Global W2,p Solutions of GBBM Equations on Unbounded Domain

In this paper we study the initial boundary value problem of GBBM equations on unbounded domain u_t - Δu_t = div f(u) u(x,0) = u_0(x) u|_{∂Ω} = 0 and corresponding Cauchy problem. Under the conditions: f( s) ∈ C^sup1 and satisfies (H)\qquad |f'(s)| ≤ C|s|^ϒ, 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3; 0 ≤ ϒ < ∞ if n = 2 u_0(x) ∈ W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω)(W^{2,p}(R^n) ∩ W^{2,2}(R^n) for Cauchy problem), 2 ≤ p < ∞, we obtain the existence and uniqueness of global solution u(x, t) ∈ W^{1,∞}(0, T; W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω))(W^{1,∞}(0, T; W^{2,p}(R^n) ∩ W^{2,2} (R^n)) for Cauchy problem), so the results of [1] and [2] are generalized and improved in essential.     

The atomic decomposition in

In this paper we present an atomic decomposition of integrable functions. As an application we compute the distance of in to the Hardy space .

     

Optimal embeddings and compact embeddings of Bessel-potential-type spaces

First, we establish necessary and sufficient conditions for embeddings of Bessel potential spaces \({H^{\sigma}X(\mathbb R^n)}\) with order of smoothness less than one, modelled upon rearrangement invariant Banach function spaces \({X(\mathbb R^n)}\), into generalized Hölder spaces. To this end, we derive a sharp estimate of modulus of smoothness of the convolution of a function \({f\in X(\mathbb R^n)}\) with the Bessel potential kernel g _σ , 0 < σ < 1. Such an estimate states that if \({g_{\sigma}}\) belongs to the associate space of X, then

$\omega(f*g_{\sigma},t)\precsim \int\limits_0^{t^n}s^{\frac{\sigma}{n}-1}f^*(s)\,ds \quad {\rm for\,all} \quad t\in(0,1) \quad {\rm and\,every}\quad f\in X(\mathbb R^n).$

Second, we characterize compact subsets of generalized Hölder spaces and then we derive necessary and sufficient conditions for compact embeddings of Bessel potential spaces \({H^{\sigma}X(\mathbb R^n)}\) into generalized Hölder spaces. We apply our results to the case when \({X(\mathbb R^n)}\) is the Lorentz–Karamata space \({L_{p,q;b}(\mathbb R^n)}\). In particular, we are able to characterize optimal embeddings of Bessel potential spaces \({H^{\sigma}L_{p,q;b}(\mathbb R^n)}\) into generalized Hölder spaces and also compact embeddings of spaces in question. Applications cover both superlimiting and limiting cases.

     

Generalized Moser–Trudinger inequality for unbounded domains and its application

We give a version of the Moser–Trudinger inequality for Orlicz–Sobolev spaces embedded into exponential and multiple exponential spaces on unbounded domains in ${\mathbb R^n, n \geq 2}$ . Applying this result and the Mountain Pass Theorem we study the existence of non-trivial weak solutions to the problem $$\begin{array}{ll}u \in W^1 L^{\Phi}(\mathbb R^n)\quad{\rm and}\\\quad -{\rm div} \left(\Phi ' (|\nabla u|)\frac{\nabla u}{|\nabla u|}\right)+V(x)\Phi'(|u|)\frac{u}{|u|} =f(x,u)\quad{\rm in}\, \mathbb R^n,\end{array}$$ where Φ is a Young function such that the space ${W^1 L^{\Phi}(\mathbb R^n)}$ is embedded into an Orlicz space of the exponential or multiple exponential type, the nonlinearity f(x, t) has the corresponding critical growth and V(x) is a continuous potential.     

