Nonanalyticity of solutions to $∂_{t}u = ∂²_{x}u + u²$

Grzegorz Łysik

Displaying similar documents to “Nonanalyticity of solutions to $\partial_{t} u = \partial ²_{x} u + u ²$ ”

An integral operator on the classes $𝒮^{*} (α)$ and $𝒞𝒱ℋ (β)$

Nicoleta Ularu, Nicoleta Breaz (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

The purpose of this paper is to study some properties related to convexity order and coefficients estimation for a general integral operator. We find the convexity order for this operator, using the analytic functions from the class of starlike functions of order $α$ and from the class $𝒞𝒱ℋ (β)$ and also we estimate the first two coefficients for functions obtained by this operator applied on the class $𝒞𝒱ℋ (β)$ .

Total blow-up of a quasilinear heat equation with slow-diffusion for non-decaying initial data

Amy Poh Ai Ling, Masahiko Shimojō (2019)

Mathematica Bohemica

Similarity:

We consider solutions of quasilinear equations $u_{t} = Δ u^{m} + u^{p}$ in $ℝ^{N}$ with the initial data $u_{0}$ satisfying $0 < u_{0} < M$ and ${lim}_{| x | \to \infty} u_{0} (x) = M$ for some constant $M > 0$ . It is known that if $0 < m < p$ with $p > 1$ , the blow-up set is empty. We find solutions $u$ that blow up throughout $ℝ^{N}$ when $m > p > 1$ .

On operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

Colloquium Mathematicae

Similarity:

We show that every operator from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ is compact when 1 ≤ p,q < s and that every operator from $ℓ_{s}$ to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

$Σ_{s}$ -products revisited

Reynaldo Rojas-Hernández (2015)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We show that any $Σ_{s}$ -product of at most $𝔠$ -many $L Σ (\leq ω)$ -spaces has the $L Σ (\leq ω)$ -property. This result generalizes some known results about $L Σ (\leq ω)$ -spaces. On the other hand, we prove that every $Σ_{s}$ -product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every $Σ_{s}$ -product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

On spirallike functions of order $α$ and type $β$ with fixed second coefficient

M. K. Aouf (1988)

Matematički Vesnik

Similarity:

Lucas factoriangular numbers

Bir Kafle, Florian Luca, Alain Togbé (2020)

Mathematica Bohemica

Similarity:

We show that the only Lucas numbers which are factoriangular are $1$ and $2$ .

Property C for ODE and Applications to an Inverse Problem for a Heat Equation

A. G. Ramm (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let $ℓ_{j} : = - d ² / d x ² + k ² q_{j} (x)$ , k = const > 0, j = 1,2, $0 < e s s i n f q_{j} (x) \leq e s s s u p q_{j} (x) < \infty$ . Suppose that (*) $\int_{0}^{1} p (x) u ₁ (x, k) u ₂ (x, k) d x = 0$ for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0,1], which changes sign finitely many times, and $u_{j}$ solves the problem $ℓ_{j} u_{j} = 0$ , 0 ≤ x ≤ 1, $u_{j}^{'} (0, k) = 0$ , $u_{j} (0, k) = 1$ . It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.

A localization property for $B_{p q}^{s}$ and $F_{p q}^{s}$ spaces

Hans Triebel (1994)

Studia Mathematica

Similarity:

Let $f^{j} = \sum_{k} a_{k} f (2^{j + 1} x - 2 k)$ , where the sum is taken over the lattice of all points k in $ℝ^{n}$ having integer-valued components, j∈ℕ and $a_{k} \in ℂ$ . Let $A_{p q}^{s}$ be either $B_{p q}^{s}$ or $F_{p q}^{s}$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on $ℝ^{n} .$ The aim of the paper is to clarify under what conditions $∥ f^{j} | A_{p q}^{s} ∥$ is equivalent to $2^{j (s - n / p)} (\sum_{k} | a_{k} |^{p})^{1 / p} ∥ f | A_{p q}^{s} ∥$ .

On the distribution of $(k, r)$ -integers in Piatetski-Shapiro sequences

Teerapat Srichan (2021)

Czechoslovak Mathematical Journal

Similarity:

A natural number $n$ is said to be a $(k, r)$ -integer if $n = a^{k} b$ , where $k > r > 1$ and $b$ is not divisible by the $r$ th power of any prime. We study the distribution of such $(k, r)$ -integers in the Piatetski-Shapiro sequence ${⌊ n^{c} ⌋}$ with $c > 1$ . As a corollary, we also obtain similar results for semi- $r$ -free integers.

