Existence and symmetry results for competing variational systems

In this paper we consider a class of gradient systems of type $$\begin{array}{ll} -c_{i} \Delta u_{i} + V_{i}(x)u_{i} = P_{u_i}(u),\qquad u_{1}, \ldots, u_{k} >\; 0\; \text{in}\; \Omega,\\ \quad u_{1} = \cdots = u_{k} = 0 \text{ on } \partial \Omega, \end{array}$$ in a bounded domain ${\Omega \subseteq \mathbb{R}^N}$ . Under suitable assumptions on V _i and P, we prove the existence of ground-state solutions for this problem. Moreover, for k = 2, assuming that the domain Ω and the potentials V _i are radially symmetric, we prove that the ground state solutions are foliated Schwarz symmetric with respect to antipodal points. We provide several examples for our abstract framework.     

Symmetry results for systems involving fractional Laplacian

In this paper we investigate symmetry results for positive solutions of systems involving the fractional Laplacian (1) $\left\{ \begin{gathered} ( - \Delta )^{\alpha _1 } u_1 (x) = f_1 (u_2 (x)),x \in \mathbb{R}^\mathbb{N} , \hfill \\ ( - \Delta )^{\alpha _2 } u_2 (x) = f_2 (u_1 (x)),x \in \mathbb{R}^\mathbb{N} , \hfill \\ \lim _{|x| \to \infty } u_1 (x) = \lim _{|x| \to \infty } u_2 (x) = 0 \hfill \\ \end{gathered} \right. $ where N ≥ 2 and α ₁, α ₂ ∈ (0, 1). We prove symmetry properties by the method of moving planes.     

On oscillation of solutions of second-order systems of deviated differentaial equations

Sufficient conditions are found for the oscillation of proper solutions of the system of differential equations $$\begin{array}{*{20}c} {u'_1 (t) = f_1 (t,u_1 (\tau _1 (t)),...,u_1 (\tau _m (t)),u_2 (\sigma _1 (t)),...,u_2 (\sigma _m (t))),} \\ {u'_2 (t) = f_2 (t,u_1 (\tau _1 (t)),...,u_1 (\tau _m (t)),u_2 (\sigma _1 (t)),...,u_2 (\sigma _m (t))),} \\ \end{array}$$ wheref _i: R₊×R^2m→R (i=1,2) satisfy the local Carathéodory conditions andσ _i , τ _i :R ₊ →R (i=1,...,m) are continuous functions such that $\sigma _i (t) \leqslant t for t \in R_ + ,\mathop {\lim }\limits_{t \to + \infty } \sigma _i (t) = + \infty ,\mathop {\lim }\limits_{t \to + \infty } \tau _i (t) = + \infty (i = 1,...,m)$      

The extended Bitsadze-Lavrent’ev-Tricomi boundary value problem

F. G. Tricomi ([5], [6]) originated the theory of boundary of value problems for mixed type equations by establishing the first mixed type equation known asthe Tricomi equation \(y \cdot u_{xx} + u_{yy} = 0\) which is hyperbolic fory<0, elliptic fory>0, and parabolic fory=0 and then observed that this equation could be applied in Aerodynamics and in general in Fluid Dynamics (transonic flows). See: M. Cribario [1], G. Fichera [2], and our doctoral dissertation [4]. Then M. A. Lavrent’ev and A. V. Bitsadze [3] established together a new mixed type boundary value problem for the equation \(\operatorname{sgn} (y) \cdot u_{xx} + u_{yy} = 0\) where sgn (y)=1 fory>0, =?1 fory<0, fory=0, which involved thediscontinuous coefficient K=sgn (y) ofu _xx while in the case of Tricomi equation the corresponding coefficientT=y wascontinuous. In this paper we establish another mixed type boundary value problem forthe extended Bitsadze-Lavrent’ev-Tricomi equation \(L u = \operatorname{sgn} (y) \cdot u_{xx} + \operatorname{sgn} (x) \cdot u_{yy} + r (x,y) \cdot u = f (x,y)\) where both coefficientsK=sgn (y),M=sgn (x) ofu _xx ,u _yy , respectively are discontinous,r=r (x, y) is once continuously differentiable,f=f (x, y) continuous, and then we prove a uniqueness theorem for quasi-regular solutions.     