Two inequalities for pseudo-differential operators

Изучается ограничен ность псевдодиффере нциальных операторов на \(L^2 (R^n )\) и на пр остранствах Харди в \(R^n \) . Пусть \(D_k = \{ \xi \in R^n :2^{k - 1} \leqq \left| \xi \right|< 2^k \} , k = 1,2,3, \ldots ,\) и \(D_0 = \{ \xi \in R^n :\left| \xi \right|< 1\} \) . Псевдодиффер енциальный операторP с символом p определяется соотно шением $$Pf(x) = \int\limits_{R^n } {e^{ix \cdot \xi } p(x,\xi )\hat f(\xi )d\xi ,x \in R^n .} $$ Будем говорить, что p пр инадлежит классу \(\bar S_{\varrho ,} {}_\delta (M,N), 0 \leqq \delta ,\varrho \leqq 1\) , ес ли $$\left| {D_x^a p(x,\xi )} \right| \leqq C_a (1 + \left| \xi \right|)^{\delta \left| a \right|} , x,\xi \in R^n ,\left| a \right| \leqq M,$$ и $$\int\limits_{D_k } {\left| {D_x^a D_\xi ^\beta p(x,\xi )} \right|d\xi \leqq C_{a\beta } 2^{kn} 2^{k(\delta |a| - \varrho |\beta |)} , x} \in R^n , k = 0,1,2, \ldots ;|a| \leqq M, |\beta | \leqq N.$$ Изучаются условия, ко торым должны удовлет ворять ?. δ,M иN, чтобы для каждого символа \(p \in \bar S_\varrho , {}_\delta (M,N)\) соответствующий оп ераторP был ограниче н на \(L^2 (R^n )\) . Далее, пусть \(p \in S_\varrho , {}_\delta \) , если дл я всех мультииндексо в а и β выполнено условие $$|D_x^a D_\xi ^\beta p(x,\xi )| \leqq C_{a\beta } (1 + |\xi |)^{\delta |\alpha | - \varrho |\beta |} , x,\xi \in R^n .$$ Доказывается, что при 0≦δ<1 операторP отображ ает пространство Харди \(H^p (R^n )\) в локальное пространство Харди ? ^p , если символp принадл ежит классуS ₁, δ.     

Cauchy Problem for a Class of Totally Characteristic Hyperbolic Operators with Characteristics of Variable Multiplicity in Gevrey Classes

This paper studies tho Cauchy problem of totally characteristic hyperbolic operator (1.1) in Gevrey classes, and obtains the following main result: Under the conditions (I) - (VI), if 1 ≤ s < \frac{σ}{σ-1} (σ is definded by (1.7)). then the Cauchy problem (1.1) is wellposed in B ([0, T], G^s_{L²}, (R^n)); if s = \frac{σ}{σ-1}, then the Cauchy problem (1.1) is wellpooed in B ([0, e], G^{\frac{σ}{σ-1}}_{L²}(R^n)) (where e > 0, small enough).     

Optimal Sobolev Type Inequalities in Lorentz Spaces

It is well known that the classical Sobolev embeddings may be improved within the framework of Lorentz spaces L ^p,q : the space $\mathcal{D}^{1,p}(\mathbb R^n)$ , 1?<?p?<?n, embeds into $L^{p^*,q}(\mathbb R^n)$ , p?≤?q?≤?∞. However, the value of the best possible embedding constants in the corresponding inequalities is known just in the case $L^{p^*,p}(\mathbb R^n)$ . Here, we determine optimal constants for the embedding of the space $\mathcal{D}^{1,p}(\mathbb R^n)$ , 1?<?p?<?n, into the whole Lorentz space scale $L^{p^{\ast}, q}(\mathbb R^n)$ , p?≤?q?≤?∞, including the limiting case q?=?p of which we give a new proof. We also exhibit extremal functions for these embedding inequalities by solving related elliptic problems.     

Sharp maximal estimates for doubly oscillatory integrals

We study doubly oscillatory integrals

and prove a sharp maximal estimate which is an immediate consequence of a well-known conjecture in Fourier analysis on .

     

On Singular Functional Differential Inequalities

Classical theorems on differential inequalities [1, 2, 3] are generalized for initial value problems of the kind and where is a singular Volterra operator, is continuous and positive on ]a, b], is a norm in R ⁿ, and [u]₊ and [u]_– are respectively the positive and the negative part of the vector u R ⁿ.     

