Well‐posedness and regularity results for a 2m‐th order parabolic equation in symmetric conical domains of ℝN+1

In this paper, we use the domain decomposition method to prove well‐posedness and smoothness results in anisotropic weighted Sobolev spaces for a multidimensional high‐order parabolic equation set in conical time‐dependent domains of . Copyright © 2017 John Wiley & Sons, Ltd.     

Well‐posedness for the Cauchy problem of the modified Hunter–Saxton equation in the Besov spaces

This paper is concerned with the Cauchy problem of the modified Hunter‐Saxton equation. The local well‐posedness of the model equation is obtained in Besov spaces (which generalize the Sobolev spaces H^s) by using Littlewood‐Paley decomposition and transport equation theory. Moreover, the local well‐posedness in critical case (with ) is considered.     

Nonstationary Stokes system in anisotropic Sobolev spaces

The nonstationary Stokes system with slip boundary conditions is considered in a bounded domain . We prove the existence and uniqueness of solutions to the problem in anisotropic Sobolev spaces . Thanks to the slip boundary conditions, the Stokes problem is transformed to the Poisson and the heat equation. In this way, difficult calculations that must be performed in considerations of boundary value problems for the Stokes system are avoided. This approach does not work for the Dirichlet and the Neumann boundary conditions. Because solvability of the Poisson and the heat equation is carried out by the regularizer technique, we have that σ > 3,α > 0. Copyright © 2014 John Wiley & Sons, Ltd.     

Nonlinear anisotropic parabolic equations in with locally integrable data

In this paper, we prove existence and regularity of weak solutions for a class of nonlinear anisotropic parabolic problems in with locally integrable data. Our approach is based on the anisotropic Sobolev inequality, a smoothness, and compactness results. Copyright © 2012 John Wiley & Sons, Ltd.     

Galerkin finite element method for generalized Forchheimer equation of slightly compressible fluids in porous media

We consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for the density. We study Galerkin finite elements method for the initial boundary value problem. The existence and uniqueness of the approximation are proved. A prior estimates for the solutions in , time derivative in and gradient in , with a∈(0,1) are established. Error estimates for the density variable are derived in several norms for both continuous and discrete time procedures. Numerical experiments using backward Euler scheme confirm the theoretical analysis regarding convergence rates. Copyright © 2017 John Wiley & Sons, Ltd.     

Global well‐posedness and blow‐up criterion for the periodic quasi‐geostrophic equations in Lei‐Lin‐Gevrey spaces

In this paper we consider a periodic 2‐dimensional quasi‐geostrophic equations with subcritical dissipation. We show the global existence and uniqueness of the solution for small initial data in the Lei‐Lin‐Gevrey spaces . Moreover, we establish an exponential type explosion in finite time of this solution.     

On the absence of global weak solutions for some differential inequalities of Sobolev type in an exterior domain

In this paper, using the nonlinear capacity method, we derive sufficient conditions for the nonexistence of global weak solutions for some differential inequalities of Sobolev type posed in an exterior domain of , N ≥ 3. The obtained results are extensions of those established by Korpusov and Sveshnikov in the case of the entire space .     

Bilinear estimate on tent‐type spaces with application to the well‐posedness of fluid equations

We introduce a class of tent‐type spaces and establish a Poisson extension result of Triebel–Lizorkin spaces . As an application, we get the well‐posedness of Navier–Stokes equations and magnetohydrodynamic equations with initial data in critical Triebel–Lizorkin spaces , . Copyright © 2016 John Wiley & Sons, Ltd.     

On the solutions of a higher‐order difference equation in terms of generalized Fibonacci sequences

This paper deals with the solutions, stability character, and asymptotic behavior of the difference equation where and the initial values x_?k,x_{?k + 1},…,x₀ are nonzero real numbers, such that their solutions are associated to Horadam numbers. Copyright © 2015 John Wiley & Sons, Ltd.     

Low Mach number limit of the compressible magnetohydrodynamic equations with zero thermal conductivity coefficient

The low Mach number limit for classical solutions of the compressible magnetohydrodynamic equations without thermal conductivity is, here, studied. A uniform existence result for the Cauchy problem in is proved under the assumption that the initial data are uniformly bounded with respect to the Mach number in and are well‐prepared in . Copyright © 2011 John Wiley & Sons, Ltd.     