Die multivariate inkomplette Beta-Funktion

The integral $$\int_{\Delta (x)} {u_1^{p_1 ^{ - 1} } ...u_n^{p_n ^{ - 1} } (1 - u_1 - ... - u_n )^{q - 1} du_1 \wedge ... \wedge du_n } $$ extended over then-simplex $$\Delta (x) = \{ (u_1 ,...,u_n ) \in R_ + ^n :u_1 /x_1 + ... + u_n /x_n< 1\} $$ is considered as a natural generalization of the classical incomplete Beta-functionB _x (p, q). The partial differential equations of this multivariate incomplete Beta-function lead to a kind of ‘minimum principle’ for its restriction to the hypersurface with the equation:p ₁/x ₁+...+ +p _n /x _n =c=const.     

Solvability of the Dirichlet problem for degenerate quasilinear elliptic equations

In a bounded domain of the n -dimensional (n?2) space one considers a class of degenerate quasilinear elliptic equations, whose model is the equation $$\sum\limits_{i = 1}^n {\frac{{\partial F}}{{\partial x_i }}} (a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i - 2} u_{x_i } ) = f(x),$$ where x =(x₁,..., x_r), l_i?0, m_i>1, the function f is summable with some power, the nonnegative continuous function a(u) vanishes at a finite number of points and satisfies \(\frac{{lim}}{{\left| u \right| \to \infty }}a(u) > 0\) . One proves the existence of bounded generalized solutions with a finite integral $$\int\limits_\Omega {\sum\limits_{i = 1}^n {a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i } dx} }$$ of the Dirichlet problem with zero boundary conditions.     

Schwache Lösungen des Frankl-Morawetz-Problems im ℝ3

In a bounded simple connected region G ? ?³ we consider the equation $$L\left[ u \right]: = k\left( z \right)\left( {u_{xx} + u_{yy} } \right) + u_{zz} + d\left( {x,y,z} \right)u = f\left( {x,y,z} \right)$$ where k(z)? 0 whenever z ? 0.G is surrounded forz≥0 by a smooth surface Γ₀ with S:=Γ₀ ? {(x,y,z)|=0} and forz<0 by the characteristic \(\Gamma _2 :---(x^2 + y^2 )^{{\textstyle{1 \over 2}}} + \int\limits_z^0 {(---k(t))^{{\textstyle{1 \over 2}}} dt = 0} \) and a smooth surface Γ₁ which intersect the planez=0 inS and where the outer normal n=(n_x, n_y, n_z) fulfills \(k(z)(n_x^2 + n_y^2 ) + n_z^2 |_{\Gamma _1 } > 0\) . Under conditions on Γ₁ and the coefficientsk(z), d(x,y,z) we prove the existence of weak solutions for the boundary value problemL[u]=f inG with \(u|_{\Gamma _0 \cup \Gamma _1 } = 0\) . The uniqueness of the classical solution for this problem was proved in [1].     

Nonlocal problem for a parabolic-hyperbolic equation in a rectangular domain

For an equation of mixed type, namely, $$ \left( {1 - \operatorname{sgn} t} \right)u_{tt} + \left( {1 - \operatorname{sgn} t} \right)u_t - 2u_{xx} = 0 $$ in the domain {(x, t) | 0 < x < 1, ?α < t < β}, where α, β are given positive real numbers, we study the problem with boundary conditions $$ u\left( {0,t} \right) = u\left( {1,t} \right) = 0, - \alpha \leqslant t \leqslant \beta , u\left( {x, - \alpha } \right) - u\left( {x,\beta } \right) = \phi \left( x \right), 0 \leqslant x \leqslant 1. $$ . We establish a uniqueness criterion for the solution constructed as the sum of Fourier series. We establish the stability of the solution with respect to its nonlocal condition φ(x).     

