Displaying similar documents to “Bi-spaces global attractors in abstract parabolic equations”

The regularity of the positive part of functions in L 2 ( I ; H 1 ( Ω ) ) H 1 ( I ; H 1 ( Ω ) * ) with applications to parabolic equations

Daniel Wachsmuth (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let u L 2 ( I ; H 1 ( Ω ) ) with t u L 2 ( I ; H 1 ( Ω ) * ) be given. Then we show by means of a counter-example that the positive part u + of u has less regularity, in particular it holds t u + L 1 ( I ; H 1 ( Ω ) * ) in general. Nevertheless, u + satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.

On higher-order semilinear parabolic equations with measures as initial data

Victor Galaktionov (2004)

Journal of the European Mathematical Society

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We consider 2 m th-order ( m 2 ) semilinear parabolic equations u t = ( Δ ) m u ± | u | p 1 u in N × + ( p > 1 ) , with Dirac’s mass δ ( x ) as the initial function. We show that for p < p 0 = 1 + 2 m / N , the Cauchy problem admits a solution u ( x , t ) which is bounded and smooth for small t > 0 , while for p p 0 such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.

Existence results for a class of nonlinear parabolic equations with two lower order terms

Ahmed Aberqi, Jaouad Bennouna, M. Hammoumi, Mounir Mekkour, Ahmed Youssfi (2014)

Applicationes Mathematicae

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We investigate the existence of renormalized solutions for some nonlinear parabolic problems associated to equations of the form ⎧ ( e β u - 1 ) / t - d i v ( | u | p - 2 u ) + d i v ( c ( x , t ) | u | s - 1 u ) + b ( x , t ) | u | r = f in Q = Ω×(0,T), ⎨ u(x,t) = 0 on ∂Ω ×(0,T), ⎩ ( e β u - 1 ) ( x , 0 ) = ( e β u - 1 ) ( x ) in Ω. with s = (N+2)/(N+p) (p-1), c ( x , t ) ( L τ ( Q T ) ) N , τ = (N+p)/(p-1), r = (N(p-1) + p)/(N+2), b ( x , t ) L N + 2 , 1 ( Q T ) and f ∈ L¹(Q).

Absence of global solutions to a class of nonlinear parabolic inequalities

M. Guedda (2002)

Colloquium Mathematicae

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We study the absence of nonnegative global solutions to parabolic inequalities of the type u t - ( - Δ ) β / 2 u - V ( x ) u + h ( x , t ) u p , where ( - Δ ) β / 2 , 0 < β ≤ 2, is the β/2 fractional power of the Laplacian. We give a sufficient condition which implies that the only global solution is trivial if p > 1 is small. Among other properties, we derive a necessary condition for the existence of local and global nonnegative solutions to the above problem for the function V satisfying V ( x ) a | x | - b , where a ≥ 0, b > 0, p > 1 and V₊(x): = maxV(x),0....

On a new normalization for tractor covariant derivatives

Matthias Hammerl, Petr Somberg, Vladimír Souček, Josef Šilhan (2012)

Journal of the European Mathematical Society

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A regular normal parabolic geometry of type G / P on a manifold M gives rise to sequences D i of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative ω on the corresponding tractor bundle V , where ω is the normal Cartan connection. The first operator D 0 in the sequence is overdetermined and it is well known that ω yields the prolongation of this operator in the homogeneous case M = G / P . Our first...

Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source

Xiangdong Zhao (2024)

Czechoslovak Mathematical Journal

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We study the chemotaxis system with singular sensitivity and logistic-type source: u t = Δ u - χ · ( u v / v ) + r u - μ u k , 0 = Δ v - v + u under the non-flux boundary conditions in a smooth bounded domain Ω n , χ , r , μ > 0 , k > 1 and n 1 . It is shown with k ( 1 , 2 ) that the system possesses a global generalized solution for n 2 which is bounded when χ > 0 is suitably small related to r > 0 and the initial datum is properly small, and a global bounded classical solution for n = 1 .

Curved thin domains and parabolic equations

M. Prizzi, M. Rinaldi, K. P. Rybakowski (2002)

Studia Mathematica

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Consider the family uₜ = Δu + G(u), t > 0, x Ω ε , ν ε u = 0 , t > 0, x Ω ε , ( E ε ) of semilinear Neumann boundary value problems, where, for ε > 0 small, the set Ω ε is a thin domain in l , possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of l . If G is dissipative, then equation ( E ε ) has a global attractor ε . We identify a “limit” equation for the family ( E ε ) , prove convergence of trajectories and establish an upper semicontinuity result for the family ε as ε → 0⁺. ...

Existence and nonexistence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities

Boumediene Abdellaoui, Eduardo Colorado, Ireneo Peral (2004)

Journal of the European Mathematical Society

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In this work we study the problem u t div ( | x | 2 γ u ) = λ u α | x | 2 ( γ + 1 ) + f in Ω × ( 0 , T ) , u 0 in Ω × ( 0 , T ) , u = 0 on Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , Ω N ( N 2 ) is a bounded regular domain such that 0 Ω , λ > 0 , α > 0 , - < γ < ( N 2 ) / 2 , f and u 0 are positive functions such that f L 1 ( Ω × ( 0 , T ) ) and u 0 L 1 ( Ω ) . The main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case α = 1 , 1 + γ > 0 ; (ii) the nonexistence of solutions if α > 1 , 1 + γ > 0 ; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions...

Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities

Philippe Souplet, Slim Tayachi (2001)

Colloquium Mathematicae

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Consider the nonlinear heat equation (E): u t - Δ u = | u | p - 1 u + b | u | q . We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates C ( T - t ) - 1 / ( p - 1 ) | | u ( t ) | | C ( T - t ) - 1 / ( p - 1 ) . Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality u t - u x x u p . More general inequalities of the form u t - u x x f ( u ) with, for instance, f ( u ) = ( 1 + u ) l o g p ( 1 + u ) are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions...

Non supercyclic subsets of linear isometries on Banach spaces of analytic functions

Abbas Moradi, Karim Hedayatian, Bahram Khani Robati, Mohammad Ansari (2015)

Czechoslovak Mathematical Journal

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Let X be a Banach space of analytic functions on the open unit disk and Γ a subset of linear isometries on X . Sufficient conditions are given for non-supercyclicity of Γ . In particular, we show that the semigroup of linear isometries on the spaces S p ( p > 1 ), the little Bloch space, and the group of surjective linear isometries on the big Bloch space are not supercyclic. Also, we observe that the groups of all surjective linear isometries on the Hardy space H p or the Bergman space L a p ( 1 < p < , p 2 )...

L₁-uniqueness of degenerate elliptic operators

Derek W. Robinson, Adam Sikora (2011)

Studia Mathematica

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Let Ω be an open subset of d with 0 ∈ Ω. Furthermore, let H Ω = - i , j = 1 d i c i j j be a second-order partial differential operator with domain C c ( Ω ) where the coefficients c i j W l o c 1 , ( Ω ̅ ) are real, c i j = c j i and the coefficient matrix C = ( c i j ) satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If 0 d s s d / 2 e - λ μ ( s ) ² < for some λ > 0 where μ ( s ) = 0 s d t c ( t ) - 1 / 2 then we establish that H Ω is L₁-unique, i.e. it has a unique L₁-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L₂-extension which generates a submarkovian semigroup....

-sums and the Banach space / c

Christina Brech, Piotr Koszmider (2014)

Fundamenta Mathematicae

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This paper is concerned with the isomorphic structure of the Banach space / c and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that / c does not have an orthogonal -decomposition, that is, it is not of the form ( X ) for any Banach space X. The main local result is that it is consistent that ( c ( ) ) does not embed isomorphically into / c , where is the cardinality of the continuum,...

Partial Hölder continuity for quasilinear parabolic systems of higher order with strictly controlled growth

Mario Marino, Antonino Maugeri (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Sfruttando i risultati di [1], si prova che le derivate spaziali D α u di ordine | α | con | α | < m - 1 delle soluzioni in Q di un sistema parabolico quasilineare di ordine 2 m con andamenti strettamente controllati, sono parzialmente hölderiane in Q con esponente di hölderianità decrescente al crescere di | α | .

Existence and regularity results for abstract non autonomous parabolic equations

Paolo Acquistapace, Brunello Terreni (1982)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si studiano esistenza, unicità e regolarità delle soluzioni strette, classiche e forti dell’equazione di evoluzione non autonoma u ( t ) A ( t ) u ( t ) = f ( t ) , con il dato iniziale u ( 0 ) = x , in spazi di Banach. I dominii degli operatori A ( t ) variano in t e non sono necessariamente densi in E . Si danno condizioni necessarie e sufficienti per l'esistenza e la regolarità holderiana della soluzione e della sua derivata.

On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets

Juan Rivera-Letelier (2001)

Fundamenta Mathematicae

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Given d ≥ 2 consider the family of polynomials P c ( z ) = z d + c for c ∈ ℂ. Denote by J c the Julia set of P c and let d = c | J c i s c o n n e c t e d be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters c d : those for which the critical point 0 is not recurrent by P c and without parabolic cycles. The Hausdorff dimension of J c , denoted by H D ( J c ) , does not depend continuously on c at such c d ; on the other hand the function c H D ( J c ) is analytic in - d . Our first result asserts that there is still some...

On the Aronszajn property for integral equations in Banach space

Stanisław Szufla (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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For the integral equation (1) below we prove the existence on an interval J = [ 0 , a ] of a solution x with values in a Banach space E , belonging to the class L p ( J , E ) , p > 1 . Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

The weak Gelfand-Phillips property in spaces of compact operators

Ioana Ghenciu (2017)

Commentationes Mathematicae Universitatis Carolinae

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For Banach spaces X and Y , let K w * ( X * , Y ) denote the space of all w * - w continuous compact operators from X * to Y endowed with the operator norm. A Banach space X has the w G P property if every Grothendieck subset of X is relatively weakly compact. In this paper we study Banach spaces with property w G P . We investigate whether the spaces K w * ( X * , Y ) and X ϵ Y have the w G P property, when X and Y have the w G P property.

