Global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth

The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}The aim of this study is to prove global existence of classical solutions for systems of the form \frac?u?t -a Du=-f(u,v){\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)} , \frac?v?t -b Dv=g(u,v){\frac{\partial v}{\partial t} -b \Delta v=g(u,v)} in (0, +∞) × Ω where Ω is an open bounded domain of class C ¹ in \mathbbRⁿ{\mathbb{R}^n}, a > 0, b > 0 and f, g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(x,h) ￡ C j(x)e^ahb{g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}}} and g(ξ, η) ≤ ψ(η)f(ξ, η) for some constants C > 0, α > 0 and β ≥ 1 where j{\varphi} and ψ are any nonnegative continuously differentiable functions on [0, +∞) such that j(0)=0{\varphi(0)=0} and lim_{h? +￥}h^b-1y(h) = l{ \lim_{\eta \rightarrow +\infty}\eta^{\beta -1}\psi(\eta)= \ell} where ℓ is a nonnegative constant. The asymptotic behavior of the global solutions as t goes to +∞ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates and Lyapunov functional methods.     

The Jacobi-Dunkl transform on ℝ and the convolution product on new spaces of distributions

In this work, we consider the Jacobi-Dunkl operator Λ _α,β , a 3 b 3 \frac-12\alpha\geq\beta\geq\frac{-1}{2} , a 1 \frac-12\alpha\neq\frac{-1}{2} , on ℝ. The eigenfunction Y_l^a,b\Psi_{\lambda}^{\alpha,\beta} of this operator permits to define the Jacobi-Dunkl transform. The main idea in this paper is to introduce and study the Jacobi-Dunkl transform and the Jacobi-Dunkl convolution product on new spaces of distributions     

Bi-Parametric Potentials, Relevant Function Spaces and Wavelet-Like Transforms

We introduce new potential type operators J^a_b = (E+(-D)^b/2)^-a/bJ^{\alpha}_{\beta} = (E+(-\Delta)^{\beta/2})^{-\alpha/\beta}, (α > 0, β > 0), and bi-parametric scale of function spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta , p}({\mathbb{R}}^n) associated with J^α_β. These potentials generalize the classical Bessel potentials (for β = 2), and Flett potentials (for β = 1). A characterization of the spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta, p}({\mathbb{R}}^n) is given with the aid of a special wavelet–like transform associated with a β-semigroup, which generalizes the well-known Gauss-Weierstrass semigroup (for β = 2) and the Poisson one (for β = 1).     

A convexity theorem for multiplicative functions

We generalize a well known convexity property of the multiplicative potential function. We prove that, given any convex function g : \mathbbR^m ? [0, ￥]{g : \mathbb{R}^m \rightarrow [{0}, {\infty}]}, the function ${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}, is convex if β ≥ 0 and α ≥ β ₁ + ··· + β _n . We also provide further generalization to functions of the form (x,y₁, . . . , y_n)? g(x)^1+af₁(y₁)^-b₁ ···f_n(y_n)^-b_n{({\rm \bf x},{\rm \bf y}_1, . . . , {y_n})\mapsto g({\rm \bf x})^{1+\alpha}f_1({\rm \bf y}_1)^{-\beta_1} \cdot \cdot \cdot f_n({\rm \bf y}_n)^{-\beta_n} } with the f _k concave, positively homogeneous and nonnegative on their domains.     

On the global well-posedness of a class of Boussinesq�CNavier�CStokes systems

In this paper we consider the following 2D Boussinesq–Navier–Stokes systems

Weighted least-squares estimators of parametric functions of the regression coefficients under a general linear model

The weighted least-squares estimator of parametric functions K β under a general linear regression model { y, X b, s²S }{\{ {\bf y},\,{\bf X \beta}, \sigma^2{\bf \Sigma} \}} is defined to be K[^(b)]{{\bf K}{\hat{\bf {\beta}}}}, where [^(b)]{\hat{{\bf \beta}}} is a vector that minimizes (y − X β)′V(y − X β) for a given nonnegative definite weight matrix V. In this paper, we study some algebraic and statistical properties of K[^(b)]{{\bf K}\hat{{\bf \beta}}} and the projection matrix associated with the estimator, such as, their ranks, unbiasedness, uniqueness, as well as equalities satisfied by the projection matrices.     