Global existence for the Cauchy problem of the parabolic–parabolic Keller–Segel system on the plane

This paper is concerned with the Cauchy problem for the Keller–Segel system $$\left\{\begin{array}{l@{\quad}l}u_t = \nabla \cdot (\nabla u - u \nabla v) & \hbox{in } {\bf R}^{2} \times(0,\infty),\\v_t = \Delta v - \lambda v + u & \hbox{ in } {\bf R}^2 \times(0,\infty),\\u(x,0) = u_0 (x) \geq 0, \; v(x,0) = v_0 (x) \geq 0 & \hbox{ in} {\bf R}^2\end{array}\right.$$ with a constant λ ≥ 0, where ${(u_0, v_0) \in (L^1 ({\bf R}^2) \cap L^\infty ({\bf R}^2) ) \times (L^1 ({\bf R}^2) \cap H^1 ({\bf R}^2))}$ . Let $$m (u_0;{\bf R}^2) = \int\limits_{{\bf R}^2} u_0 (x) dx$$ . The same method as in [9] yields the existence of a blowup solution with m (u ₀; R ²) > 8π. On the other hand, it was recently shown in [7] that under additional hypotheses ${u_0 \log (1 + |x|^2) \in L^1 ({\bf R}^2)}$ and ${u_0 \log u_0 \in L^1 ({\bf R}^2)}$ , any solution with m(u ₀; R ²) < 8π exists globally in time. In[18], the extra assumptions were taken off, but the condition on mass was restricted to m (u ₀; R ²) < 4π. In this paper, we prove that any solution with m (u ₀; R ²) < 8π exists globally in time under no extra conditions. Furthermore the global existence of solutions is obtained under some condition on u ₀ also in the critical case m (u ₀; R ²) = 8π.     

Existence and Concentration of Bound States of a Class of Nonlinear SchrSdinger Equations in R2 with Potential Tending to Zero at Infinity

In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schr?dinger equation $$- \varepsilon ^2 \Delta u_\varepsilon + V\left( x \right)u_\varepsilon = K\left( x \right)\left| {u_\varepsilon } \right|^{p - 2} u_\varepsilon e^{\alpha _0 \left| {u_\varepsilon } \right|^\gamma } , u_\varepsilon > 0, u_\varepsilon \in H^1 \left( {\mathbb{R}^2 } \right),$$ where 2 < p < ∞, α ₀ > 0, 0 < γ < 2. When the potential function V (x) decays at infinity like (1 + |x|)^?α with 0 < α ≤ 2 and K(x) > 0 are permitted to be unbounded under some necessary restrictions, we will show that a positive H ¹(?²)-solution u _? exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schr?dinger equation $- \Delta u + V\left( \xi \right)u = K\left( \xi \right)\left| u \right|^{p - 2} ue^{\alpha _0 \left| u \right|^\gamma }$ has local minimum points. Furthermore, the concentration property of u _? is also established as ? tends to zero.     

Nash equilibria on topological semilattices

A function ${u : X \to \mathbb{R}}$ defined on a partially ordered set is quasi-Leontief if, for all ${x \in X}$ , the upper level set ${\{x\prime \in X : u(x\prime) \geq u(x)\}}$ has a smallest element; such an element is an efficient point of u. An abstract game ${u_{i} : \prod^{n}_{j=1} X_j \to \mathbb{R}, i \in \{1, \ldots , n\}}$ , is a quasi-Leontief game if, for all i and all ${(x_{j})_{j \neq i} \in \prod_{j \neq i} X_{j}, u_{i}((x_{j})_{j \neq i};-) : X_{i} \to \mathbb{R}}$ is quasi-Leontief; a Nash equilibrium x* of an abstract game ${u_{i} :\prod^{n}_{j=1} X_{j} \to \mathbb{R}}$ is efficient if, for all ${i, x^{*}_{i}}$ is an efficient point of the partial function ${u_{i}((x^{*}_{j})_{j \neq i};-) : X_{i} \to \mathbb{R}}$ . We establish the existence of efficient Nash equilibria when the strategy spaces X _i are topological semilattices which are Peano continua and Lawson semilattices.     