A NECESSARY AND SUFFICIENT CONDITION OF EXISTENCE OF GLOBAL SOLUTIONS FOR SOME NONLINEAR HYPERBOLIC EQUATIONS

ＡＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＯＮＤＩＴＩＯＮＯＦＥＸＩＳＴＥＮＣＥＯＦＧＬＯＢＡＬＳＯＬＵＴＩＯＮＳＦＯＲＳＯＭＥＮＯＮＬＩＮＥＡＲＨＹＰＥＲＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＺＨＡＮＧＱＵＡＮＤＥ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅ...     

On a cubic-quintic Ginzburg-Landau equation with global coupling

We study standing wave solutions in a Ginzburg-Landau equation which consists of a cubic-quintic equation stabilized by global coupling

We classify the existence and stability of all possible standing wave solutions.

     

Global W2,p (2<=p

This paper studies the initial-boundary value problem of GBBM equations u_t - Δu_t = div f(u) \qquad\qquad\qquad(a) u(x, 0) = u_0(x)\qquad\qquad\qquad(b) u |∂Ω = 0 \qquad\qquad\qquad(c) in arbitrary dimensions, Ω ⊂ R^n. Suppose that. f(s) ∈ C¹ and |f'(s)| ≤ C (1+|s|^ϒ), 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3, 0 ≤ ϒ < ∞ if n = 2, u_0 (x) ∈ W^{2⋅p}(Ω) ∩ W^{1⋅p}_0(Ω) (2 ≤ p < ∞), then ∀T > 0 there exists a unique global W^{2⋅p} solution u ∈ W^{1,∞}(0, T; W{2⋅p}(Ω)∩ W^{1⋅p}_0(Ω)), so the known results are generalized and improved essentially.     

Existence of Nontrivial Weak Solutions to Quasi-linear Elliptic Equations with Exponential Growth

In this paper, we study the existence of nontrivial weak solutions to the following quasi-linear elliptic equations $$-Δ_nu+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^β}, x ∈ R^n(n ≥ 2),$$ where $-Δ_nu=-div(|∇u|^{n-2}∇u), 0 ≤β < n, V:R^n→R$ is a continuous function, f (x,u) is continuous in $R^n×R$ and behaves like $e^{αu^{\frac{n}{n-1}}}$ as $u→+∞$.     

与截面相关的Morrey空间与奇异积分

In this paper, we define the Morrey spaces M_F~(p,q) (Rn) and the Campanato spaces E_F~(p,q) (R~n) associated with a family F of sections and a doubling measure μ, where F is closely related to the Monge-Amp`ere equation. Furthermore, we obtain the boundedness of the Hardy-Littlewood maximal function associated to F, Monge-Amp`ere singular integral operators and fractional integrals on M_F~(p,q)(R~n). We also prove that the Morrey spaces M_F~(p,q) (R~n)and the Campanato spaces E_F~(p,q) (R~n) are equivalent with 1 ≤ q ≤ p ∞.     

A mixed boundary-value problem for a hyperbolic-parabolic equation

Let Ω be a bounded domain in the n-dimensional Euclidean space. In the cylindrical domain QT=Ω x [0, T] we consider a hyperbolic-parabolic equation of the form (1) $$Lu = k(x,t)u_{tt} + \sum\nolimits_{i = 1}^n {a_i u_{tx_i } - } \sum\nolimits_{i,j = 1}^n {\tfrac{\partial }{{\partial x_i }}} (a_{ij} (x,t)u_{x_j } ) + \sum\nolimits_{i = 1}^n {t_i u_{x_i } + au_t + cu = f(x,t),} $$ where \(k(x,t) \geqslant 0,a_{ij} = a_{ji} ,\nu |\xi |^2 \leqslant a_{ij} \xi _i \xi _j \leqslant u|\xi |^2 ,\forall \xi \in R^n ,\nu > 0\) . The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces W ₂ ¹ (QT) and W ₂ ² (QT).     

