Fujita type conditions to heat equation with variable source

This paper studies heat equation with variable exponent u _t = Δu + u^p(x) + u ^q in ? ^N × (0, T), where p(x) is a nonnegative continuous, bounded function, 0 < p_? = inf p(x) ≤ p(x) ≤ sup p(x) = p₊. It is easy to understand for the problem that all nontrivial nonnegative solutions must be global if and only if max {p₊, q} ≤ 1. Based on the interaction between the two sources with fixed and variable exponents in the model, some Fujita type conditions are determined that that all nontrivial nonnegative solutions blow up in finite time if 0 < q ≤ 1 with p₊ > 1, or 1 < q < 1 + \(\frac{2}{N}\). In addition, if q > 1 + \(\frac{2}{N}\), then (i) all solutions blow up in finite time with 0 < p_? ≤ p₊ ≤ 1 + \(\frac{2}{N}\); (ii) there are both global and nonglobal solutions for p_? > 1 + \(\frac{2}{N}\); and (iii) there are functions p(x) such that all solutions blow up in finite time, and also functions p(x) such that the problem possesses global solutions when p_? < 1 + \(\frac{2}{N}\) < p₊.     

An indefinite concave-convex equation under a Neumann boundary condition I

We investigate the problem (P _λ) ?Δu = λb(x)|u| ^q?2 u + a(x)|u| ^p?2 u in Ω, ?u/?n = 0 on ?Ω, where Ω is a bounded smooth domain in R ^N (N ≥ 2), 1 < q < 2 < p, λ ∈ R, and a, b ∈ \({C^\alpha }\left( {\overline \Omega } \right)\) with 0 < α < 1. Under certain indefinite type conditions on a and b, we prove the existence of two nontrivial nonnegative solutions for small |λ|. We then characterize the asymptotic profiles of these solutions as λ → 0, which in some cases implies the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type component in the non-negative solutions set. We prove the existence of such a component in certain cases, via a bifurcation and a topological analysis of a regularized version of (P _λ).     

Nikol’skii inequality between the uniform norm and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>-norm with jacobi weight of algebraic polynomials on an interval

We study the Nikol’skii inequality for algebraic polynomials on the interval [?1, 1] between the uniform norm and the norm of the space L _q ^(α,β) , 1 ≤ q < ∞, with the Jacobi weight ?(α,β)(x) = (1 ? x) ^α (1 + x) ^β , α ≥ β > ?1. We prove that, in the case α > β ≥ ?1/2, the polynomial with unit leading coefficient that deviates least from zero in the space L _q ^(α+1,,β) with the Jacobi weight ? ^(α+1,β)(x) = (1?x) ^α+1(1+x) ^β is the unique extremal polynomial in the Nikol’skii inequality. To prove this result, we use the generalized translation operator associated with the Jacobi weight. We describe the set of all functions at which the norm of this operator in the space L _q ^(α,β) for 1 ≤ q < ∞ and α > β ≥ ?1/2 is attained.     

Boundary blow-up solutions of <Emphasis Type="Italic">p</Emphasis>-Laplacian elliptic equations with lower order terms

We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation of the type ?Δ _p u = a(x)u ^m ?b(x)f(u) with p >  1 and 0 <  m <  p?1. The main technical tool is a new comparison principle that enables us to extend arguments for semilinear equations to quasilinear ones. Indeed, this paper is an attempt to generalize all available results for the semilinear case with p =  2 to the quasilinear case with p >  1.     

Integral averaging technique for oscillation of damped half-linear oscillators

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator (a(t)?_p(x′))′ + b(t)?_p(x′) + c(t)?_p(x) = 0, where ?p(x) = |x|^p?1 sgn x for x ∈ ? and p > 1. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if p ≠ 2 is presented.     

Bessel Property of the System of Root Functions of a Second-Order Singular Operator on an Interval

For the system of root functions of an operator defined by the differential operation ?u″ + p(x)u′ + q(x)u, x ∈ G = (0, 1), with complex-valued singular coefficients, sufficient conditions for the Bessel property in the space L₂(G) are obtained and a theorem on the unconditional basis property is proved. It is assumed that the functions p(x) and q(x) locally belong to the spaces L₂ and W₂^?1, respectively, and may have singularities at the endpoints of G such that q(x) = q_R(x) +q′S(x) and the functions q_S(x), p(x), q ₂ ^S (x)w(x), p²(x)w(x), and q_R(x)w(x) are integrable on the whole interval G, where w(x) = x(1 ? x).     

