Existence of blowup solutions for nonlinear problems with a gradient term

Whole Space ◽

We prove the existence of positive explosive solutions for the equationΔu+λ(|x|)|∇u(x)|=ϕ(x,u(x))in the whole spaceℝN(N≥3), whereλ:[0,∞)→[0,∞)is a continuous function andϕ:ℝN×[0,∞)→[0,∞)is required to satisfy some hypotheses detailed below. More precisely, we will give a necessary and sufficient condition for the existence of a positive solution that blows up at infinity.

Semi-smooth Points in Some Classical Function Spaces

Results in Mathematics ◽

10.1007/s00025-021-01545-9 ◽

2021 ◽

Vol 77 (1) ◽

Author(s):

Beata Derȩgowska ◽

Beata Gryszka ◽

Karol Gryszka ◽

Paweł Wójcik

Keyword(s):

Continuous Function ◽

Metric Space ◽

Continuous Functions ◽

Sufficient Condition ◽

Compact Metric Space ◽

Spaces Of Continuous Functions ◽

Classical Function ◽

Necessary And Sufficient ◽

Vector Valued

AbstractThe investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

‎INSERTION OF A CONTRA-CONTINUOUS FUNCTION BETWEEN TWO ‎‎COMPARABLE CONTRA-$\alpha-$CONTINUOUS (CONTRA-$C-$CONTINUOUS) FUNCTIONS

Facta Universitatis Series Mathematics and Informatics ◽

10.22190/fumi1901013m ◽

2019 ◽

pp. 13

Author(s):

Majid Mirmiran ◽

Binesh Naderi

Keyword(s):

Continuous Function ◽

Continuous Functions ◽

Topological Spaces ◽

Sufficient Condition ◽

‎A necessary and sufficient condition in terms of lower cut sets ‎are given for the insertion of a contra-continuous function ‎between two comparable real-valued functions on such topological ‎spaces that kernel of sets are open‎.

On Weighted Polynomial Approximation With Gaps

Nagoya Mathematical Journal ◽

10.1017/s0027763000009119 ◽

2005 ◽

Vol 178 ◽

pp. 55-61 ◽

Cited By ~ 10

Author(s):

Guantie Deng

Keyword(s):

Banach Space ◽

Continuous Function ◽

Polynomial Approximation ◽

Continuous Functions ◽

Sufficient Condition ◽

Weighted Polynomial Approximation ◽

Weighted Banach Space ◽

Nonnegative Continuous Function ◽

Let α be a nonnegative continuous function on ℝ. In this paper, the author obtains a necessary and sufficient condition for polynomials with gaps to be dense in Cα, where Cα is the weighted Banach space of complex continuous functions ƒ on ℝ with ƒ(t) exp(−α(t)) vanishing at infinity.

A blow-up dichotomy for semilinear fractional heat equations

Mathematische Annalen ◽

10.1007/s00208-020-02078-2 ◽

2020 ◽

Author(s):

Robert Laister ◽

Mikołaj Sierżęga

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Positive Solutions ◽

Finite Time ◽

Blow Up ◽

Heat Equations ◽

Sufficient Condition ◽

Whole Space ◽

Abstract We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive solutions to become unbounded in finite time. Moreover, we show that this condition is equivalent to blow-up of all positive solutions of a closely-related scalar ordinary differential equation.

Dunkl harmonic analysis and fundamental sets of continuous functions on the unit sphere

Advances in Pure and Applied Mathematics ◽

10.1515/apam-2014-0053 ◽

2015 ◽

Vol 6 (3) ◽

Author(s):

Roman A. Veprintsev

Keyword(s):

Continuous Function ◽

Harmonic Analysis ◽

Unit Sphere ◽

Continuous Functions ◽

Sufficient Condition ◽

The Family ◽

AbstractWe establish a necessary and sufficient condition on a continuous function on [-1,1] under which the family of functions on the unit sphere 𝕊