Multiplicity of solutions for a fourth order equation with power-type nonlinearity

Let B be the unit ball in ${\mathbb{R}^N}Let B be the unit ball in \mathbbR^N{\mathbb{R}^N}, N ≥ 3 and n be the exterior unit normal vector on the boundary. We consider radial solutions to

D2 u = l(1+ sign(p)u)p     in  B,     u = 0,     \frac?u?n = 0     on  ?B\Delta^2 u = \lambda(1+ {\rm sign}(p)u)^{p} \quad {\rm in} \, B, \quad u = 0, \quad \frac{\partial{u}}{\partial{n}} = 0 \quad {\rm on} \, \partial B      8. A Generalized Radon Transform on the Plane      Zhongkai Li  Futao Song 《Constructive Approximation》2011,33(1):93-123 A new generalized Radon transform R α, β on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator Δ α, β , and the Jacobi polynomials Pk(b, a)(t)P_{k}^{(\beta,\,\alpha)}(t). The transform R α, β and its dual Ra, b*R_{\alpha,\,\beta}^{\ast} are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for R α, β for functions in La, bp(\mathbbR2+)L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+}) are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform R α, β is used to represent the solutions of the partial differential equations Lu:=?j=1majDa, bju=fLu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f with constant coefficients a j and the Cauchy problem for the generalized wave equation associated with the operator Δ α, β . Another application is that, by an invariant property of R α, β , a new product formula for the Jacobi polynomials of the type Pk(b, a)(s)C2ka+b+1(t)=còòPk(b, a)P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)} is obtained.      9. Regularity of weak solutions to the Navier–Stokes equations in exterior domains      Reinhard Farwig  Christian Komo 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(3):303-321 Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR3{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u 0, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u 0 and f or on the solution u itself such as Serrin’s condition || u ||Ls(0,T; Lq(W)) < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||Ls￠(t-d,t; Lq(W)){\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||22{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.      10. Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function      V. A. Kofanov 《Ukrainian Mathematical Journal》2011,63(7):1118-1135 For arbitrary [α, β] ⊂ R and p > 0, we solve the extremal problem òab | x(k)(t) |qdt ? sup,     q 3 p,    k = 0    \textor    q 3 1,    k 3 1, \int\limits_\alpha^\beta {{{\left| {{x^{(k)}}(t)} \right|}^q}dt \to \sup, \quad q \geq p,\quad k = 0\quad {\text{or}}\quad q \geq 1,\quad k \geq 1},      11. Existence of Solutions to a Singular Elliptic Equation      Marcelo Montenegro 《Milan Journal of Mathematics》2011,79(1):293-301 We study the equation ${{-{\Delta}u = (-\frac{1}{u^{\beta}}+\lambda{u}^{p})\chi\{u >0 }\}}${{-{\Delta}u = (-\frac{1}{u^{\beta}}+\lambda{u}^{p})\chi\{u >0 }\}} in Ω with Dirichlet boundary condition, where 0 < p < 1 and 0 < β < 1. We regularize the term 1/u β near u ~ 0 by using a function g ε (u) which pointwisely tends to 1/u β as ε → 0. When the parameter λ > 0 is large enough, the corresponding energy functional has critical points u ε . Letting ε → 0, then u ε converges to a solution of the original problem, which is nontrivial, nonnegative and vanishes at some portion of Ω. There are two nontrivial solutions.      12. Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential      Y. Belaud  A. Shishkov 《Journal of Evolution Equations》2010,10(4):857-882 We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ut + Lu + a(x) |u|q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb RN{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò01 s-1 (meas\nolimits {x ? W: |a(x)| ￡ s })q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.      13. On the time continuity of entropy solutions      Cl��ment Canc��s  Thierry Gallou?t 《Journal of Evolution Equations》2011,11(1):43-55 We show that any entropy solution u of a convection diffusion equation ?t u + div F(u)-Df(u) = b{\partial_t u + {\rm div} F(u)-\Delta\phi(u) =b} in Ω × (0, T) belongs to C([0,T),L1loc(W)){C([0,T),L^1_{\rm loc}({\Omega}))} . The proof does not use the uniqueness of the solution.      14. Maximal regularity for the Lamé system in certain classes of non-smooth domains      Marius Mitrea  Sylvie Monniaux 《Journal of Evolution Equations》2010,10(4):811-833 The aim of this article is twofold. On the one hand, we study the well-posedness of the Lamé system ${-\mu\Delta-\mu'\nabla{\rm div} }The aim of this article is twofold. On the one hand, we study the well-posedness of the Lamé system -mD-m￠?div{-\mu\Delta-\mu'\nabla{\rm div} } in L q (Ω), where Ω is an open subset of \mathbbRn{{\mathbb{R}}^n} satisfying mild regularity assumptions and the Lamé moduli μ, μ′ are such that μ > 0 and μ + μ′ > 0. On the other hand, we prove the analyticity of the semigroup generated by the Lamé operator as well as the maximal regularity property for the time-dependent Lamé system equipped with a homogeneous Dirichlet boundary condition based on off-diagonal estimates.      15. Minimal rotational hypersurfaces in Minkowski (<Emphasis Type="Italic">α</Emphasis>, <Emphasis Type="Italic">β</Emphasis>)-space      Ningwei?CuiEmail author  Yi-Bing?Shen 《Geometriae Dedicata》2011,151(1):27-39 In Finsler geometry, minimal surfaces with respect to the Busemann-Hausdorff measure and the Holmes-Thompson measure are called BH-minimal and HT-minimal surfaces, respectively. In this paper, we give the explicit expressions of BH-minimal and HT-minimal rotational hypersurfaces generated by plane curves rotating around the axis in the direction of [(b)\tilde]\sharp{\tilde{\beta}^{\sharp}} in Minkowski (α, β)-space (\mathbbVn+1,[(Fb)\tilde]){(\mathbb{V}^{n+1},\tilde{F_b})} , where \mathbbVn+1{\mathbb{V}^{n+1}} is an (n+1)-dimensional real vector space, [(Fb)\tilde]=[(a)\tilde]f([(b)\tilde]/[(a)\tilde]), [(a)\tilde]{\tilde{F_b}=\tilde{\alpha}\phi(\tilde{\beta}/\tilde{\alpha}), \tilde{\alpha}} is the Euclidean metric, [(b)\tilde]{\tilde{\beta}} is a one form of constant length b:=||[(b)\tilde]||[(a)\tilde], [(b)\tilde]\sharp{b:=\|\tilde{\beta}\|_{\tilde{\alpha}}, \tilde{\beta}^{\sharp}} is the dual vector of [(b)\tilde]{\tilde{\beta}} with respect to [(a)\tilde]{\tilde{\alpha}} . As an application, we first give the explicit expressions of the forward complete BH-minimal rotational surfaces generated around the axis in the direction of [(b)\tilde]\sharp{\tilde{\beta}^{\sharp}} in Minkowski Randers 3-space (\mathbbV3,[(a)\tilde]+[(b)\tilde]){(\mathbb{V}^{3},\tilde{\alpha}+\tilde{\beta})} .      