On -estimates for solutions of the Cauchy problem for parabolic differential equations
P. Besala (1971)
Annales Polonici Mathematici
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P. Besala (1971)
Annales Polonici Mathematici
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Eugene Fabes (1972)
Studia Mathematica
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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...
Zofia Szmydt, Bogdan Ziemian (1979)
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).
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.
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....
David Borthwick, Colin Guillarmou (2016)
Journal of the European Mathematical Society
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On geometrically finite hyperbolic manifolds , including those with non-maximal rank cusps, we give upper bounds on the number of resonances of the Laplacian in disks of size as . In particular, if the parabolic subgroups of satisfy a certain Diophantine condition, the bound is .
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.
Adrian Karpowicz (2010)
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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We consider the following Darboux problem for the functional differential equation a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b]where the function is defined by for (s,t) ∈ [-a₀,0]×[-b₀,0]. We prove a theorem on existence of the Carathéodory solutions of the above problem.
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...
Mordechay B. Levin (2013)
Colloquium Mathematicae
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We prove the central limit theorem for the multisequence where , are reals, are partially hyperbolic commuting s × s matrices, and x is a uniformly distributed random variable in . The main tool is the S-unit theorem.
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. ...
Hiroshi Matano, Fabio Punzo, Alberto Tesei (2015)
Journal of the European Mathematical Society
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We study the Cauchy problem in the hyperbolic space for the semilinear heat equation with forcing term, which is either of KPP type or of Allen-Cahn type. Propagation and extinction of solutions, asymptotical speed of propagation and asymptotical symmetry of solutions are addressed. With respect to the corresponding problem in the Euclidean space new phenomena arise, which depend on the properties of the diffusion process in . We also investigate a family of travelling wave solutions,...
Xiliang Bao, Francis Bonahon (2002)
Bulletin de la Société Mathématique de France
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A hyperideal polyhedron is a non-compact polyhedron in the hyperbolic -space which, in the projective model for , is just the intersection of with a projective polyhedron whose vertices are all outside and whose edges all meet . We classify hyperideal polyhedra, up to isometries of , in terms of their combinatorial type and of their dihedral angles.
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.
Grzegorz Karch (1997)
Annales Polonici Mathematici
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We study the decay in time of the spatial -norm (1 ≤ p ≤ ∞) of solutions to parabolic conservation laws with dispersive and dissipative terms added uₜ - uₓₓₜ - νuₓₓ + buₓ = f(u)ₓ or uₜ + uₓₓₓ - νuₓₓ + buₓ = f(u)ₓ, and we show that under general assumptions about the nonlinearity, solutions of the nonlinear equations have the same long time behavior as their linearizations at the zero solution.
Christopher J. Leininger, Saul Schleimer (2014)
Journal of the European Mathematical Society
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We prove, for any , that there is a closed connected orientable surface so that the hyperbolic space almost-isometrically embeds into the Teichmüller space of , with quasi-convex image lying in the thick part. As a consequence, quasi-isometrically embeds in the curve complex of .
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...
Ajoy Jana, M. Thamban Nair (2019)
Czechoslovak Mathematical Journal
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It is known that the nonlinear nonhomogeneous backward Cauchy problem , with , where is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on and , that a solution of the above problem satisfies an integral equation involving the spectral representation of , which is also ill-posed. Spectral truncation...
A. Kruse (1967)
Acta Arithmetica
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Martine Babillot, Marc Peigné (2006)
Bulletin de la Société Mathématique de France
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We consider a large class of non compact hyperbolic manifolds with cusps and we prove that the winding process generated by a closed -form supported on a neighborhood of a cusp , satisfies a limit theorem, with an asymptotic stable law and a renormalising factor depending only on the rank of the cusp and the Poincaré exponent of . No assumption on the value of is required and this theorem generalises previous results due to Y. Guivarc’h, Y. Le Jan, J. Franchi and N. Enriquez. ...
Gabriel Vigny (2014)
Annales de l’institut Fourier
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Let be a dominant rational map of such that there exists with for all . Under mild hypotheses, we show that, for outside a pluripolar set of , the map admits a hyperbolic measure of maximal entropy with explicit bounds on the Lyapunov exponents. In particular, the result is true for polynomial maps hence for the homogeneous extension of to . This provides many examples where non uniform hyperbolic dynamics is established. One of the key tools is to approximate...