Universal solutions of a nonlinear heat equation on $\mathbb {R}^N$

Thierry Cazenave; Flávio Dickstein; Fred B. Weissler

Displaying similar documents to “Universal solutions of a nonlinear heat equation on $ℝ^{N}$ ”

Classification of initial data for the Riccati equation

N. Chernyavskaya, L. Shuster (2002)

Bollettino dell'Unione Matematica Italiana

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We consider a Cauchy problem $y^{'} (x) + y^{2} (x) = q (x), {y (x)|}_{x = x_{0}} = y_{0}$ where $x_{0}$ , $y_{0} \in R$ and $q (x) \in L_{1}^{loc} (R)$ is a non-negative function satisfying the condition: $\int_{- \infty}^{x} q (t) d t > 0, \int_{x}^{\infty} q (t) d t > 0 for x \in R .$ We obtain the conditions under which $y (x)$ can be continued to all of $R$ . This depends on $x_{0}$ , $y_{0}$ and the properties of $q (x)$ .

Short-time heat flow and functions of bounded variation in $R^{N}$

Michele Miranda, Diego Pallara, Fabio Paronetto, Marc Preunkert (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

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We prove a characterisation of sets with finite perimeter and $B V$ functions in terms of the short time behaviour of the heat semigroup in $R^{N}$ . For sets with smooth boundary a more precise result is shown.

The ratio and generating function of cogrowth coefficients of finitely generated groups

Ryszard Szwarc (1998)

Studia Mathematica

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Let G be a group generated by r elements $g_{1}, \dots, g_{r}$ . Among the reduced words in $g_{1}, \dots, g_{r}$ of length n some, say $γ_{n}$ , represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of $γ_{2 n}$ has a limit, called the cogrowth exponent with respect to the generators $g_{1}, \dots, g_{r}$ . We show by analytic methods that the numbers $γ_{n}$ vary regularly, i.e. the ratio $γ_{2 n + 2} / γ_{2 n}$ is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function...

An inverse problem for the equation $▵ u = - c u - d$

Michael Vogelius (1994)

Annales de l'institut Fourier

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Let $Ω$ be a bounded, convex planar domain whose boundary has a not too degenerate curvature. In this paper we provide partial answers to an identification question associated with the boundary value problem $▵ u = - c u - d in Ω, u = 0 on \partial Ω .$ We prove two results: 1) If $Ω$ is not a ball and if one considers only solutions with $- c u - d \leq 0$ , then there exist at most finitely many pairs of coefficients $(c, d)$ so that the normal derivative $\frac{\partial u}{\partial ν} |_{\partial Ω}$ equals a given $ψ \neq 0$ . 2) If one imposes no sign condition on the...

Cramér's formula for Heisenberg manifolds

Mahta Khosravi, John A. Toth (2005)

Annales de l'institut Fourier

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Let $R (λ)$ be the error term in Weyl’s law for a 3-dimensional Riemannian Heisenberg manifold. We prove that $\int_{1}^{T} {| R (t) |}^{2} d t = c T^{\frac{5}{2}} + O_{δ} (T^{\frac{9}{4} + δ})$ , where $c$ is a specific nonzero constant and $δ$ is an arbitrary small positive number. This is consistent with the conjecture of Petridis and Toth stating that $R (t) = O_{δ} (t^{\frac{3}{4} + δ})$ .The idea of the proof is to use the Poisson summation formula to write the error term in a form which can be estimated by the method of the stationary phase. The similar result will be also proven in the $2 n + 1$ -dimensional case. ...

On equations with reflection

D. Przeworska-Rolewicz (1969)

Studia Mathematica

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Displaying similar documents to “Universal solutions of a nonlinear heat equation on ℝ N ”

Classification of initial data for the Riccati equation

Short-time heat flow and functions of bounded variation in R N

The ratio and generating function of cogrowth coefficients of finitely generated groups

An inverse problem for the equation ▵ u = - c u - d

Cramér's formula for Heisenberg manifolds

On equations with reflection

Displaying similar documents to “Universal solutions of a nonlinear heat equation on $ℝ^{N}$ ”

Short-time heat flow and functions of bounded variation in $R^{N}$

An inverse problem for the equation $▵ u = - c u - d$