16. From Ginzburg-Landau to Gross-Pitaevskii      Alberto Farina 《Monatshefte für Mathematik》2003,179(2):265-269 (w, c) ? R2, u ? Lloc3 (RN, C)\font\Opr=msbm10 at 8pt \def\Op#1{\hbox{\Opr{#1}}}(\omega, c)\in {\Op R}^2, {\upsilon} \in L_{\rm loc}^3 ({\Op R}^N, {\bf C}) and x||j||L￥(RN×R)2 ￡ max{0, 1-w+[(c2)/4]}.\font\Opr=msbm10 at 8pt \def\Op#1{\hbox{\Opr{#1}}}\Vert\varphi\Vert_{L^\infty({\Op R}^N\times{\Op R})}^2 \le \max\bigg\{0, 1-\omega+{c^2\over 4}\bigg\}.      17. Bounds and monotonicity for the generalized Robin problem      Tiziana Giorgi  Robert Smits 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,124(1):600-618 We consider the principal eigenvalue λ 1Ω(α) corresponding to Δu = λ (α) u in W, \frac?u?v = au \Omega, \frac{\partial u}{\partial v} = \alpha u on ∂Ω, with α a fixed real, and W ì Rn\Omega \subset {\mathcal{R}}^n a C 0,1 bounded domain. If α > 0 and small, we derive bounds for λ 1Ω(α) in terms of a Stekloff-type eigenvalue; while for α > 0 large we study the behavior of its growth in terms of maximum curvature. We analyze how domain monotonicity of the principal eigenvalue depends on the geometry of the domain, and prove that domains which exhibit domain monotonicity for every α are calibrable. We conjecture that a domain has the domain monotonicity property for some α if and only if it is calibrable.      18. Decay of mass for nonlinear equation with fractional Laplacian      Ahmad Fino  Grzegorz Karch 《Monatshefte für Mathematik》2010,148(1):375-384 The large time behavior of non-negative solutions to the reaction–diffusion equation ?t u=-(-D)a/2u - up{\partial_t u=-(-\Delta)^{\alpha/2}u - u^p}, ${(\alpha\in(0,2], \;p > 1)}${(\alpha\in(0,2], \;p > 1)} posed on \mathbbRN{\mathbb{R}^N} and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptotics for p > 1 + α/N, while nonlinear effects win if p ≤ 1 + α/N.      19. Regularity of the Anosov distributions of Euler-Lagrange deformations of geodesic flows      Yong Fang 《Geometriae Dedicata》2010,145(1):139-150 Let g be a negatively curved Riemannian metric of a closed C ∞ manifold M of dimension at least three. Let L λ be a C ∞ one-parameter convex superlinear Lagrangian on TM such that L0(v) = \frac12 g(v, v){L_0(v)= \frac{1}{2} g(v, v)} for any v ∈ TM. We denote by jl{\varphi^\lambda} the restriction of the Euler-Lagrange flow of L λ on the \frac12{\frac{1}{2}} -energy level. If λ is small enough then the flow jl{\varphi^\lambda} is Anosov. In this paper we study the geometric consequences of different assumptions about the regularity of the Anosov distributions of jl{\varphi^\lambda} . For example, in the case that the initial Riemannian metric g is real hyperbolic, we prove that for λ small, jl{\varphi^\lambda} has C 3 weak stable and weak unstable distributions if and only if jl{\varphi^\lambda} is C ∞ orbit equivalent to the geodesic flow of g.      20. A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations      Xiaochun Chen  Sadek Gala  Zhengguang Guo 《Acta Appl Math》2011,113(2):207-213 In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L\frac21-r( 0,T;[(X)\dot]r( \mathbbR3) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.

