Parabolic potentials and wavelet transforms with the generalized translation

Ilham A. Aliev; Boris Rubin

Displaying similar documents to “Parabolic potentials and wavelet transforms with the generalized translation”

Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities

Philippe Souplet, Slim Tayachi (2001)

Colloquium Mathematicae

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Consider the nonlinear heat equation (E): $u_{t} - Δ u = {| u |}^{p - 1} u + b {| \nabla u |}^{q}$ . We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates $C ₁ {(T - t)}^{- 1 / (p - 1)} {\leq | | u (t) | |}_{\infty} \leq C ₂ {(T - t)}^{- 1 / (p - 1)}$ . Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality $u_{t} - u_{x x} \geq u^{p}$ . More general inequalities of the form $u_{t} - u_{x x} \geq f (u)$ with, for instance, $f (u) = (1 + u) l o g^{p} (1 + u)$ are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions...

The initial value problem for parabolic equations with data in $L^{p} (R^{n})$

Eugene Fabes (1972)

Studia Mathematica

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Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales

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 $L^{2} (0, T; H_{0}^{1} (Ω))$ , fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation $ε^{p} \partial_{t} u_{ε} (x, t) - \nabla \cdot (a (x ε^{- 1}, x ε^{- 2}, t ε^{- q}, t ε^{- r}) \nabla u_{ε} (x, t)) = f (x, t)$ , where $0 < p < q < r$ . 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 $p$ , compared to the standard matching that gives rise...

Absence of global solutions to a class of nonlinear parabolic inequalities

M. Guedda (2002)

Colloquium Mathematicae

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We study the absence of nonnegative global solutions to parabolic inequalities of the type $u_{t} \geq - {(- Δ)}^{β / 2} u - V (x) u + h (x, t) u^{p}$ , where ${(- Δ)}^{β / 2}$ , 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 ${V ₊ (x) \sim a | x |}^{- b}$ , where a ≥ 0, b > 0, p > 1 and V₊(x): = maxV(x),0....

On $L^{p}$ -estimates for solutions of the Cauchy problem for parabolic differential equations

P. Besala (1971)

Annales Polonici Mathematici

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Inequivalence of Wavelet Systems in $L ₁ (ℝ^{d})$ and $B V (ℝ^{d})$

Paweł Bechler (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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Theorems stating sufficient conditions for the inequivalence of the d-variate Haar wavelet system and another wavelet system in the spaces $L ₁ (ℝ^{d})$ and $B V (ℝ^{d})$ are proved. These results are used to show that the Strömberg wavelet system and the system of continuous Daubechies wavelets with minimal supports are not equivalent to the Haar system in these spaces. A theorem stating that some systems of smooth Daubechies wavelets are not equivalent to the Haar system in $L ₁ (ℝ^{d})$ is also shown.

Existence results for a class of nonlinear parabolic equations with two lower order terms

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 ⎧ $\partial (e^{β u} - 1) / \partial t - {d i v (| \nabla u |}^{p - 2} \nabla u) + d i v (c (x, t) {| u |}^{s - 1} u) + b (x, t) {| \nabla u |}^{r} = f$ in Q = Ω×(0,T), ⎨ u(x,t) = 0 on ∂Ω ×(0,T), ⎩ $(e^{β u} - 1) (x, 0) = (e^{β u ₀} - 1) (x)$ in Ω. with s = (N+2)/(N+p) (p-1), $c (x, t) \in {(L^{τ} (Q T))}^{N}$ , τ = (N+p)/(p-1), r = (N(p-1) + p)/(N+2), $b (x, t) \in L^{N + 2, 1} (Q T)$ and f ∈ L¹(Q).

On the long-time behaviour of solutions of the p-Laplacian parabolic system

Paweł Goldstein (2008)

Colloquium Mathematicae

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Convergence of global solutions to stationary solutions for a class of degenerate parabolic systems related to the p-Laplacian operator is proved. A similar result is obtained for a variable exponent p. In the case of p constant, the convergence is proved to be $¹_{l o c}$ , and in the variable exponent case, L² and $W^{1, p (x)}$ -weak.

Asymptotically self-similar solutions for the parabolic system modelling chemotaxis

Yūki Naito (2006)

Banach Center Publications

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We consider a nonlinear parabolic system modelling chemotaxis $u_{t} = \nabla \cdot (\nabla u - u \nabla v)$ , $v_{t} = Δ v + u$ in ℝ², t > 0. We first prove the existence of time-global solutions, including self-similar solutions, for small initial data, and then show the asymptotically self-similar behavior for a class of general solutions.

Strong maximum and minimum principles for parabolic functional-differential problems with nonlocal inequalities $[u^{j} (t_{0}, x) - K^{j}] + Σ_{i} h_{i} (x) [u^{j} (T_{i}, x) - K^{j}] \leq (\geq) 0$

Ludwik Byszewski (1990)

Annales Polonici Mathematici

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The regularity of the positive part of functions in $L^{2} (I; H^{1} (Ω)) \cap H^{1} (I; H^{1} {(Ω)}^{*})$ with applications to parabolic equations

Daniel Wachsmuth (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $u \in L^{2} (I; H^{1} (Ω))$ with $\partial_{t} u \in L^{2} (I; H^{1} {(Ω)}^{*})$ be given. Then we show by means of a counter-example that the positive part $u^{+}$ of $u$ has less regularity, in particular it holds $\partial_{t} u^{+} \notin L^{1} (I; H^{1} {(Ω)}^{*})$ in general. Nevertheless, $u^{+}$ satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.