Nonexistence results for the Cauchy problem of some systems of hyperbolic equations

Mokhtar Kirane, Salim Messaoudi (2002)

Annales Polonici Mathematici

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We consider the systems of hyperbolic equations ⎧ u = Δ ( a ( t , x ) u ) + Δ ( b ( t , x ) v ) + h ( t , x ) | v | p , t > 0, x N , (S1) ⎨ ⎩ v = Δ ( c ( t , x ) v ) + k ( t , x ) | u | q , t > 0, x N u = Δ ( a ( t , x ) u ) + h ( t , x ) | v | p , t > 0, x N , (S2) ⎨ ⎩ v = Δ ( c ( t , x ) v ) + l ( t , x ) | v | m + k ( t , x ) | u | q , t > 0, x N , (S3) ⎧ u = Δ ( a ( t , x ) u ) + Δ ( b ( t , x ) v ) + h ( t , x ) | u | p , t > 0, x N , ⎨ ⎩ v = Δ ( c ( t , x ) v ) + k ( t , x ) | v | q , t > 0, x N , in ( 0 , ) × N with u(0,x) = u₀(x), v(0,x) = v₀(x), uₜ(0,x) = u₁(x), vₜ(0,x) = v₁(x). We show that, in each case, there exists a bound B on N such that for 1 ≤ N ≤ B solutions to the systems blow up in finite time.

Pisier's inequality revisited

Tuomas Hytönen, Assaf Naor (2013)

Studia Mathematica

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Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies - 1 , 1 | | j = 1 n j f j ( ε ) | | p d μ ( ε ) p - 1 , 1 - 1 , 1 | | j = 1 n δ j Δ f j ( ε ) | | p d μ ( ε ) d μ ( δ ) , where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and j j = 1 n and Δ = j = 1 n j are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by p ( X ) , we show that p ( X ) k = 1 n 1 / k for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...

On the H-property and rotundity of Cesàro direct sums of Banach spaces

Saard Youyen, Suthep Suantai (2008)

Banach Center Publications

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In this paper, we define the direct sum ( i = 1 n X i ) c e s p of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ( i = 1 n X i ) c e s p has the H-property if and only if each X i has the H-property, and ( i = 1 n X i ) c e s p has the Schur property if and only if each X i has the Schur property. Moreover, we also show that ( i = 1 n X i ) c e s p is rotund if and only if each X i is rotund.

A local Landau type inequality for semigroup orbits

Gerd Herzog, Peer Christian Kunstmann (2014)

Studia Mathematica

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Given a strongly continuous semigroup ( S ( t ) ) t 0 on a Banach space X with generator A and an element f ∈ D(A²) satisfying | | S ( t ) f | | e - ω t | | f | | and | | S ( t ) A ² f | | e - ω t | | A ² f | | for all t ≥ 0 and some ω > 0, we derive a Landau type inequality for ||Af|| in terms of ||f|| and ||A²f||. This inequality improves on the usual Landau inequality that holds in the case ω = 0.

Hydrodynamical behavior of symmetric exclusion with slow bonds

Tertuliano Franco, Patrícia Gonçalves, Adriana Neumann (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the exclusion process in the one-dimensional discrete torus with N points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance N - β , with β [ 0 , ) . We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter β . If β [ 0 , 1 ) , the hydrodynamic limit is given by the usual heat equation. If β = 1 , it is given by a parabolic equation involving an operator...

H calculus and dilatations

Andreas M. Fröhlich, Lutz Weis (2006)

Bulletin de la Société Mathématique de France

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We characterise the boundedness of the H calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if - A generates a bounded analytic C 0 semigroup ( T t ) on a UMD space, then the H calculus of A is bounded if and only if ( T t ) has a dilation to a bounded group on L 2 ( [ 0 , 1 ] , X ) . This generalises a Hilbert space result of C.LeMerdy. If X is an L p space we can choose another L p space in place of L 2 ( [ 0 , 1 ] , X ) .

Self-similar solutions in reaction-diffusion systems

Joanna Rencławowicz (2003)

Banach Center Publications

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In this paper we examine self-similar solutions to the system u i t - d i Δ u i = k = 1 m u k p k i , i = 1,…,m, x N , t > 0, u i ( 0 , x ) = u 0 i ( x ) , i = 1,…,m, x N , where m > 1 and p k i > 0 , to describe asymptotics near the blow up point.

Unicellularity of the multiplication operator on Banach spaces of formal power series

B. Yousefi (2001)

Studia Mathematica

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Let β ( n ) n = 0 be a sequence of positive numbers and 1 ≤ p < ∞. We consider the space p ( β ) of all power series f ( z ) = n = 0 f ̂ ( n ) z such that n = 0 | f ̂ ( n ) | p | β ( n ) | p < . We give some sufficient conditions for the multiplication operator, M z , to be unicellular on the Banach space p ( β ) . This generalizes the main results obtained by Lu Fang [1].