scholarly journals On the local well-posedness of the nonlinear heat equation associated to the fractional Hermite operator in modulation spaces

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Elena Cordero
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Frequency Analysis ◽  
Pseudodifferential Operators ◽  
Local Solvability ◽  
Modulation Spaces ◽  
Nonlinear Heat Equation ◽  
Well Posedness ◽  
Hermite Operator ◽  
Time Frequency

AbstractIn this note we consider the nonlinear heat equation associated to the fractional Hermite operator $$H^\beta =(-\Delta +|x|^2)^\beta $$ H β = ( - Δ + | x | 2 ) β , $$0<\beta \le 1$$ 0 < β ≤ 1 . We show the local solvability of the related Cauchy problem in the framework of modulation spaces. The result is obtained by combining tools from microlocal and time-frequency analysis. As a byproduct, we compute the Gabor matrix of pseudodifferential operators with symbols in the Hörmander class $$S^m_{0,0}$$ S 0 , 0 m , $$m\in \mathbb {R}$$ m ∈ R .

On the Cauchy problem for a generalized nonlinear heat equation

2018 ◽  
Vol 25 (2) ◽  
pp. 169-180
Author(s):  
Franka Baaske ◽  
Hans-Jürgen Schmeißer
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Space Variable ◽  
Strong Solutions ◽  
Nonlinear Heat Equation ◽  
Well Posedness ◽  
Weierstrass Semigroup ◽  
Smooth Functions ◽  
The Cauchy Problem ◽  
Generalized Heat Equation

Abstract The paper is concerned with the Cauchy problem for a nonlinear generalized heat equation which is related to the generalized Gauss–Weierstrass semigroup via Duhamel’s principle. For the initial data we assume that they belong to some fractional Sobolev spaces. We study the existence and uniqueness of mild and strong solutions which are local in time. Moreover, they are smooth functions and belong to Lebesgue spaces with respect to the space variable. We use both fixed point arguments and mapping properties of the generalized Gauss–Weierstrass semigroup. Finally, we study the well-posedness of the problem.

Local well‐posedness and blow‐up for an inhomogeneous nonlinear heat equation

2020 ◽  
Vol 43 (8) ◽  
pp. 5264-5272
Author(s):  
Rasha Alessa ◽  
Aisha Alshehri ◽  
Haya Altamimi ◽  
Mohamed Majdoub
Keyword(s):  
Heat Equation ◽  
Blow Up ◽  
Nonlinear Heat Equation ◽  
Well Posedness

scholarly journals Dilation Properties for Weighted Modulation Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Elena Cordero ◽  
Kasso A. Okoudjou
Keyword(s):  
Cauchy Problem ◽  
Besov Spaces ◽  
Sharp Estimate ◽  
Modulation Spaces ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
Plate Equations ◽  
Unweighted Case ◽  
Vibrating Plate

We give a sharp estimate on the norm of the scaling operatorUλf(x)=f(λx)acting on the weighted modulation spacesMs,tp,q(ℝd). In particular, we recover and extend recent results by Sugimoto and Tomita in the unweighted case. As an application of our results, we estimate the growth in time of solutions of the wave and vibrating plate equations, which is of interest when considering the well-posedness of the Cauchy problem for these equations. Finally, we provide new embedding results between modulation and Besov spaces.

scholarly journals Large Time Behavior of Solutions to the Nonlinear Heat Equation with Absorption with Highly Singular Antisymmetric Initial Values

2020 ◽  
Vol 20 (2) ◽  
pp. 311-337
Author(s):  
Hattab Mouajria ◽  
Slim Tayachi ◽  
Fred B. Weissler
Keyword(s):  
Heat Equation ◽  
Large Time ◽  
Large Time Behavior ◽  
Nonlinear Heat Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Time Behavior ◽  
Initial Values ◽  
Self Similar ◽  
Similar Solutions

AbstractIn this paper, we study global well-posedness and long-time asymptotic behavior of solutions to the nonlinear heat equation with absorption, {u_{t}-\Delta u+\lvert u\rvert^{\alpha}u=0}, where {u=u(t,x)\in\mathbb{R}}, {(t,x)\in(0,\infty)\times\mathbb{R}^{N}} and {\alpha>0}. We focus particularly on highly singular initial values which are antisymmetric with respect to the variables {x_{1},x_{2},\ldots,x_{m}} for some {m\in\{1,2,\ldots,N\}}, such as {u_{0}=(-1)^{m}\partial_{1}\partial_{2}\cdots\partial_{m}\lvert\,{\cdot}\,% \rvert^{-\gamma}\in\mathcal{S}^{\prime}(\mathbb{R}^{N})}, {0<\gamma<N}. In fact, we show global well-posedness for initial data bounded in an appropriate sense by {u_{0}} for any {\alpha>0}. Our approach is to study well-posedness and large time behavior on sectorial domains of the form {\Omega_{m}=\{x\in\mathbb{R}^{N}:x_{1},\ldots,x_{m}>0\}}, and then to extend the results by reflection to solutions on {\mathbb{R}^{N}} which are antisymmetric. We show that the large time behavior depends on the relationship between α and {\frac{2}{\gamma+m}}, and we consider all three cases, α equal to, greater than, and less than {\frac{2}{\gamma+m}}. Our results include, among others, new examples of self-similar and asymptotically self-similar solutions.

