Nonexistence Theorems Research Articles

A non-existence theorem of lacunas for hyperbolic differential operators with constant coefficients

We prove a theorem on non-existence of lacunas of the fundamental solution for hyperbolic differential operators with constant coefficients.

Blow-up of solutions for a class of semilinear reaction diffusion equations with mixed boundary conditions

The Hopf's maximum principles are utilized to deal with the problem on blow-up of the solutions for a class of semilinear reaction diffusion equations u, t = Δ u + f( x, u, q, t) ( q = |▽ u| 2), subject to mixed boundary conditions. Some nonexistence theorems of global solutions and the bounds of “blow-up time” are obtained.

Global existence and non-existence theorems for nonlinear wave equations

In this article we focus on the global well-posedness of an initial-boundary value problem for a nonlinear wave equation in all space dimensions. The nonlinearity in the equation features the damping term |u| k |u t | m sgn(u t ) and a source term of the form |u| p-1 u, where k, p > 1 and 0 k + m and the initial energy is negative, then local weak solutions blow-up in finite time, regardless of the size of the initial data.

A Nonexistence Theorem of Proper Harmonic Morphisms Between Hyperbolic Spaces

We provide a nonexistence theorem of harmonic morphisms between hyperbolic spaces which are proper and C2-smooth to the boundary.

Some existence and nonexistence theorems for compact graphs of constant mean curvature with boundary in parallel planes

Given H≥0 and bounded convex curves α1, ...,⇌n, α in the plane z=0 bounding domains D1, …, Dn, D, respectively, with\(\overline D _i \cap \overline D _j = \emptyset \) if i ∈ j and with Di ⊂ D, we obtain several results proving the existence of a constanth depending only on H and on the geometry of the curves αi, α such that the Dirichlet problem for the constant mean curvature H equation:\(\left\{ \begin{gathered} u \in C^2 (\Omega ) \cap C^0 (\bar \Omega ) \hfill \\ div\frac{{\nabla u}}{{\sqrt {1 + \left| {\nabla u} \right|^2 } }} + 2H = 0 \hfill \\ u\left| {\alpha = 0, } \right.u\left| {\alpha _i = h, i = 1, \ldots ,n } \right. \hfill \\ \end{gathered} \right.\) where\(\Omega = D\backslash ( \cup \overline D _i )\) may accept or not a solution.

Application of representation formulae to comparison and nonexistence theorems for elliptic boundary value problems

Using the representation formulas obtained earlier, new comparison theorems for elliptic boundary value problems are developed. Properties of support function of convex domain are applied for proofs and for obtaining nonexistence theorems for solutions of capillary problems in the absence of gravity.

Nonexistence Theorems for Systems of Quasilinear Parabolic Inequalities

Nonexistence Theorems for Systems of Quasilinear Parabolic Inequalities

Instability and nonexistence theorems for $F$-harmonic maps

In this paper we study the unstability and nonexistence of $F$-harmonic maps. We introduce the notion of $F$-strongly unstable and $F$-unstable manifolds and discuss properties of such manifolds. We classify all compact irreducible $F$-unstable symmetric spaces.

The balance between existence and nonexistence theorems in differential geometry

We discuss the delicate balance between existence and nonexistence theorems in differential geometry. Studying their interplay yields some information about $ p $-harmonic maps, $ p $-SSU manifolds, geometric $ k_p $-connected manifolds, minimal hypersurfaces and Gauss maps, and manifolds admitting essential positive supersolutions of certain nonlinear PDE. As an application of the theory developed, we obtain a topological theorem for minimal submanifolds in complete manifolds with nonpositive sectional curvature.

A nonexistence result on difference sets, partial difference sets and divisible difference sets

Abstract A new nonexistence theorem is proved for abelian difference sets. The proof technique generalizes to yield similar theorems for partial and divisible difference sets. As applications, we establish the nonexistence of two previously open difference sets: (243, 121, 60) difference sets in Z 9 × Z 27 and (351, 126, 45) difference sets in Z 351 , thereby filling two missing entries in a recent table of Vera Lopez and Sanchez (1997) both with answer ‘no’.

Nonexistence theorems for weak solutions of quasilinear elliptic equations

New nonexistence results are obtained for entire bounded (either from above or from below) weak solutions of wide classes of quasilinear elliptic equations and inequalities. It should be stressed that these solutions belong only locally to the corresponding Sobolev spaces. Important examples of the situations considered herein are the following: ,,, where n ≥ 1, p > 1, q > 0 are fixed real numbers, and a(x) is a nonnegative measurable locally bounded function. The methods involve the use of capacity theory in connection with special types of test functions and new integral inequalities. Various results, involving mainly classical solutions, are improved and/or extended to the present cases.