Liouville Systems of Mean Field Equations

In this article, we discuss the recent work of Lin and Zhang on the Liouville system of mean field equations: $$\Delta{u}_i+\sum_{j}a_{ij}\rho_{j} ({\frac{{h_j}e^{u_{j}}}{\int_{M}{h_{j}e^{u_{j}}}}-{\frac{1}{|M|}}})=0\,\, \quad{\rm on}\, M,$$ where M is a compact Riemann surface and |M| is the area, or $$\Delta{u}_i+\sum_{j}a_{ij}\rho_{j} \frac{{h_j}e^{u_{j}}}{\int_{\Omega}{h_{j}e^{u_{j}}}}=0\,\, \quad{\rm in}\, \Omega,$$ $${u_j}=0,\,\, \quad{\rm on}\, \partial\Omega, $$ where ?? is a bounded domain in ${\mathbb{R}^2}$ . Among other things, we completely determine the set of non-critical parameters and derive a degree counting formula for these systems.     

Removable Singularities for Anisotropic Elliptic Equations

We study a class of quasi-linear elliptic equations with model representative \(\sum _{i=1}^{n}(|u_{x_{i}}|^{p_{i}-2}u_{x_{i}})_{x_{i}}=0\) , which solutions have singularities on a smooth manifold. We establish the condition for removability of singularity on a manifold for solutions of such equations.     

Approximation of a linear system of second-order differential equations

In a Hilbert space H we consider the approximation by systems $$\frac{{d^2 u_1 }}{{dt^2 }} = A_{11} u_1 + A_{12} u_2 + f_1 ,\varepsilon \frac{{d^2 u_2 }}{{dt^2 }} = A_{21} u_1 + A_{22} u_2 + f_2 ,\varepsilon > 0,$$ of the semievolutionary system obtained from (1) when ∈=0. Under certain conditions on the solutions of the Cauchy problem for system (1) and the existence of a bounded linear operator A ₂₂ ^?1 we establish the convergence of the solutions u^∈(∈ → 0) to a solution of the corresponding problem for system (1) with ∈=0. We also establish the uniform correctness of the Cauchy problem for the above system.     

On the new critical exponent for the nonlinear Schrödinger equations

We consider the Cauchy problem for the pth order nonlinear Schrödinger equation in one space dimension $$\left\{\begin{array}{ll}iu_{t} + \frac{1}{2} u_{xx} = |u|^{p}, x \in {\bf R}, \, t > 0, \\ \qquad u(0, x) = u_{0} (x), \; x \in {\bf R},\end{array}\right.$$ where \({p > p_{s} = \frac{3 + \sqrt{17}}{2}}\) . We reveal that p = 4 is a new critical exponent with respect to the large time asymptotic behavior of solutions. We prove that if p _s < p < 4, then the large time asymptotics of solutions essentially differs from that for the linear case, whereas it has a quasilinear character for the case of p > 4.     

The integral representation of the exponential product of stochastic semigroups

An integral representation is obtained for the exponential product of stochastic semigroups $$X_s^t \otimes Z_s^t = X_s^t + \mathop \smallint \limits_{s< u< t} X_u^t dV_u X_s^u + \mathop {\smallint \smallint }\limits_{s< u_1< u_2< t} X_{u_2 }^t dV_{u_2 } X_{u_1 }^{u_2 } dV_{u_1 } X_s^{u_1 } + \cdots ,$$ whereV _t is the generating process of the semigroupZ _s ^t and the integrals are understood in the sense of mean-square limits of the Riemann-Stieltjes sums. This representation is different from the traditional representation $$X_s^t \otimes Z_s^t = E + \mathop \smallint \limits_{s< u< t} dW_u + \mathop {\smallint \smallint }\limits_{s< u_1< u_2< t} dW_{u_2 } dW_{u_1 } + \cdots ,$$ in which the integration extends over the processW _t=Y_t+V_t that is the generating process of the exponential productX _s ^t ?Z _s ^t andY _t is the generator of the semigroupX _s ^t .     

Asymptotic Behaviour for a Nonlinear Schrödinger Equation in Domains with Moving Boundaries

We consider a nonlinear Schrödinger equation in a time-dependent domain Q _τ of ?² given by $$u_{\tau} - i u_{\varepsilon\varepsilon} + |u|^{2} u + \gamma v=0. $$ We prove the well-posedness of the above model and analyze the behaviour of the solution as t→+∞. We consider two situations: the conservative case (γ=0) and the dissipative case (γ>0). In both situations the existence of solutions are proved using the Galerkin method and the stabilization of solutions are obtained considering multiplier techniques.     