Morrey Regularity of Solutions to Degenerate Elliptic Equations in Rn

In this paper, we study the Morrey regularity of solutions to the de- generate elliptic equation -(a_{ij}u_{xi})_{xj} = -(f_j)_{xj} in R^n. For this purpose, we introduce four weighted Morrey spaces in R^n.     

Conditions of Existence and Uniqueness of Solutions of the Multipoint Boundary Value Problem for a System of Generalized Ordinary Differential Equations

Effective sufficient conditions are established for the solvability and unique solvability of the boundary value problem where is a matrix-function with bounded variation components, is a vector-function belonging to the Carathéodory class corresponding to are continuous functionals (in general nonlinear) defined on the set of all vector-functions of bounded variation.     

Existence of Bounded Solutions for Quasilinear Subelliptic Dirichlet Problems

This paper proves the existence of solution for the following quasilinear subelliptic Dirichlet problem: {Σ^m_{j=1}X^∗_ja_j(X, v, Xv)+ a_o(x, v, Xv) + H(x,v, Xv) = 0 v ∈ M^{1,p}_0(Ω) ∩ L^∞(Ω) Here X = {X_1 , …, X_m} is a system of vector fields defined in an open domain M of R^n, n ≥ 2, Ω ⊂ ⊂ M, and X satisfies the so-called Hormander's condition at the order of r > 1 on M. M_{1,p}_0(Ω) is the weighted Sobolev's space associated with the system X . The Hamiltonian H grows at most like |Xv|^p.     

Interpolation Between $$H^{p(\cdot )}({\mathbb {R}}^n)$$ and $$L^\infty ({\mathbb {R}}^n)$$L∞(Rn): Real Method

Let \(p(\cdot ):\ {\mathbb {R}}^n\rightarrow (0,\infty )\) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into “good” and “bad” parts and then prove the following real interpolation theorem between the variable Hardy space \(H^{p(\cdot )}({\mathbb {R}}^n)\) and the space \(L^{\infty }({\mathbb {R}}^n)\): \((H^{p(\cdot )}(\mathbb R^n),L^{\infty }({\mathbb {R}}^n))_{\theta ,\infty }= WH^{p(\cdot )/(1-\theta )}({\mathbb {R}}^n),\quad \mathrm{where}~\theta \in (0,1), \mathrm{and}\) \(WH^{p(\cdot )/(1-\theta )}({\mathbb {R}}^n)\) denotes the variable weak Hardy space. As an application, the variable weak Hardy space \(WH^{p(\cdot )}({\mathbb {R}}^n)\) with \(p_-:=\mathop {\text {ess inf}}\limits _{x\in {{{\mathbb {R}}}^n}}p(x)\in (1,\infty )\) is proved to coincide with the variable Lebesgue space \(WL^{p(\cdot )}({\mathbb {R}}^n)\).     

A reverse weighted inequality for the Hardy-Littlewood maximal function in Orlicz spaces

Let Φ(t)= ∫_0^t a(s) ds and Ψ(t)= ∫_0^t b(s) ds, where a(s) is a positive continuous function such that ∫_0^1 \frac{a(s)}{s} ds < ∞and ∫_1^{\∞}\frac{a(s)}{s} ds= +\∞, and b(s) is an increasing function such that \lim_{s\to\∞}b(s)= +\∞. Letw be a weight function and suppose that w∈A₁\∩ A_∞'. Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent:(I) there exist positive constants C ₁ and C ₂ such that $$\int_0^s {\frac{{a\left( t \right)}}{t}dt \geqq } C_1 b\left( {C_2 s} \right)foralls > 0;$$ (II) there exist positive constants C ₃ and C ₄ such that $$\int {_{R^n } } \Psi \left( {C_3 \left| {f\left( x \right)} \right|} \right)w\left( x \right)dx \leqq C_4 \int {_{R^n } } \Phi \left( {Mf\left( x \right)} \right)w\left( x \right)dxforallf \in {\mathcal{R}}_0 \left( w \right)$$