scholarly journals Estimates for Unimodular Multipliers on Modulation Hardy Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Jiecheng Chen ◽  
Dashan Fan ◽  
Lijing Sun ◽  
Chunjie Zhang
Keyword(s):  
Cauchy Problem ◽  
Hardy Spaces ◽  
Asymptotic Estimate ◽  
Nonlinear Partial Differential Equations ◽  
Modulation Spaces ◽  
Mixed Norm ◽  
Well Posedness ◽  
Cauchy Problems ◽  
Norm Estimates ◽  
The Cauchy Problem

It is known that the unimodular Fourier multiplierseit|Δ|α/2,α>0,are bounded on all modulation spacesMp,qsfor1≤p,q≤∞. We extend such boundedness to the case of all0<p,q≤∞and obtain its asymptotic estimate astgoes to infinity. As applications, we give the grow-up rate of the solution for the Cauchy problems for the free Schrödinger equation with the initial data in a modulation space, as well as some mixed norm estimates. We also study theMp1,qs→Mp2,qsboundedness for the operatoreit|Δ|α/2, for the case0<α≤2andα≠1.Finally, we investigate the boundedness of the operatoreit|Δ|α/2forα>0and obtain the local well-posedness for the Cauchy problem of some nonlinear partial differential equations with fundamental semigroupeit|Δ|α/2.

scholarly journals $$\alpha $$-Modulation Spaces for Step Two Stratified Lie Groups

2022 ◽  
Vol 32 (2) ◽  
Author(s):  
Eirik Berge
Keyword(s):  
Frequency Analysis ◽  
Group Structure ◽  
Modulation Spaces ◽  
Metric Geometry ◽  
Rational Structure ◽  
Open Problems ◽  
Stratified Lie Group ◽  
Time Frequency ◽  
Stratified Lie Groups ◽  
Structure Constants

AbstractWe define and investigate $$\alpha $$ α -modulation spaces $$M_{p,q}^{s,\alpha }(G)$$ M p , q s , α ( G ) associated to a step two stratified Lie group G with rational structure constants. This is an extension of the Euclidean $$\alpha $$ α -modulation spaces $$M_{p,q}^{s,\alpha }({\mathbb {R}}^n)$$ M p , q s , α ( R n ) that act as intermediate spaces between the modulation spaces ($$\alpha = 0$$ α = 0 ) in time-frequency analysis and the Besov spaces ($$\alpha = 1$$ α = 1 ) in harmonic analysis. We will illustrate that the group structure and dilation structure on G affect the boundary cases $$\alpha = 0,1$$ α = 0 , 1 where the spaces $$M_{p,q}^{s}(G)$$ M p , q s ( G ) and $${\mathcal {B}}_{p,q}^{s}(G)$$ B p , q s ( G ) have non-standard translation and dilation symmetries. Moreover, we show that the spaces $$M_{p,q}^{s,\alpha }(G)$$ M p , q s , α ( G ) are non-trivial and generally distinct from their Euclidean counterparts. Finally, we examine how the metric geometry of the coverings $${\mathcal {Q}}(G)$$ Q ( G ) underlying the $$\alpha = 0$$ α = 0 case $$M_{p,q}^{s}(G)$$ M p , q s ( G ) allows for the existence of geometric embeddings $$\begin{aligned} F:M_{p,q}^{s}({\mathbb {R}}^k) \longrightarrow {} M_{p,q}^{s}(G), \end{aligned}$$ F : M p , q s ( R k ) ⟶ M p , q s ( G ) , as long as k (that only depends on G) is small enough. Our approach naturally gives rise to several open problems that is further elaborated at the end of the paper.

scholarly journals Klein-Gordon Equations on Modulation Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Guoping Zhao ◽  
Jiecheng Chen ◽  
Weichao Guo
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Stability Theory ◽  
Scattering Theory ◽  
Modulation Spaces ◽  
Well Posedness ◽  
Blowup Criterion ◽  
The Cauchy Problem ◽  
Klein Gordon

We consider the Cauchy problem for a family of Klein-Gordon equations with initial data in modulation spacesMp,1a. We develop the well-posedness, blowup criterion, stability of regularity, scattering theory, and stability theory.

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