the low regularity case by using the oscillatory integral theory directly on the representation formula provided by the Fokas method. Global wellposedness of the defocusing model is established up to cubic nonlinearities by using the multiplier technique and proving hidden trace regularities.
We study the regularity of solution and of sonic boundary to a degenerate Goursat problem originated from the two-dimensional Riemann problem of the compressible isothermal Euler equations. By using the ideas of characteristic decomposition and the bootstrap method, we show that the solution is uniformly
This paper mainly investigates a class of almost periodic Nicholson's blowflies model involving a nonlinear density-dependent mortality term and time-varying delays. Combining Lyapunov function method and differential inequality approach, some novel assertions are established to guarantee the existence and exponential stability of positive almost periodic solutions for the addressed model, which generalize and refine the corresponding results in some recent published literatures. Particularly, an example and its numerical simulations are given to support the proposed approach.
In this paper we are concerned with the multiplicity of solutions near resonance for the following nonlinear equation:
associated with the Dirichlet boundary condition, where
In this article, we consider a quasilinear hyperbolic system of partial differential equations governing the dynamics of a thin film of a perfectly soluble anti-surfactant liquid. We construct elementary waves of the corresponding Riemann problem and study their interactions. Further, we provide exact solution of the Riemann problem along with numerical examples. Finally, we show that the solution of the Riemann problem is stable under small perturbation of the initial data.
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