Solutions for perturbed biharmonic equations with critical nonlinearity

In this paper, we study the perturbed biharmonic equations where Δ² is the biharmonic operator, is the Sobolev critical exponent, p ∈ (2,2 ^{* *} ), P(x), and Q(x) are bounded positive functions. Under some given conditions on V, we prove that the problem has at least one nontrivial solution provided that and that for any , it has at least n ^* pairs solutions if , where and are sufficiently small positive numbers. Moreover, these solutions u_ε → 0 in as ε → 0. Copyright © 2013 The authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd.     

Time‐harmonic and asymptotically linear Maxwell equations in anisotropic media

This paper is focused on following time‐harmonic Maxwell equation: where is a bounded Lipschitz domain, is the exterior normal, and ω is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that f is asymptotically linear as , we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor and permittivity tensor , ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.     

Low regularity solutions,blowup, and global existence for a generalization of Camassa–Holm‐type equation

We consider a generalization of Camassa–Holm‐type equation including the Camassa–Holm equation and the Novikov equation. We mainly establish the existence of solutions in lower order Sobolev space with . Then, we present a precise blowup scenario and give a global existence result of strong solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

Local well‐posedness and zero‐α limit for the Euler‐α equations

In this paper, we prove the local‐in‐time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler‐α equations of inviscid incompressible fluid flows in . We also establish the convergence rate of the solutions of the Euler‐α equations to the corresponding solutions of the Euler equations as the regularization parameter α approaches 0 in . Copyright © 2016 John Wiley & Sons, Ltd.     

Ground state of solutions for a class of fractional Schrödinger equations with critical Sobolev exponent and steep potential well

In this paper, we study the following fractional Schrödinger equations: (1) where (?△)^α is the fractional Laplacian operator with , 0≤s ≤2α , λ >0, κ and β are real parameter. is the critical Sobolev exponent. We prove a fractional Sobolev‐Hardy inequality and use it together with concentration compact theory to get a ground state solution. Moreover, concentration behaviors of nontrivial solutions are obtained when the coefficient of the potential function tends to infinity.     

Nonlinear wave equations of carrier type on manifolds

In this paper, we investigate the existence and uniqueness of a solution for differential equations of the carrier type on lateral boundary Σ of the cylinder Q, cf. (1). The main point is to transform this initial value problem into a differential operator equation of the type , cf. (6). The operator is defined in Section 2, and it acts in Sobolev spaces on Γ, boundary of Ω. The initial value problem (6) is investigated in Section 3 by the method of Faedo–Galerkin. Thus, we obtain the existence of a weak solution for (6), and in Section 4, we prove its uniqueness. Copyright © 2012 John Wiley & Sons, Ltd.     

Multiple non semi‐trivial solutions of systems of Kirchhoff‐type equations with discontinuous nonlinearities in RN

In the present paper, we study the problem of multiple non semi‐trivial solutions for the following systems of Kirchhoff‐type equations with discontinuous nonlinearities (1.1) where F∈C¹(R^N×R⁺×R⁺,R),V∈C(R^N,R), and By establishing a new index theory, we obtain some multiple critical point theorems on product spaces, and as applications, three multiplicity results of non semi‐trivial solutions for (1.1) are obtained. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence results for a class of nonlocal problems involving p(x)‐Laplacian

This paper is concerned with the existence of solutions to a class of p(x)‐Kirchhoff‐type equations with Dirichlet boundary data as follows: By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish some conditions on the existence of solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

Polynomial stability for system of 3 wave equations with infinite memories

Here, a system of 3 wave equations in with infinite memories acting in the first 2 equations is considered. Using weighted spaces, we prove the polynomial stability of the system under some conditions on μ₁,μ₂, and ϕ as .     

A nonhomogeneous boundary value problem for the Kuramoto–Sivashinsky equation in a quarter plane

We study the initial boundary value problem for the one‐dimensional Kuramoto–Sivashinsky equation posed in a half line with nonhomogeneous boundary conditions. Through the analysis of the boundary integral operator, and applying the known results of the Cauchy problem of the Kuramoto–Sivashinsky equation posed on the whole line , the initial boundary value problem of the Kuramoto–Sivashinsky equation is shown to be globally well‐posed in Sobolev space for any s >?2. Copyright © 2017 John Wiley & Sons, Ltd.