Displaying similar documents to “Nonlinear diffusion equations with perturbation terms on unbounded domains”

Existence theorems for nonlinear differential equations having trichotomy in Banach spaces

Adel Mahmoud Gomaa (2017)

Czechoslovak Mathematical Journal

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We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation x ˙ ( t ) = ( t ) x ( t ) + f ( t , x ( t ) ) , t ( P ) where { ( t ) : t } is a family of linear operators from a Banach space E into itself and f : × E E . By L ( E ) we denote the space of linear operators from E into itself. Furthermore, for a < b and d > 0 , we let C ( [ - d , 0 ] , E ) be the Banach space of continuous functions from [ - d , 0 ] into E and f d : [ a , b ] × C ( [ - d , 0 ] , E ) E . Let ^ : [ a , b ] L ( E ) be a strongly measurable and Bochner integrable operator on [ a , b ] and for t [ a , b ] define τ t x ( s ) = x ( t + s ) for each s [ - d , 0 ] . We prove that, under certain...

Spreading and vanishing in nonlinear diffusion problems with free boundaries

Yihong Du, Bendong Lou (2015)

Journal of the European Mathematical Society

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We study nonlinear diffusion problems of the form u t = u x x + f ( u ) with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special f ( u ) of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any f ( u ) which is C 1 and satisfies f ( 0 ) = 0 , we show that the omega limit set ω ( u ) of every bounded positive solution is determined by a stationary...

Further generalized versions of Ilmanen’s lemma on insertion of C 1 , ω or C loc 1 , ω functions

Václav Kryštof (2021)

Commentationes Mathematicae Universitatis Carolinae

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The author proved in 2018 that if G is an open subset of a Hilbert space, f 1 , f 2 : G continuous functions and ω a nontrivial modulus such that f 1 f 2 , f 1 is locally semiconvex with modulus ω and f 2 is locally semiconcave with modulus ω , then there exists f C loc 1 , ω ( G ) such that f 1 f f 2 . This is a generalization of Ilmanen’s lemma (which deals with linear modulus and functions on an open subset of n ). Here we extend the mentioned result from Hilbert spaces to some superreflexive spaces, in particular to L p spaces, p [ 2 , ) . We...

Involutivity degree of a distribution at superdensity points of its tangencies

Silvano Delladio (2021)

Archivum Mathematicum

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Let Φ 1 , ... , Φ k + 1 (with k 1 ) be vector fields of class C k in an open set U N + m , let 𝕄 be a N -dimensional C k submanifold of U and define 𝕋 : = { z 𝕄 : Φ 1 ( z ) , ... , Φ k + 1 ( z ) T z 𝕄 } where T z 𝕄 is the tangent space to 𝕄 at z . Then we expect the following property, which is obvious in the special case when z 0 is an interior point (relative to 𝕄 ) of 𝕋 : If z 0 𝕄 is a ( N + k ) -density point (relative to 𝕄 ) of 𝕋 then all the iterated Lie brackets of order less or equal to k Φ i 1 ( z 0 ) , [ Φ i 1 , Φ i 2 ] ( z 0 ) , [ [ Φ i 1 , Φ i 2 ] , Φ i 3 ] ( z 0 ) , ... ( h , i h k + 1 ) belong to T z 0 𝕄 . Such a property has been proved in [9] for k = 1 and its proof in the...

On compactness and connectedness of the paratingent

Wojciech Zygmunt (2016)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this note we shall prove that for a continuous function ϕ : Δ n , where Δ ,  the paratingent of ϕ at a Δ is a non-empty and compact set in n if and only if ϕ satisfies Lipschitz condition in a neighbourhood of a . Moreover, in this case the paratingent is a connected set.

Admissible spaces for a first order differential equation with delayed argument

Nina A. Chernyavskaya, Lela S. Dorel, Leonid A. Shuster (2019)

Czechoslovak Mathematical Journal

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We consider the equation - y ' ( x ) + q ( x ) y ( x - ϕ ( x ) ) = f ( x ) , x , where ϕ and q ( q 1 ) are positive continuous functions for all x and f C ( ) . By a solution of the equation we mean any function y , continuously differentiable everywhere in , which satisfies the equation for all x . We show that under certain additional conditions on the functions ϕ and q , the above equation has a unique solution y , satisfying the inequality y ' C ( ) + q y C ( ) c f C ( ) , where the constant c ( 0 , ) does not depend on the choice of f .

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

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Let Ω be a bounded domain of class C 2 in N and let K be a compact subset of Ω . Assume that q ( N + 1 ) / ( N 1 ) and denote by U K the maximal solution of Δ u + u q = 0 in Ω which vanishes on Ω K . We obtain sharp upper and lower estimates for U K in terms of the Bessel capacity C 2 / q , q ' and prove that U K is σ -moderate. In addition we describe the precise asymptotic behavior of U K at points σ K , which depends on the “density” of K at σ , measured in terms of the capacity C 2 / q , q ' .

The Rothberger property on C p ( Ψ ( 𝒜 ) , 2 )

Daniel Bernal-Santos (2016)

Commentationes Mathematicae Universitatis Carolinae

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A space X is said to have the Rothberger property (or simply X is Rothberger) if for every sequence 𝒰 n : n ω of open covers of X , there exists U n 𝒰 n for each n ω such that X = n ω U n . For any n ω , necessary and sufficient conditions are obtained for C p ( Ψ ( 𝒜 ) , 2 ) n to have the Rothberger property when 𝒜 is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family 𝒜 for which the space C p ( Ψ ( 𝒜 ) , 2 ) n is Rothberger for all n ω .