Divergent solutions to the <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-supercritical NLS equations

We investigate the nonlinear Schrödinger equation iu _t +Δu+|u| ^p?1 u = 0with 1+ 4/N < p < 1+ 4/N?2 (when N = 1, 2, 1 + 4/N < p < ∞) in energy space H ¹ and study the divergent property of infinite-variance and nonradial solutions. If \(M{\left( u \right)^{\frac{{1 - {s_C}}}{{{s_C}}}}}E\left( u \right) \prec M{\left( Q \right)^{\frac{{1 - {s_C}}}{{{s_C}}}}}E\left( Q \right)\) and \(\left\| {{u_0}} \right\|_2^{\frac{{1 - {s_c}}}{{{s_c}}}}\left\| {\nabla {u_0}} \right\|_2^{\frac{{1 - {s_c}}}{{{s_c}}}}{\left\| {\nabla Q} \right\|_2}\), then either u(t) blows up in finite forward time or u(t) exists globally for positive time and there exists a time sequence t _n → +∞ such that \({\left\| {\nabla u\left( {{t_n}} \right)} \right\|_2} \to + \infty \). Here Q is the ground state solution of ?(1?s _c )Q+ΔQ+Q ^p?1 Q = 0. A similar result holds for negative time. This extend the result of the 3D cubic Schrödinger equation obtained by Holmer to the general mass-supercritical and energy-subcritical case.     

Liouville type results for a <Emphasis Type="Italic">p</Emphasis>-Laplace equation with negative exponent

Positive entire solutions of the equation \(\Delta _p u = u^{ - q} in \mathbb{R}^N (N \geqslant 2)\) where 1 < p ≤ N, q > 0, are classified via their Morse indices. It is seen that there is a critical power q = q _c such that this equation has no positive radial entire solution that has finite Morse index when q > q _c but it admits a family of stable positive radial entire solutions when 0 < q ≤ q _c. Proof of the stability of positive radial entire solutions of the equation when 1 < p < 2 and 0 < q ≤ q _c relies on Caffarelli–Kohn–Nirenberg’s inequality. Similar Liouville type result still holds for general positive entire solutions when 2 < p ≤ N and q > q _c. The case of 1 < p < 2 is still open. Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p = 2 obtained previously. Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator.     

Seifert manifolds and (1, 1)-knots

The aim of this paper is to investigate the relations between Seifert manifolds and (1, 1)-knots. In particular, we prove that each orientable Seifert manifold with invariants

$\{ Oo,0| - 1;\underbrace {(p,q),...,(p,q)}_{n times},(l,l - 1)\} $

has the fundamental group cyclically presented by G _n ((x ₁ ^q ...x _n ^{q l} x _n ^?p ) and, moreover, it is the n-fold strongly-cyclic covering of the lens space L(|nlq ? p|, q) which is branched over the (1, 1)-knot K(q, q(nl ? 2), p ? 2q, p ? q) if p ≥ 2q and over the (1, 1)-knot K(p? q, 2q ? p, q(nl ? 2), p ? q) if p< 2q.

     

On multiple zeros of a partial theta function

We consider the partial theta function θ(q, x) := ∑_j=0^∞q^j(j+1)/2x^j, where x ∈ ? is a variable and q ∈ ?, 0 < |q| < 1, is a parameter. We show that, for any fixed q, if ζ is a multiple zero of the function θ(q, · ), then |ζ| ≤ 8¹¹.     

Estimates of the Root Functions of a One-Dimensional Schrödinger Operator with a Strong Boundary Singularity

For any operator defined by the differential operation Lu = ?u″ + q(x)u on the interval G = (0, 1) with complex-valued potential q(x) locally integrable on G and satisfying the inequalities \(\int_{{x_1}}^{{x_2}} {\zeta |(q(\zeta ))|d\zeta \leqslant ln({x_1}/{x_2})} \) and \(\int_{{x_1}}^{{x_2}} {\zeta |(q(1 - \zeta ))|d\zeta \leqslant \gamma ln({x_1}/{x_2})} \) with some constant γ for all sufficiently small 0 < x₁ < x₂, we estimate the norms of root functions in the Lebesgue spaces L _p (G), 1 ≤ p < ∞. We show that for sufficiently small γ these norms satisfy the same estimates asymptotic in the spectral parameter as in the unperturbed case.     

On a singular minimizing problem

We investigate the non-homogeneous modular Dirichlet problem Δ _p (·)u(x) = f (x) (where Δ _p (·)u(x) = div(|?u|^p(x-2)?u(x)) from the functional analytic point of view and we prove the stability of the solutions \({\left( {{u_{{p_i}}}} \right)_i}\) of the equation \({\Delta _{{p_i}\left( \cdot \right)}}{u_{{p_i}\left( \cdot \right)}} = f\) as p _i (·) → q(·) via Gamma-convergence of sequence of appropriate functionals.     

