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.
Tayeb Benhamoud, Elmehdi Zaouche, Mahmoud Bousselsal (2024)
Mathematica Bohemica
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This paper is concerned with the study of a nonlocal nonlinear parabolic problem associated with the equation in , where is a bounded domain of , is a positive number, is an matrix of variable coefficients depending on and , , are given functions. We consider two different assumptions on . The existence of a weak solution for this problem is proved using the Schauder fixed point theorem for each of these assumptions. Moreover, if depends only on...
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.
Tatiana Danielsson, Pernilla Johnsen (2021)
Mathematica Bohemica
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In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in , fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation , where . The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for which the local problem is parabolic is shifted by , compared to the standard matching that gives rise...
Nakayashiki, Ryota
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In this paper, we consider a class of initial-boundary value problems for quasilinear PDEs, subject to the dynamic boundary conditions. Each initial-boundary problem is denoted by (S) with a nonnegative constant , and for any , (S) can be regarded as a vectorial transmission system between the quasilinear equation in the spatial domain , and the parabolic equation on the boundary , having a sufficient smoothness. The objective of this study is to establish a mathematical method,...
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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