$C^{*}$ -points vs $P$ -points and $P^{♭}$ -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In a Tychonoff space $X$ , the point $p \in X$ is called a $C^{*}$ -point if every real-valued continuous function on $C ∖ {p}$ can be extended continuously to $p$ . Every point in an extremally disconnected space is a $C^{*}$ -point. A classic example is the space $𝐖^{*} = ω_{1} + 1$ consisting of the countable ordinals together with $ω_{1}$ . The point $ω_{1}$ is known to be a $C^{*}$ -point as well as a $P$ -point. We supply a characterization of $C^{*}$ -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

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

Mokhtar Kirane, Salim Messaoudi (2002)

Annales Polonici Mathematici

Similarity:

We consider the systems of hyperbolic equations ⎧ $u ₜ ₜ = Δ (a (t, x) u) + Δ (b (t, x) v) + {h (t, x) | v |}^{p}$ , t > 0, $x \in ℝ^{N}$ , (S1) ⎨ ⎩ $v ₜ ₜ = Δ (c (t, x) v) + {k (t, x) | u |}^{q}$ , t > 0, $x \in ℝ^{N}$ ⎧ $u ₜ ₜ = Δ (a (t, x) u) + {h (t, x) | v |}^{p}$ , t > 0, $x \in ℝ^{N}$ , (S2) ⎨ ⎩ $v ₜ ₜ = Δ (c (t, x) v) + {l (t, x) | v |}^{m} + k (t, x) {| u |}^{q}$ , t > 0, $x \in ℝ^{N}$ , (S3) ⎧ $u ₜ ₜ = Δ (a (t, x) u) + Δ (b (t, x) v) + {h (t, x) | u |}^{p}$ , t > 0, $x \in ℝ^{N}$ , ⎨ ⎩ $v ₜ ₜ = Δ (c (t, x) v) + {k (t, x) | v |}^{q}$ , t > 0, $x \in ℝ^{N}$ , in $(0, \infty) \times ℝ^{N}$ 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.

Repdigits in generalized Pell sequences

Jhon J. Bravo, Jose L. Herrera (2020)

Archivum Mathematicum

Similarity:

For an integer $k \geq 2$ , let ${(n)}_{n}$ be the $k -$ generalized Pell sequence which starts with $0, ..., 0, 1$ ( $k$ terms) and each term afterwards is given by the linear recurrence $n = 2 n - 1 + n - 2 + \dots + n - k$ . In this paper, we find all $k$ -generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence ${(P_{n}^{(2)})}_{n}$ . ...

Classical boundary value problems for integrable temperatures in a $C^{1}$ domain

Anna Grimaldi Piro, Francesco Ragnedda (1991)

Annales Polonici Mathematici

Similarity:

Abstract. We study a Neumann problem for the heat equation in a cylindrical domain with $C^{1}$ -base and data in $h_{c}^{1}$ , a subspace of $L$ 1. We derive our results, considering the action of an adjoint operator on $B_{T} M O C$ , a predual of $h_{c}^{1}$ , and using known properties of this last space.

Hydrodynamical behavior of symmetric exclusion with slow bonds

Tertuliano Franco, Patrícia Gonçalves, Adriana Neumann (2013)

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

Similarity:

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 $β \in [0, \infty)$ . 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 $β \in [0, 1)$ , the hydrodynamic limit is given by the usual heat equation. If $β = 1$ , it is given by a parabolic equation involving an operator...

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

Similarity:

Let $Ω$ be a bounded domain of class $C^{2}$ in $ℝ$ N and let $K$ be a compact subset of $\partial Ω$ . Assume that $q \geq (N + 1) / (N - 1)$ and denote by $U_{K}$ the maximal solution of $- Δ u + u^{q} = 0$ in $Ω$ which vanishes on $\partial Ω ∖ K$ . We obtain sharp upper and lower estimates for $U_{K}$ in terms of the Bessel capacity $C_{2 / q, q^{'}}$ and prove that $U_{K}$ is $σ$ -moderate. In addition we describe the precise asymptotic behavior of $U_{K}$ at points $σ \in K$ , which depends on the “density” of $K$ at $σ$ , measured in terms of the capacity $C_{2 / q, q^{'}}$ .

An application of the method wherein the $p h i$ - and $c h i$ - methods are combined for the determination of the grating constant. [II.]

Swami Jnanananda (1936)

Časopis pro pěstování matematiky a fysiky

Similarity:

$L_{p, q}$ spaces

Joseph Kupka

Similarity:

CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the $L_{p, q}$ spaces................................. 114. Integral representation of bounded linear functionals on $L_{p, q} (B)$ ........ 235. Examples in $L_{p, q}$ theory...................................................................................

Generalized versions of Ilmanen lemma: Insertion of $C^{1, ω}$ or $C_{loc}^{1, ω}$ functions

Václav Kryštof (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We prove that for a normed linear space $X$ , if $f_{1} : X \to ℝ$ is continuous and semiconvex with modulus $ω$ , $f_{2} : X \to ℝ$ is continuous and semiconcave with modulus $ω$ and $f_{1} \leq f_{2}$ , then there exists $f \in C^{1, ω} (X)$ such that $f_{1} \leq f \leq f_{2}$ . Using this result we prove a generalization of Ilmanen lemma (which deals with the case $ω (t) = t$ ) to the case of an arbitrary nontrivial modulus $ω$ . This generalization (where a $C_{l o c}^{1, ω}$ function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

Displaying similar documents to “Nonanalyticity of solutions to ∂ t u = ∂ ² x u + u ² ”

Displaying similar documents to “Nonanalyticity of solutions to $\partial_{t} u = \partial ²_{x} u + u ²$ ”