Strong maximum and minimum principles for parabolic functional-differential problems with nonlocal inequalities
Ludwik Byszewski (1990)
Annales Polonici Mathematici
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Ludwik Byszewski (1990)
Annales Polonici Mathematici
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M. Guedda (2002)
Colloquium Mathematicae
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We study the absence of nonnegative global solutions to parabolic inequalities of the type , where , 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 , where a ≥ 0, b > 0, p > 1 and V₊(x): = maxV(x),0....
Philippe Souplet, Slim Tayachi (2001)
Colloquium Mathematicae
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Consider the nonlinear heat equation (E): . We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates . Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality . More general inequalities of the form with, for instance, are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions...
Eugene Fabes (1972)
Studia Mathematica
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Dorota Krassowska, Janusz Matkowski (2005)
Annales Polonici Mathematici
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It is shown that, under some general algebraic conditions on fixed real numbers a,b,α,β, every solution f:ℝ → ℝ of the system of functional inequalities f(x+a) ≤ f(x)+α, f(x+b) ≤ f(x)+β that is continuous at some point must be a linear function (up to an additive constant). Analogous results for three other similar simultaneous systems are presented. An application to a characterization of -norm is given.
A. Koldobsky, A. Pajor, V. Yaskin (2008)
Studia Mathematica
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We extend Kahane-Khinchin type inequalities to the case p > -2. As an application we verify the slicing problem for the unit balls of finite-dimensional spaces that embed in , p > -2.
P. Besala (1971)
Annales Polonici Mathematici
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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 ⎧ in Q = Ω×(0,T), ⎨ u(x,t) = 0 on ∂Ω ×(0,T), ⎩ in Ω. with s = (N+2)/(N+p) (p-1), , τ = (N+p)/(p-1), r = (N(p-1) + p)/(N+2), and f ∈ L¹(Q).
Adrian Karpowicz (2008)
Annales Polonici Mathematici
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We consider the Darboux problem for a functional differential equation: a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b]∖(0,a]×(0,b], where the function is defined by for (s,t) ∈ [-a₀,0]×[-b₀,0]. We give a few theorems about weak and strong inequalities for this problem. We also discuss the case where the right-hand side of the differential equation is linear.
László Simon (2015)
Mathematica Bohemica
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We consider second order quasilinear evolution equations where also the main part contains functional dependence on the unknown function. First, existence of solutions in is proved and examples satisfying the assumptions of the existence theorem are formulated. Then a uniqueness theorem is proved. Finally, existence and some qualitative properties of the solutions in (boundedness and stabilization as ) are shown.
Daniel Wachsmuth (2016)
Commentationes Mathematicae Universitatis Carolinae
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Let with be given. Then we show by means of a counter-example that the positive part of has less regularity, in particular it holds in general. Nevertheless, satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.
J. W. Cholewa, T. Dłotko (2003)
Banach Center Publications
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An abstract semilinear parabolic equation in a Banach space X is considered. Under general assumptions on nonlinearity this problem is shown to generate a bounded dissipative semigroup on . This semigroup possesses an -global attractor that is closed, bounded, invariant in , and attracts bounded subsets of in a ’weaker’ topology of an auxiliary Banach space Z. The abstract approach is finally applied to the scalar parabolic equation in Rⁿ and to the partly dissipative system. ...
Victor Galaktionov (2004)
Journal of the European Mathematical Society
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We consider th-order semilinear parabolic equations in , with Dirac’s mass as the initial function. We show that for , the Cauchy problem admits a solution which is bounded and smooth for small , while for such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.
Itai Shafrir, Gershon Wolansky (2005)
Journal of the European Mathematical Society
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We prove several optimal Moser–Trudinger and logarithmic Hardy–Littlewood–Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere , on a bounded domain and on all of . In some cases we also address the question of existence of minimizers.
Ronan Pouliquen (1999)
Studia Mathematica
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We prove trace inequalities of type where , under suitable hypotheses on the sequences and , with the first sequence increasing and the second bounded.
Anne-Laure Dalibard (2011)
Journal of the European Mathematical Society
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This article investigates the long-time behaviour of parabolic scalar conservation laws of the type , where and the flux is periodic in . More specifically, we consider the case when the initial data is an disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between u and the stationary solution behaves in norm like a self-similar profile for large times. The proof uses a time and space change of variables...
Z. Kominek (1974)
Annales Polonici Mathematici
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Alessandra Lunardi (1998)
Studia Mathematica
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We study existence, uniqueness, and smoothing properties of the solutions to a class of linear second order elliptic and parabolic differential equations with unbounded coefficients in . The main results are global Schauder estimates, which hold in spite of the unboundedness of the coefficients.
C. T. Ng (1973)
Annales Polonici Mathematici
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H. Światak (1967)
Annales Polonici Mathematici
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M. Malenica (1982)
Matematički Vesnik
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Boumediene Abdellaoui, Eduardo Colorado, Ireneo Peral (2004)
Journal of the European Mathematical Society
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In this work we study the problem in , in , on , in , is a bounded regular domain such that , , , , and are positive functions such that and . The main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case , ; (ii) the nonexistence of solutions if , ; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions...
Albert Baernstein II, Robert C. Culverhouse (2002)
Studia Mathematica
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Let , , where the are independent random vectors, each uniformly distributed on the unit sphere in ℝⁿ, and are real constants. We prove that if is majorized by in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝⁿ → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts...
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 on a manifold gives rise to sequences of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative on the corresponding tractor bundle , where is the normal Cartan connection. The first operator in the sequence is overdetermined and it is well known that yields the prolongation of this operator in the homogeneous case . Our first...