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.

scholarly journals The Cauchy problem for the Finsler heat equation

2020 ◽  
Vol 13 (3) ◽  
pp. 257-278 ◽  
Author(s):  
Goro Akagi ◽  
Kazuhiro Ishige ◽  
Ryuichi Sato
Keyword(s):  
Cauchy Problem ◽  
Linear Operator ◽  
Heat Equation ◽  
Laplace Operator ◽  
Sufficient Condition ◽  
Smooth Functions ◽  
Dual Norm ◽  
The Cauchy Problem

AbstractLet H be a norm of {\mathbb{R}^{N}} and {H_{0}} the dual norm of H. Denote by {\Delta_{H}} the Finsler–Laplace operator defined by {\Delta_{H}u:=\operatorname{div}(H(\nabla u)\nabla_{\xi}H(\nabla u))}. In this paper we prove that the Finsler–Laplace operator {\Delta_{H}} acts as a linear operator to {H_{0}}-radially symmetric smooth functions. Furthermore, we obtain an optimal sufficient condition for the existence of the solution to the Cauchy problem for the Finsler heat equation\partial_{t}u=\Delta_{H}u,\quad x\in\mathbb{R}^{N},\,t>0,where {N\geq 1} and {\partial_{t}:=\frac{\partial}{\partial t}}.

scholarly journals On the Blow-Up of Solutions of a Weakly Dissipative Modified Two-Component Periodic Camassa-Holm System

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu ◽  
Weian Tao
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Strong Solutions ◽  
Well Posedness ◽  
Holm Equation ◽  
Two Component ◽  
The Cauchy Problem ◽  
Blow Up Rate

We study the Cauchy problem of a weakly dissipative modified two-component periodic Camassa-Holm equation. We first establish the local well-posedness result. Then we derive the precise blow-up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow-up results for strong solutions to the system.

HOMOGENEOUS HYPERBOLIC EQUATIONS WITH COEFFICIENTS DEPENDING ON ONE SPACE VARIABLE

2007 ◽  
Vol 04 (03) ◽  
pp. 533-553 ◽  
Author(s):  
SERGIO SPAGNOLO ◽  
GIOVANNI TAGLIALATELA
Keyword(s):  
Cauchy Problem ◽  
Space Variable ◽  
Hyperbolic Equations ◽  
Diagonal Matrix ◽  
Well Posedness ◽  
Characteristic Roots ◽  
The Cauchy Problem

We investigate the Cauchy problem for homogeneous equations of order m in the (t,x)-plane, with coefficients depending only on x. Assuming that the characteristic roots satisfy the condition [Formula: see text] we succeed in constructing a smooth symmetrizer which behaves like a diagonal matrix: this allows us to prove the well-posedness in [Formula: see text].

scholarly journals The Cauchy Problem for a Dissipative Periodic 2-Component Degasperis-Procesi System

2014 ◽  
Vol 2014 ◽  
pp. 1-17
Author(s):  
Sen Ming ◽  
Han Yang ◽  
Ls Yong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Transport Equation ◽  
Wave Breaking ◽  
A Priori Estimates ◽  
A Priori ◽  
Strong Solutions ◽  
Well Posedness ◽  
Priori Estimates ◽  
The Cauchy Problem

The dissipative periodic 2-component Degasperis-Procesi system is investigated. A local well-posedness for the system in Besov space is established by using the Littlewood-Paley theory and a priori estimates for the solutions of transport equation. The wave-breaking criterions for strong solutions to the system with certain initial data are derived.

scholarly journals Solution concepts, well-posedness, and wave breaking for the Fornberg–Whitham equation

2020 ◽  
Author(s):  
Günther Hörmann
Keyword(s):  
Cauchy Problem ◽  
Traveling Waves ◽  
Wave Breaking ◽  
Blow Up ◽  
Strong Solutions ◽  
Well Posedness ◽  
Solution Concepts ◽  
Nonlinear Operators ◽  
The Cauchy Problem ◽  
Whitham Equation

AbstractWe discuss concepts and review results about the Cauchy problem for the Fornberg–Whitham equation, which has also been called Burgers–Poisson equation in the literature. Our focus is on a comparison of various strong and weak solution concepts as well as on blow-up of strong solutions in the form of wave breaking. Along the way we add aspects regarding semiboundedness at blow-up, from semigroups of nonlinear operators to the Cauchy problem, and about continuous traveling waves as weak solutions.

scholarly journals On non-resistive limit of 1D MHD equations with no vacuum at infinity

2021 ◽  
Vol 11 (1) ◽  
pp. 702-725
Author(s):  
Zilai Li ◽  
Huaqiao Wang ◽  
Yulin Ye
Keyword(s):  
Cauchy Problem ◽  
A Priori ◽  
Strong Solutions ◽  
Mhd Equations ◽  
Well Posedness ◽  
Large Initial Data ◽  
One Dimensional ◽  
The Cauchy Problem ◽  
Global Strong Solutions ◽  
The One

Abstract In this paper, the Cauchy problem for the one-dimensional compressible isentropic magnetohydrodynamic (MHD) equations with no vacuum at infinity is considered, but the initial vacuum can be permitted inside the region. By deriving a priori ν (resistivity coefficient)-independent estimates, we establish the non-resistive limit of the global strong solutions with large initial data. Moreover, as a by-product, the global well-posedness of strong solutions for the compressible resistive MHD equations is also established.

scholarly journals Well-Posedness, Blow-Up Phenomena, and Asymptotic Profile for a Weakly Dissipative Modified Two-Component Camassa-Holm Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Strong Solutions ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Asymptotic Profile ◽  
Holm Equation ◽  
Two Component ◽  
The Cauchy Problem ◽  
Blow Up Rate

We study the Cauchy problem of a weakly dissipative modified two-component Camassa-Holm equation. We firstly establish the local well-posedness result. Then we present a precise blow-up scenario. Moreover, we obtain several blow-up results and the blow-up rate of strong solutions. Finally, we consider the asymptotic behavior of solutions.

On the Cauchy problem for the nonlinear heat equation

2019 ◽  
Author(s):  
Elena Nikolova ◽  
Mirko Tarulli ◽  
George Venkov
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Nonlinear Heat Equation ◽  
The Cauchy Problem

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 .

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