Unified Approach to Stabilization of Waves on Compact Surfaces by Simultaneous Interior and Boundary Feedbacks of Unrestricted Growth

Let $\mathcal{M}\subset\mathbb{R}^{3}$ be an oriented compact surface on which we consider the system: $$\left \{ \begin{array}{l@{\quad}l} u_{tt} - \Delta_{\mathcal{M}} u + a(x)g_{0}(u_{t})=0 & \text{in } \mathcal{M}\times\mathopen{]} 0,\infty[ ,\\ \partial_{\nu_{co}}u +u + b(x)g(u_t)=0 & \text{on } \partial \mathcal {M}\times\mathopen{]}0,+\infty[. \end{array} \right . $$ If $\mathcal{M}$ along with the localizers a, b and the nonlinear feedbacks g,g ₀ satisfy certain conditions then uniform (but not necessarily exponential) decay rates of the finite energy of solutions can be established. We present a unified approach that bridges and extends a number of earlier results on stabilization of 2nd-order hyperbolic equations on manifolds. The methodology captures geometric requirements for damping acting simultaneously on subsets of the interior and of the boundary, and shows how placements of these feedbacks can complement each other depending on the underlying surface. In addition, the results conveniently incorporate the existing theory that allows elimination of geometric conditions from the controlled boundary (in absence of nearby interior damping), and elimination of damping entirely from certain boundary neighborhoods. The model also admits feedbacks that grow sub- or super-linearly not only at the origin, but also at infinity and demonstrates an interplay between the regularity of solutions and asymptotic energy decay rates.     

The analytic spread of codimension two monomial varieties

Let ${\rm} A=k[{u_{1}^{a_{1}}},{u_{2}^{a_{2}}},\dots,{u_{n}^{a_{n}}},{u_{1}^{c_{1}}} \dots {u_{n}^{c_{n}}},{u_{1}^{b_{1}}} \dots {u_{n}^{b_{n}}}]\ \subset k[{u_{1}}, \dots {u_{n}}],$ where, a_j, b_j, C_j ∈ ?, a_j > 0, (b_j, C_j) ≠ (0,0) for 1 ≤ j ≤ n, and, further ${\underline b}:=\ ({b_{1}}, \dots,{b_{n}})\ \not=\ 0 $ and ${\underline c}:=\ ({c_{1}}, \dots,{c_{n}})\ \not=\ 0 $ . The main result says that the defining ideal I ? m = (x₁,…, x_n, y, z) ? k[x₁,…, x_n, y, z] of the semigroup ring A has analytic spread ?(I_m) at most three.     

Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions

We consider a linear system of PDEs of the form 1 $$\begin{aligned} & \begin{aligned} u_{tt} - c\Delta u_t - \Delta u &= 0 \quad\text{in } \varOmega\times (0,T)\\ u_{tt} + \partial_n (u+cu_t) - \Delta_\varGamma(c \alpha u_t + u)& = 0 \quad\text{on } \varGamma_1 \times(0,T)\\ u &= 0 \quad\text{on } \varGamma_0 \times(0,T) \end{aligned} \\ &\quad (u(0),u_t(0),u|_{\varGamma_1}(0),u_t|_{\varGamma_1}(0)) \in {\mathcal{H}} \end{aligned}$$ on a bounded domain Ω with boundary Γ=Γ ₁∪Γ ₀. We show that the system generates a strongly continuous semigroup T(t) which is analytic for α>0 and of Gevrey class for α=0. In both cases the flow exhibits a regularizing effect on the data. In particular, we prove quantitative time-smoothing estimates of the form ∥(d/dt)T(t)∥?|t|^?1 for α>0, ∥(d/dt)T(t)∥?|t|^?2 for α=0. Moreover, when α=0 we prove a novel result which shows that these estimates hold under relatively bounded perturbations up to 1/2 power of the generator.