scholarly journals A calculus approach to hyperfunctions III

1990 ◽  
Vol 118 ◽  
pp. 133-153 ◽  
Author(s):  
Tadato Matsuzawa
Keyword(s):  
Heat Equation ◽  
Heat Kernel ◽  
Compact Subset ◽  
Compact Support ◽  
Initial Values ◽  
Analytic Functionals

In the previous papers, [18] and [19], we have given some basis of a calculus approach to hyperfunctions. We have taken hyperfunctions with the compact support as initial values of the solutions of the heat equation. More precisely, let A′[K] be the space of analytic functionals supported by a compact subset K of Rn and let E(x, t) be the n-dimensional heat kernel given by.

The Resolvent Operator

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Laplace Transform ◽  
Heat Kernel ◽  
Resolvent Operator ◽  
Contour Integration ◽  
Resolvent Kernel ◽  
The Laplace Transform ◽  
Elliptic Estimates ◽  
The Resolvent Operator

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) is a right inverse for (μ‎-L). In Hölder spaces, these are the natural elliptic estimates for generalized Kimura diffusions. The chapter first constructs the resolvent kernel using an induction over the maximal codimension of bP, and proves various estimates on it, along with corresponding estimates for the solution operator for the homogeneous Cauchy problem. It then considers holomorphic semi-groups and uses contour integration to construct the solution to the heat equation, concluding with a discussion of Kimura diffusions where all coefficients have the same leading homogeneity.

scholarly journals Brownian Motion and Harmonic Analysis on Sierpinski Carpets

1999 ◽  
Vol 51 (4) ◽  
pp. 673-744 ◽  
Author(s):  
Martin T. Barlow ◽  
Richard F. Bass
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Heat Kernel ◽  
Lower Bounds ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Upper And Lower Bounds ◽  
Isotropic Diffusion ◽  
Sierpinski Carpets

AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

Rough estimates on the scalar heat kernel

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Wave Equation ◽  
Heat Equation ◽  
Heat Kernel ◽  
Heat Kernels ◽  
Hypoelliptic Operator ◽  
Uniform Bounds ◽  
Standard Heat ◽  
Probabilistic Construction

This chapter establishes rough estimates on the heat kernel rb,tX for the scalar hypoelliptic operator AbX on X defined in the preceding chapter. By rough estimates, this chapter refers to just the uniform bounds on the heat kernel. The chapter also obtains corresponding bounds for the heat kernels associated with operators AbX and another AbX over ̂X. Moreover, it gives a probabilistic construction of the heat kernels. This chapter also explains the relation of the heat equation for the hypoelliptic Laplacian on X to the wave equation on X and proves that as b → 0, the heat kernel rb,tX converges to the standard heat kernel of X.

scholarly journals Integral maximum principle and its applications

1994 ◽  
Vol 124 (2) ◽  
pp. 353-362 ◽  
Author(s):  
Alexander Grigor'yan
Keyword(s):  
Riemannian Manifold ◽  
Maximum Principle ◽  
Heat Equation ◽  
Heat Kernel ◽  
Integral Maximum Principle ◽  
Double Integrals

The integral maximum principle for the heat equation on a Riemannian manifold is improved and applied to obtain estimates of double integrals of the heat kernel.

scholarly journals On the Operator ⨁BkRelated to Bessel Heat Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Wanchak Satsanit
Keyword(s):  
Heat Equation ◽  
Heat Kernel ◽  
Positive Constant ◽  
Nonnegative Integer ◽  
Unknown Function ◽  
Generalized Function ◽  
Initial Condition

We study the equation(∂/∂t)u(x,t)=c2⊕Bku(x,t)with the initial conditionu(x,0)=f(x)forx∈Rn+.The operator⊕Bkis the operator iterated k-times and is defined by⊕Bk=((∑i=1pBxi)4-(∑j=p+1p+qBxi)4)k, wherep+q=nis the dimension of theRn+,Bxi=∂2/∂xi2+(2vi/xi)(∂/∂xi),2vi=2αi+1,αi>-1/2,i=1,2,3,…,n, andkis a nonnegative integer,u(x,t)is an unknown function for(x,t)=(x1,x2,…,xn,t)∈Rn+×(0,∞),f(x)is a given generalized function, andcis a positive constant. We obtain the solution of such equation, which is related to the spectrum and the kernel, which is so called Bessel heat kernel. Moreover, such Bessel heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.

scholarly journals On an asymptotically log-periodic solution to the graphical curve shortening flow equation

2022 ◽  
Vol 4 (3) ◽  
pp. 1-14
Author(s):  
Dong-Ho Tsai ◽  
◽  
Xiao-Liu Wang ◽  
Keyword(s):  
Periodic Solution ◽  
Heat Equation ◽  
Periodic Function ◽  
Compact Subset ◽  
Interesting Property ◽  
Flow Equation ◽  
Graphical Solution ◽  
Curve Shortening Flow ◽  
Curve Shortening

<abstract><p>With the help of heat equation, we first construct an example of a graphical solution to the curve shortening flow. This solution $ y\left(x, t\right) \ $has the interesting property that it converges to a log-periodic function of the form</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ A\sin \left( \log t\right) +B\cos \left( \log t\right) $\end{document} </tex-math></disp-formula></p> <p>as$ \ t\rightarrow \infty, \ $where $ A, \ B $ are constants. Moreover, for any two numbers $ \alpha &lt; \beta, \ $we are also able to construct a solution satisfying the oscillation limits</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \liminf\limits_{t\rightarrow \infty}y\left( x,t\right) = \alpha,\ \ \ \limsup\limits _{t\rightarrow \infty}y\left( x,t\right) = \beta,\ \ \ x\in K $\end{document} </tex-math></disp-formula></p> <p>on any compact subset$ \ K\subset \mathbb{R}. $</p></abstract>

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.

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