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

Cesàro kernels on classical groups

10.1017/s1446788700000951 ◽

1997 ◽

Vol 63 (3) ◽

pp. 364-389

Author(s):

Dashan Fan

Keyword(s):

Continuous Function ◽

Harmonic Analysis ◽

Classical Group ◽

Sufficient Condition ◽

Classical Groups ◽

Cesàro Operator ◽

Necessary And Sufficient ◽

Cesaro Operator

AbstractWe study the Cesàro operator on the classical group G and give a necessary and sufficient condition on the index α = α(G) for which the operator is convergent to f(U) for any continuous function f as N → ∞. The result in this paper solves a question posed by Gong in the book Harmonic analysis on classical groups.

Erratum: A necessary and sufficient condition for a two-sided continuous function to be cohomologous to a one-sided continuous function

Dynamical Systems ◽

10.1080/1468936031000095672 ◽

2003 ◽

Vol 18 (3) ◽

pp. 271-278 ◽

Cited By ~ 2

Author(s):

P. Walters

Keyword(s):

Continuous Function ◽

Sufficient Condition ◽

Badly approximable functions and interpolation by Blaschke products

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500010671 ◽

1976 ◽

Vol 20 (2) ◽

pp. 159-161 ◽

Cited By ~ 1

Author(s):

L. A. Rubel ◽

A. L. Shields

Keyword(s):

Continuous Function ◽

Supremum Norm ◽

Sufficient Condition ◽

Blaschke Products ◽

Winding Number ◽

Uniform Limit ◽

Disc Algebra ◽

Necessary And Sufficient ◽

Badly Approximable

A continuous function φ on the unit circle is called badly approximable if ‖ φ − p ‖∞ ≧ ‖ φ |∞ for all polynomials p, where ‖ |∞ is the essential supremum norm. In (4), Poreda asked whether every continuous φ may be written φ = φW+φB, where φW is the uniform limit of polynomials (i.e. φW belongs to the disc algebra A) and φB is badly approximable. We call such a function φ decomposable. In (4), he characterised the badly approximable functions as those of constant non-zero modulus and negative winding number around the origin, i.e. ind (φ)<0. (See (3) for two new proofs of this result.) We show that the answer to Poreda's question is no in general, but give a necessary and sufficient condition for a given φ to have such a decomposition. Then we apply this criterion to solve an interpolation problem.

Insertion of a Contra-continuous Function between Two Comparable Real-valued Functions

Earthline Journal of Mathematical Sciences ◽

10.34198/ejms.3120.2135 ◽

2019 ◽

pp. 21-35 ◽

Cited By ~ 1

Author(s):

Majid Mirmiran ◽

Binesh Naderi

Keyword(s):

Continuous Function ◽

Topological Spaces ◽

Sufficient Condition ◽

A necessary and sufficient condition in terms of lower cut sets are given for the insertion of a contra-continuous function between two comparable real-valued functions on such topological spaces that kernel of sets are open.

INHOMOGENEOUS PERIODIC PARABOLIC PROBLEMS WITH INDEFINITE DATA

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972711002553 ◽

2011 ◽

Vol 84 (3) ◽

pp. 516-524 ◽

Cited By ~ 1

Author(s):

T. GODOY ◽

U. KAUFMANN

Keyword(s):

Positive Solution ◽

Bounded Domain ◽

Parabolic Problems ◽

Sufficient Condition ◽

Smooth Bounded Domain ◽

Unbounded Function ◽

Existence Of Positive Solutions ◽

Carathéodory Function ◽

AbstractLet Ω⊂ℝN be a smooth bounded domain and let f⁄≡0 be a possibly discontinuous and unbounded function. We give a necessary and sufficient condition on f for the existence of positive solutions for all λ>0 of Dirichlet periodic parabolic problems of the form Lu=h(x,t,u)+λf(x,t), where h is a nonnegative Carathéodory function that is sublinear at infinity. When this condition is not fulfilled, under some additional assumptions on h we characterize the set of λs for which the aforementioned problem possesses some positive solution. All results remain true for the corresponding elliptic problems.