Boundary singularities of solutions of semilinear elliptic equations in the half-space with a Hardy potential

We study a nonlinear equation in the half-space {x ₁ > 0} with a Hardy potential, specifically

$$ - \Delta u - \frac{\mu }{{x_1^2}}u + {u^p} = 0in\mathbb{R}_ + ^n,$$

where p > 1 and ?∞ < μ < 1/4. The admissible boundary behavior of the positive solutions is either O(x ₁ ^?2/(p?1)) as x ₁ → 0, or is determined by the solutions of the linear problem \( - \Delta h - \frac{\mu }{{x_1^2}}h = 0\). In the first part we study in full detail the separable solutions of the linear equations for the whole range of μ. In the second part, by means of sub and supersolutions we construct separable solutions of the nonlinear problem which behave like O(x ₁ ^?2/(p?1)) near the origin and which, away from the origin, have exactly the same asymptotic behavior as the separable solutions of the linear problem. In the last part we construct solutions that behave like O(x ₁ ^?2/(p?1)) at some prescribed parts of the boundary, while at the rest of the boundary the solutions decay or blowup at a slower rate determined by the linear part of the equation.

     

Jacobi translation and the inequality of different metrics for algebraic polynomials on an interval

The sharp inequality of different metrics (Nikol’skii’s inequality) for algebraic polynomials in the interval [?1, 1] between the uniform norm and the norm of the space L _q ^(α,β) , 1 ≤ q < ∞, with Jacobi weight ?^(α,β)(x) = (1 ? x)^α(1 + x)^β α ≥ β > ?1, is investigated. The study uses the generalized translation operator generated by the Jacobi weight. A set of functions is described for which the norm of this operator in the space L _q ^(α,β) , 1 ≤ q < ∞, \(\alpha > \beta \geqslant - \frac{1}{2}\), is attained.     

Triebel-Lizorkin space boundedness of rough singular integrals associated to surfaces of revolution

We consider the boundedness of the rough singular integral operator T_(?,ψ,h) along a surface of revolution on the Triebel-Lizorkin space F~α_( p,q)(R~n) for ? ∈ H~1(~(Sn-1)) and ? ∈ Llog~+L(S~(n-1)) ∪_1q∞(B~((0,0))_q(S~(n-1))), respectively.     

Connecting orbits of Lagrangian systems in a nonstationary force field

We study connecting orbits of a natural Lagrangian system defined on a complete Riemannian manifold subjected to the action of a nonstationary force field with potential U(q, t) = f(t)V(q). It is assumed that the factor f(t) tends to ∞ as t→±∞ and vanishes at a unique point t ₀ ∈ ?. Let X ₊, X _? denote the sets of isolated critical points of V (x) at which U(x, t) as a function of x distinguishes its maximum for any fixed t > t ₀ and t < t ₀, respectively. Under nondegeneracy conditions on points of X _± we prove the existence of infinitely many doubly asymptotic trajectories connecting X _? and X ₊.     

Short cubic exponential sums over primes

For y ≥ x ^4/5 L ^8B+151 (where L = log(xq) and B is an absolute constant), a nontrivial estimate is obtained for short cubic exponential sums over primes of the form S ₃(α; x, y) = ∑ _x?y<n≤x Λ(n)e(αn ³), where α = a/q + θ/q ², (a, q) = 1, L ^32(B+20) < q ≤ y ⁵ x ^?2 L ^?32(B+20), |θ| ≤ 1, Λ is the von Mangoldt function, and e(t) = e ^2πit.     

Regularization of Mickelsson generators for nonexceptional quantum groups

Let g′ ? g be a pair of Lie algebras of either symplectic or orthogonal infinitesimal endomorphisms of the complex vector spaces C ^N?2 ? C ^N and U _q (g′) ? U _q (g) be a pair of quantum groups with a triangular decomposition U _q (g) = U _q (g_-)U _q (g₊)U _q (h). Let Z _q (g, g′) be the corresponding step algebra. We assume that its generators are rational trigonometric functions h ? → U _q (g_±). We describe their regularization such that the resulting generators do not vanish for any choice of the weight.     

New characterization of <Emphasis Type="Italic">S</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) by its noncommuting graph

Let G be a nonabelian group, and associate the noncommuting graph ?(G) with G as follows: the vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Let S ₄(q) be the projective symplectic simple group, where q is a prime power. We prove that if G is a group with ?(G) ? ?(S ₄(q)) then G ? S ₄(q).     

Blow-up of critical norms for the 3-D Navier-Stokes equations

Let u =(uh, u3) be a smooth solution of the 3-D Navier-Stokes equations in R3× [0, T). It was proved that if u3 ∈ L∞(0, T;˙B-1+3/p p,q(R3)) for 3 p, q ∞ and uh∈ L∞(0, T; BMO-1(R3)) with uh(T) ∈ VMO-1(R3), then u can be extended beyond T. This result generalizes the recent result proved by Gallagher et al.(2016), which requires u ∈ L∞(0, T;˙B-1+3/pp,q(R3)). Our proof is based on a new interior regularity criterion in terms of one velocity component, which is independent of interest.