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A Generalized Radon Transform on the Plane

A new generalized Radon transform R _α, β on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator Δ _α, β , and the Jacobi polynomials P_k^(b, a)(t)P_{k}^{(\beta,\,\alpha)}(t). The transform R _α, β and its dual R_a, b^*R_{\alpha,\,\beta}^{\ast} are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for R _α, β for functions in L_a, b^p(\mathbbR²₊)L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+}) are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform R _α, β is used to represent the solutions of the partial differential equations Lu:=?_j=1^ma_jD_a, b^ju=fLu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f with constant coefficients a _j and the Cauchy problem for the generalized wave equation associated with the operator Δ _α, β . Another application is that, by an invariant property of R _α, β , a new product formula for the Jacobi polynomials of the type P_k^(b, a)(s)C_2k^a+b+1(t)=còòP_k^(b, a)P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)} is obtained.     

Regularity of weak solutions to the Navier–Stokes equations in exterior domains

Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u ₀, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u ₀ and f or on the solution u itself such as Serrin’s condition || u ||_{L^s(0,T; L^q(W))} < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||_{L^s￠(t-d,t; L^q(W))}{\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||₂²{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.     

Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function

For arbitrary [α, β] ⊂ R and p > 0, we solve the extremal problem

Existence of Solutions to a Singular Elliptic Equation

We study the equation ${{-{\Delta}u = (-\frac{1}{u^{\beta}}+\lambda{u}^{p})\chi\{u >0 }\}}${{-{\Delta}u = (-\frac{1}{u^{\beta}}+\lambda{u}^{p})\chi\{u >0 }\}} in Ω with Dirichlet boundary condition, where 0 < p < 1 and 0 < β < 1. We regularize the term 1/u ^β near u ~ 0 by using a function g _ε (u) which pointwisely tends to 1/u ^β as ε → 0. When the parameter λ > 0 is large enough, the corresponding energy functional has critical points u _ε . Letting ε → 0, then u _ε converges to a solution of the original problem, which is nontrivial, nonnegative and vanishes at some portion of Ω. There are two nontrivial solutions.     

Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation u_t + Lu + a(x) |u|^q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb R^N{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò₀¹ s^-1 (meas\nolimits {x ? W: |a(x)| ￡ s })^q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.     

On the time continuity of entropy solutions

We show that any entropy solution u of a convection diffusion equation ?_t u + div F(u)-Df(u) = b{\partial_t u + {\rm div} F(u)-\Delta\phi(u) =b} in Ω × (0, T) belongs to C([0,T),L¹_loc(W)){C([0,T),L^1_{\rm loc}({\Omega}))} . The proof does not use the uniqueness of the solution.     

Maximal regularity for the Lamé system in certain classes of non-smooth domains

The aim of this article is twofold. On the one hand, we study the well-posedness of the Lamé system ${-\mu\Delta-\mu'\nabla{\rm div} }The aim of this article is twofold. On the one hand, we study the well-posedness of the Lamé system -mD-m￠?div{-\mu\Delta-\mu'\nabla{\rm div} } in L ^q (Ω), where Ω is an open subset of \mathbbRⁿ{{\mathbb{R}}^n} satisfying mild regularity assumptions and the Lamé moduli μ, μ′ are such that μ > 0 and μ + μ′ > 0. On the other hand, we prove the analyticity of the semigroup generated by the Lamé operator as well as the maximal regularity property for the time-dependent Lamé system equipped with a homogeneous Dirichlet boundary condition based on off-diagonal estimates.     

Minimal rotational hypersurfaces in Minkowski (<Emphasis Type="Italic">α</Emphasis>, <Emphasis Type="Italic">β</Emphasis>)-space

In Finsler geometry, minimal surfaces with respect to the Busemann-Hausdorff measure and the Holmes-Thompson measure are called BH-minimal and HT-minimal surfaces, respectively. In this paper, we give the explicit expressions of BH-minimal and HT-minimal rotational hypersurfaces generated by plane curves rotating around the axis in the direction of [(b)\tilde]^\sharp{\tilde{\beta}^{\sharp}} in Minkowski (α, β)-space (\mathbbVⁿ⁺¹,[(F_b)\tilde]){(\mathbb{V}^{n+1},\tilde{F_b})} , where \mathbbVⁿ⁺¹{\mathbb{V}^{n+1}} is an (n+1)-dimensional real vector space, [(F_b)\tilde]=[(a)\tilde]f([(b)\tilde]/[(a)\tilde]), [(a)\tilde]{\tilde{F_b}=\tilde{\alpha}\phi(\tilde{\beta}/\tilde{\alpha}), \tilde{\alpha}} is the Euclidean metric, [(b)\tilde]{\tilde{\beta}} is a one form of constant length b:=||[(b)\tilde]||_[(a)\tilde], [(b)\tilde]^\sharp{b:=\|\tilde{\beta}\|_{\tilde{\alpha}}, \tilde{\beta}^{\sharp}} is the dual vector of [(b)\tilde]{\tilde{\beta}} with respect to [(a)\tilde]{\tilde{\alpha}} . As an application, we first give the explicit expressions of the forward complete BH-minimal rotational surfaces generated around the axis in the direction of [(b)\tilde]^\sharp{\tilde{\beta}^{\sharp}} in Minkowski Randers 3-space (\mathbbV³,[(a)\tilde]+[(b)\tilde]){(\mathbb{V}^{3},\tilde{\alpha}+\tilde{\beta})} .     