Refinement type equations: sources and results

Rafał Kapica, Janusz Morawiec (2013)

Banach Center Publications

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It has been proved recently that the two-direction refinement equation of the form $f (x) = \sum_{n \in} c_{n, 1} f (k x - n) + \sum_{n \in ℤ} c_{n, - 1} f (- k x - n)$ can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation $f (x) = \sum_{n \in ℤ} c ₙ f (k x - n)$ , which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f (x) = \int_{ℝ} c (y) f (k x - y) d y$ has also various interesting...

Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales

Pernilla Johnsen, Tatiana Lobkova (2018)

Applications of Mathematics

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This paper is devoted to the study of the linear parabolic problem $ε \partial_{t} u_{ε} (x, t) - \nabla \cdot (a (x / ε, t / ε^{3}) \nabla u_{ε} (x, t)) = f (x, t)$ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient $ε$ in front of the time derivative. First, we have an elliptic homogenized problem although the problem studied is parabolic. Secondly, we get a parabolic local problem even though the problem has a different relation between the spatial and temporal scales than those normally giving rise to parabolic...

Single-point blow-up for a semilinear parabolic system

Ph. Souplet (2009)

Journal of the European Mathematical Society

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We consider positive solutions of the system $u_{t} - Δ u = v^{p}$ ; $v_{t} - Δ v = u^{q}$ in a ball or in the whole space, with $p, q > 1$ . Relatively little is known on the blow-up set for semilinear parabolic systems and, up to now, no result was available for this basic system except for the very special case $p = q$ . Here we prove single-point blow-up for a large class of radial decreasing solutions. This in particular solves a problem left open in a paper of A. Friedman and Y. Giga (1987). We also obtain lower pointwise estimates for...

Bi-spaces global attractors in abstract 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 $X^{α}$ . This semigroup possesses an $(X^{α} - Z)$ -global attractor that is closed, bounded, invariant in $X^{α}$ , and attracts bounded subsets of $X^{α}$ 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. ...

Polar wavelets and associated Littlewood-Paley theory

Epperson Jay, Frazier Michael

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Abstract We develop an almost orthogonal wavelet-type expansion in ℝ² which is adapted to polar coordinates. We start by defining a product Fourier-Hankel transform f̂ and proving a sampling formula for f such that f̂ is compactly supported. For general f, the sampling formula and a partition of unity lead to an identity of the form $f = \sum_{μ, k, m} ⟨ f, φ_{μ k m} ⟩ ψ_{μ k m}$ , in which each function $φ_{μ k m}$ and $ψ_{μ k m}$ is concentrated near a certain annular sector, has compactly supported product Fourier-Hankel transform, and is smooth...

On higher-order semilinear parabolic equations with measures as initial data

Victor Galaktionov (2004)

Journal of the European Mathematical Society

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We consider $2 m$ th-order $(m \geq 2)$ semilinear parabolic equations $u_{t} = - {(- Δ)}^{m} u \pm {| u |}^{p - 1} u$ in $ℛ^{N} \times ℝ_{+}$ $(p > 1)$ , with Dirac’s mass $δ (x)$ as the initial function. We show that for $p < p_{0} = 1 + 2 m / N$ , the Cauchy problem admits a solution $u (x, t)$ which is bounded and smooth for small $t > 0$ , while for $p \geq p_{0}$ such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.

Good-λ inequalities for wavelets of compact support

Sarah V. Cook (2004)

Colloquium Mathematicae

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For a wavelet ψ of compact support, we define a square function $S_{w}$ and a maximal function NΛ. We then obtain the $L_{p}$ equivalence of these functions for 0 < p < ∞. We show this equivalence by using good-λ inequalities.

Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux

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 $\partial_{t} u + {div}_{y} A (y, u) - Δ_{y} u = 0$ , where $y \in ℝ^{N}$ and the flux $A$ is periodic in $y$ . More specifically, we consider the case when the initial data is an $L^{1}$ 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 $L^{1}$ norm like a self-similar profile for large times. The proof uses a time and space change of variables...

Decomposition systems for function spaces

G. Kyriazis (2003)

Studia Mathematica

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Let $Θ : = θ_{I}^{e} : e \in E, I \in D$ be a decomposition system for $L ₂ (ℝ^{d})$ indexed over D, the set of dyadic cubes in $ℝ^{d}$ , and a finite set E, and let $Θ ̃ : = Θ ̃_{I}^{e} : e \in E, I \in D$ be the corresponding dual functionals. That is, for every $f \in L ₂ (ℝ^{d})$ , $f = \sum_{e \in E} \sum_{I \in D} ⟨ f, Θ ̃_{I}^{e} ⟩ θ_{I}^{e}$ . We study sufficient conditions on Θ,Θ̃ so that they constitute a decomposition system for Triebel-Lizorkin and Besov spaces. Moreover, these conditions allow us to characterize the membership of a distribution f in these spaces by the size of the coefficients $⟨ f, Θ ̃_{I}^{e} ⟩$ , e ∈ E, I ∈ D. Typical examples of such decomposition...

Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $ℝ^{n}$

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 $ℝ^{n}$ . The main results are global Schauder estimates, which hold in spite of the unboundedness of the coefficients.