Inscribing cubes and covering by rhombic dodecahedra via equivariant topology

First, we prove a special case of Knaster’s problem, implying that each symmetric convex body in R 3 admits an inscribed cube. We deduce it from a theorem in equivariant topology, which says that there is no S4-equivariant map from SO(3) to S 2 , where S4 acts on SO(3) as the rotation group of the cube and on S 2 as the symmetry group of the regular tetrahedron. We also give some generalizations. Second, we show how the above non-existence theorem yields Makeev’s conjecture in R 3 that each set in R 3 of diameter 1 can be covered by a rhombic dodecahedron, which has distance 1 between its opposite faces. This reveals an unexpected connection between inscribing cubes into symmetric bodies and covering sets by rhombic dodecahedra. Finally, we point out a possible application of our second theorem to the Borsuk problem in R 3 .

Non-Existence Results for a Semilinear Hyperbolic Problem with Boundary Condition of Memory Type

We consider a problem which models the evolution of sound in a compressible fluid with reflection of sound at the surface of the material. Different methods such as the concavity method of Levine, the potential well method and an argument due to Tsutsumi are used to derive global non-existence theorems.

On stable currents and positively curved hypersurfaces

We establish a nonexistence theorem for stable currents (or stable varifolds) in complete 5-pinched hypersurfaces of a real space form with nonnegative constant sectional curvature. This is a partial positive answer to the well-known conjecture of Lawson and Simons. ?

Stabilizability considerations and design of rational controllers for a class of time-delay systems

The problem of stabilization of a linear system characterized by its rational transfer function, with an input time delay, by a rational controller, is considered. If the delay is not known at all (its interval is infinite), the necessary and sufficient condition for the existence of a rational controller is stability of the open-loop rational plant. Moreover, the existence of a rational stabilizing controller implies the existence of a constant gain stabilizing controller. If the delay is known to lie in a given finite interval, two non-existence theorems are derived, one for the constant gain stabilizing controller and one for any rational stabilizing controller. A design method for stabilizing rational controllers, based on first-order all-pass filters cascaded by a constant gain and on violating the conditions for the non-existence theorem of the constant gain stabilizer, is presented.

Nonexistence of time‐periodic solutions of the Dirac equation in an axisymmetric black hole geometry

We prove that in the nonextreme Kerr-Newman black hole geometry, the Dirac equation has no normalizable, time-periodic solutions. A key tool is Chan-drasekhar's separation of the Dirac equation in this geometry. A similar nonexistence theorem is established in a more general class of stationary, axisymmetric metrics in which the Dirac equation is known to be separable. These results indicate that, in contrast to the classical situation of massive particle orbits, a quantum mechanical Dirac particle must either disappear into the black hole or escape to infinity. © 2000 John Wiley & Sons, Inc.

Blow up, global existence and growth rate estimates in nonlinear parabolic systems

We prove Fujita-type global existence and nonexistence theorems for a system of m equations (m > 1) with different diffusion coefficients, i.e. $u_{it} - d_{i} Δu_{i} = \prod_{k=1}^m u_{k}^{p_k^i}, i=1,...,m, x ∈ ℝ^{N}, t > 0,$ with nonnegative, bounded,

Nonexistence for Semilinear Equations and Systems in the Heisenberg Group

In this paper several nonexistence theorems for nonnegative solutions of semilinear subelliptic Laplace equations and systems in the Heisenberg group are established and some interesting integral identities are given.

Global Nonexistence Theorems for a Class of Evolution Equations with Dissipation

.We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the type \( \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, & (t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, & (t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es& \ea \) where \( 0$, $p>2$, $m>1\). We prove a global nonexistence theorem for positive initial value of the energy when \( 1 1\).

A non-existence theorem for a semilinear Dirichlet problem involving critical exponent on halfspaces of the Heisenberg group

Let \( \Delta _ {{\Bbb H}^n}\) be the Kohn Laplacian on the Heisenberg group \( {\Bbb H}^n \) and let Q = 2n + 2 be the homogeneous dimension of \( {\Bbb H}^n \). In this note, completing a recent result obtained with E. Lanconelli [9], we prove that, if \( \Pi \) is a halfspace of \( {\Bbb H}^n \), then the critical Dirichlet problem ¶¶\( (^*) \qquad - \Delta _{{\Bbb H}^n}u = u^{{Q+2} \over {Q-2}} \qquad {\rm in} \, \Pi, \qquad u = 0 \qquad {\rm in} \, \partial \Pi\),¶¶ has no nontrivial nonnegative weak solutions. This result enables to improve a representation theorem by Citti [2], for Palais-Smale sequences related to the equation in (*).