scholarly journals The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity

2020 ◽  
Vol 19 (7) ◽  
pp. 3625-3650
Author(s):  
Ahmad Z. Fino ◽  
◽  
Mokhtar Kirane ◽  
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Fractional Laplacian ◽  
Exponential Nonlinearity ◽  
The Cauchy Problem

scholarly journals An Application Of The Theory Of Scale Of Banach Spaces

2015 ◽  
Vol 29 (1) ◽  
pp. 51-59
Author(s):  
Łukasz Dawidowski
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Banach Spaces ◽  
Abstract Cauchy Problem ◽  
Initial Boundary ◽  
Scale Of Banach Spaces ◽  
The Cauchy Problem ◽  
Selection Of

AbstractThe abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.

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 A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2843
Author(s):  
Ángel García ◽  
Mihaela Negreanu ◽  
Francisco Ureña ◽  
Antonio M. Vargas
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Meshless Method ◽  
Existence And Uniqueness ◽  
Fractional Laplacian ◽  
Porous Medium Equation ◽  
Medium Equation ◽  
The Cauchy Problem ◽  
Irregular Meshes ◽  
Complicated Geometry

The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using the generalized finite difference method, we obtain the convergence of the numerical solution to the classical/theoretical solution of the equation for nonnegative initial data sufficiently smooth and bounded. This procedure allows us to use meshes with complicated geometry (more realistic) or with an irregular distribution of nodes (providing more accurate solutions where needed). Some numerical results are presented in arbitrary irregular meshes to illustrate the potential of the method.

scholarly journals No local L1 solutions for semilinear fractional heat equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Kexue Li
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Heat Equations ◽  
Fractional Heat Equation ◽  
The Cauchy Problem

AbstractWe study the Cauchy problem for the semilinear fractional heat equation

scholarly journals The Cauchy problem for a nonhomogeneous heat equation with reaction

2013 ◽  
Vol 33 (2) ◽  
pp. 643-662 ◽  
Author(s):  
Arturo de Pablo ◽  
◽  
Guillermo Reyes ◽  
Ariel Sánchez ◽  
◽  
...  
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
The Cauchy Problem ◽  
Nonhomogeneous Heat
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