From Ginzburg-Landau to Gross-Pitaevskii

(w, c) ? R², u ? L_loc³ (R^N, C)\font\Opr=msbm10 at 8pt \def\Op#1{\hbox{\Opr{#1}}}(\omega, c)\in {\Op R}^2, {\upsilon} \in L_{\rm loc}^3 ({\Op R}^N, {\bf C}) and x||j||_{L^￥(R^N×R)}² ￡ max{0, 1-w+[(c²)/4]}.\font\Opr=msbm10 at 8pt \def\Op#1{\hbox{\Opr{#1}}}\Vert\varphi\Vert_{L^\infty({\Op R}^N\times{\Op R})}^2 \le \max\bigg\{0, 1-\omega+{c^2\over 4}\bigg\}.     

Bounds and monotonicity for the generalized Robin problem

We consider the principal eigenvalue λ ₁^Ω(α) corresponding to Δu = λ (α) u in W, \frac?u?v = au \Omega, \frac{\partial u}{\partial v} = \alpha u on ∂Ω, with α a fixed real, and W ì Rⁿ\Omega \subset {\mathcal{R}}^n a C ^0,1 bounded domain. If α > 0 and small, we derive bounds for λ ₁^Ω(α) in terms of a Stekloff-type eigenvalue; while for α > 0 large we study the behavior of its growth in terms of maximum curvature. We analyze how domain monotonicity of the principal eigenvalue depends on the geometry of the domain, and prove that domains which exhibit domain monotonicity for every α are calibrable. We conjecture that a domain has the domain monotonicity property for some α if and only if it is calibrable.     

Decay of mass for nonlinear equation with fractional Laplacian

The large time behavior of non-negative solutions to the reaction–diffusion equation ?_t u=-(-D)^a/2u - u^p{\partial_t u=-(-\Delta)^{\alpha/2}u - u^p}, ${(\alpha\in(0,2], \;p > 1)}${(\alpha\in(0,2], \;p > 1)} posed on \mathbbR^N{\mathbb{R}^N} and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptotics for p > 1 + α/N, while nonlinear effects win if p ≤ 1 + α/N.     

Regularity of the Anosov distributions of Euler-Lagrange deformations of geodesic flows

Let g be a negatively curved Riemannian metric of a closed C ^∞ manifold M of dimension at least three. Let L _λ be a C ^∞ one-parameter convex superlinear Lagrangian on TM such that L₀(v) = \frac12 g(v, v){L_0(v)= \frac{1}{2} g(v, v)} for any v ∈ TM. We denote by j^l{\varphi^\lambda} the restriction of the Euler-Lagrange flow of L _λ on the \frac12{\frac{1}{2}} -energy level. If λ is small enough then the flow j^l{\varphi^\lambda} is Anosov. In this paper we study the geometric consequences of different assumptions about the regularity of the Anosov distributions of j^l{\varphi^\lambda} . For example, in the case that the initial Riemannian metric g is real hyperbolic, we prove that for λ small, j^l{\varphi^\lambda} has C ³ weak stable and weak unstable distributions if and only if j^l{\varphi^\lambda} is C ^∞ orbit equivalent to the geodesic flow of g.     

A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations

In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L^\frac21-r( 0,T;[(X)\dot]_r( \mathbbR³) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.